An equation for the breakage of particles under impact

Powder Technology 132 (2003) 161 – 166 www.elsevier.com/locate/powtec An equation for the breakage of particles under impact Heechan Cho a,*, Leonard G. Austin b a b School of Civil, Urban and Geosystem Engineering, Seoul National University, Seoul 151-742, South Korea College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, PA 16802, USA Received 28 March 2002; received in revised form 30 September 2002; accepted 19 February 2003 Abstract Using data from the literature, the following simple empirical equation is derived for the cumulative mass fraction of particles broken (m̄) when impacted at a specific impact energy (J/g of impacted mass) of E: 2 0; EVEmini 6 6 m̄i ðEÞ ¼ 6 6 ĀlnðE=Ki Þ; Emini VEVEmaxi 4 1; EzEmaxi where Emini ¼ Ki Emaxi ¼ Ki expð1=ĀÞ and Ki ¼ Cðxi =xo Þm : In this equation, i is an integer index of a M2 screen size interval of upper size xi; xo is a standard size of 1 mm; Emini is the minimum specific impact energy required to produce significant breakage of feed material of size i and Emaxi is the specific impact energy that breaks all feed material of size i. Ā, C and m are material-dependent constants. Values of Ā = 0.235, m = 0.58 and C = 0.13 J/g were found for particles of limestone in a thin bed impacted by a falling steel ball. The equation shows that smaller particles require a higher specific impact energy than larger particles to give the same fraction of breakage out of the feed size interval. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Grinding; Impact breakage; Mill modeling; Particle strengths 1. Introduction The ‘‘state of the art’’ in the construction of computer simulation models for the description of the process behavior of size reduction machines is well advanced, particularly for the versatile and widely used tumbling media mills such as ball mills [1,2]. However, when these models are used for designing a milling circuit [3], it is necessary to have characteristic parameters that describe the breakage behavior of the particular materials or materials being ground, for * Corresponding author. Tel.: +82-28807342; fax: +82-28732684. E-mail address: [email protected] (H. Cho). input to the simulation model. Since the range of stresses applied to particles varies with the type of mill, as does the frequency of stress application, tests have been developed to obtain the characteristic breakage parameters of the material of interest for a particular type of device, usually by crushing or grinding the material in a laboratory scale version of the device and measuring the amount of breakage and the product size distribution of the breakage. The parameters then have to be scaled up to give the values appropriate for a full-scale industrial machine, with the scale-up methods based on previous empirical experience of correlations of small-scale test results with full-scale results. 0032-5910/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0032-5910(03)00057-3 162 H. Cho, L.G. Austin / Powder Technology 132 (2003) 161–166 To date, it has not generally been possible to predict, with sufficient accuracy for simulation purposes, the breakage properties (of a particular material in a particular breakage environment) from a knowledge of the chemical, mineralogical and microstructural properties of the material. It is more common to test a material extensively before the completion of the process design simulator, using the small-scale tests appropriate to the device(s) involved. It would be of considerable value to have a simple and rapid test to give more fundamental fracture properties of a material that could then be applied to predict the desired parameters for any type of size reduction machine. Application of this concept involves several features. First, it is well known that the stress required to cause fracture of particles of a given ‘‘size’’ (defined by sieve size, for example) is not a single number, because differences in particle shape and inherent variability in microstructure lead to inherent variability in strength. There will be a distribution of ‘‘strength’’ (the applied stress required to cause disintegrative fracture) even for the same diameter of perfect spheres of a purely brittle and apparently homogeneous material such as glass. Natural materials are far more variable and it is necessary to test a sufficient number of particles to obtain a reasonably accurate probability density function of the distribution of strength. It is also well understood that this function will vary with particle size. Secondly, it is not enough to know that a particle has (or has not) broken, because it is necessary also to know the distribution of particle sizes of the mean set of fragments produced in these primary fractures, known as the primary breakage distribution function. Thirdly, it is clearly necessary to have precise description of the following: (a) the probability of a given particle receiving a given impact in a particular machine and (b) the frequency of impacts while the material is in the machine. Fourthly, it has been shown [4] that a particle of a given strength that receives an impact of insufficient force to cause observable disintegrative fracture may still suffer interval damage, so that it becomes weaker to further impacts. All of these four features of the problem are being actively investigated by a number of researchers [5 –11], but the present paper is to illustrate one method of dealing with the first feature. The data of Datta [12] is used to derive an equation for the probability distribution of disintegrative fracture on a single impact as a function of the magnitude of the impact and the size of the particles. The experimental work of Datta [12], performed at the University of Utah, was designed to give data that could be used in applying the Discrete Element Method (DEM) to the modeling of ball milling. However, the data is manipulated here to give an equation that can be applied more generally. In Datta’s test work, a drop ball apparatus was used to impact approximately four layers of particles, sized within a M2 sieve range, placed in a thin receptacle positioned on an anvil. After a single impact, the impacted mass was screened to determine the mass of undersize material. Forty to fifty beds were broken, using feed particles screened to the same M2 screen interval, and the results expressed as the grams of undersize product produced per joule of impact energy. The collected undersize material was screened to give its size distribution. This procedure was repeated for feeds of different M2 screen sizes, each for a range of impact energies, using a ball diameter of 5.08 cm (2 in.). 2. Analysis of impact breakage Datta [12] gives the fitting equation for the data obtained using a limestone (his Limestone A) as M ¼ AlnI þ B ð1Þ where M is the mass (in grams) broken out of the feed size interval, I is the impact energy in joules (calculated from the mass of the falling ball and its velocity at impact), and A and B depend on feed size. The form of this relation is shown in Fig. 1, and the values for A and B determined by Datta are given in Table 1, where we have considered I to be converted to the impact energy divided by 1 J so that the units of A and B become grams per impact. Although Datta considers this equation to be sufficient for his objectives, we need to answer the question ‘‘what fraction of mass impacted with a certain specific impact energy E ( = I/impacted mass) is broken (and hence what fraction remains unbroken) as a function of particle size, over a complete range of E values. The falling ball impacts only a part of the powder mass in the receptacle, so the Fig. 1. Production of undersize (broken) limestone A: drop ball diameter 5.08 cm, four layers of particles (Ref. [12]). 163 H. Cho, L.G. Austin / Powder Technology 132 (2003) 161–166 Table 1 Fitting constants A and B (Ref. [12]) and estimate of mo, Ā and K for impact breakage m̄i ¼ Āi lnðE=Ki Þ; Material Size (mm) A (g) B (g) mo (g) K (J/g) Ā Limestone 0.425  0.30 1.18  0.85 2.36  1.70 0.016 0.036 0.058 0.068 0.146 0.229 0.068 0.153 0.247 0.210 0.113 0.078 0.235 0.235 0.235 actual mass impacted has to be estimated. This can be done by assuming that all the actually impacted mass will be broken to less than the lower size of the feed size interval at a sufficiently high impact energy. Unfortunately for our purposes, the tests were not carried to a high enough impact energy to achieve this, and Eq. (1) extrapolates to infinity not a finite mass. However, the data for the smallest tested size, 0.425  0.30 mm, showed that the amount of undersized material appeared to be approaching a fixed amount at energies above 0.7 J. Therefore, we have used Eq. (1) with the arbitrary but reasonable assumption that breakage out of the feed size interval was complete at 1 J. Letting the actual impacted mass be mo, Table 1 shows the estimated value for this particle size range as 0.068 g. The values of mo will be larger for larger sizes because the four layers of particles will contain a higher mass in the impact zone. Thus, at a given impact energy I the stresses produced in each impacted particle will be less because, as the impact energy is converted to the strain energy of the compressed particles, the strain energy per unit mass will be correspondingly less. It is expected, therefore, that higher impact energy will be required to cause complete fracture of the impacted mass of larger particles even though the critical fracture stress generally decreases for larger particles. The correct conversion of the impact energy to the probability of breakage is to use the specific impact energy E. Eq. (1) can then be put as m̄ ¼ A ln½Emo expðB=AÞ; mo then Eq. (1a) becomes 0Vm̄V1 In order to calculate Ki, it is necessary to estimate moi. Examination of the data for breakage of the three tested sizes shown in Fig. 1 indicate that all three curves can be reduced to a single curve by appropriate choices of mo. This means that Ki becomes a scaling factor on the specific energy scale for different particle sizes, so that the mass fraction of particles broken at a given E/Ki is the same independently of i. The variation of Ki with xi will be the variation of the comparative strength of the particles, where ‘‘strength’’ is defined as the specific strain energy required to cause breakage. It is expected that Ki will decrease for larger xi values because larger particles require more force and energy to break them but less specific strain energy. Making this assumption, that the distribution of strengths is the same for all particle sizes except for a scale factor in E, Eq. (3) shows that Āi is a constant independent of i and moi ¼ moi* Ai =Ai* ð4Þ where i* is the size interval for which mo has been already estimated, 0.425  0.3 mm in this case. For example, for 1.8  0.85 mm particles, moi = 0.068  (0.036/0.016) = 0.153. Āi = 0.036/0.053 = 0.235. Ki = 1/(0.153 exp(0.146/ 0.036)) = 0.113. Table 1 shows the values of all sizes predicted using this assumption. The constraints on Eq. (3) mean that a minimum specific impact energy is required to produce significant breakage of size i material and Emini ¼ Ki ð5Þ Similarly, the specific impact energy required to produce complete breakage out of size i (m̄ = 1) is 0Vm̄V1 ð1aÞ Emaxi ¼ Ki expð1=ĀÞ where m̄ is the fraction of actually impacted mass broken at specific impact energy E J/g. Note that ‘‘broken’’ in this sense means that particles are broken to less than the lower screen size of the tested feed interval. Since the data refer to feed material in M2 sieve size intervals the measured values of A and B become Ai and Bi where, in the usual way, i is an integer of particle size interval starting with 1 for the largest size interval with a top sieve size of x1, ranging to n for the final ‘‘sink’’ interval for all material less than a sieve size xn. Letting Ai/moi = Āi and Ki ¼ 1=moi expðBi =Ai Þ; J=g ð3Þ ð2Þ ð6Þ Fig. 2 shows a plot of Ki (J/g) versus xi (mm), which can be represented by Ki ¼ Cðxi =xo Þm ; xo ¼ 1 mm ð7Þ where, for this data, C = 0.13 J/g and m = 0.58, xo being a standard size of 1 mm. The value of C is the value of K for a size of 1  0.707 mm. The dimensionless curve of Eq. (3) is shown in Fig. 3, using the value of Ā = 0.235 determined for this data. 164 H. Cho, L.G. Austin / Powder Technology 132 (2003) 161–166 Fig. 4. Predicted mass fraction left in feed size as a function of specific impact energy, showing that smaller particles are inherently stronger. Fig. 2. Variation of Ki values as a function of particle size. Combining equations, m̄i as a function of E is: 2 0; EVEmini 6 6 m̄i ðEÞ ¼ 6 6 ĀlnðE=Ki Þ; Emini VEVEmaxi 4 1; EzEmaxi Ki ¼ Cðxi =xo Þ m Emini ¼ Ki Emaxi ¼ Ki expð1=ĀÞ g m for the limestone data, expressed as the mass fraction left unbroken versus specific impact energy, for a range of feed sizes. ð8Þ Eq. (8) is the final result of the development, and Fig. 4 shows the results of this equation using the parameters Ā, C, Fig. 3. The assumed dimensionless curve of fraction broken versus reduced specific impact energy. 3. Discussion and conclusions When a powder bed is compressed in a piston-cylinder test device, the breakage of the particles can be correlated with the applied pressure (the grinding pressure) and the specific energy of compression can be calculated from the integral of pressure over the compressive deformation of the bed [13]. However, when a ball is dropped into a bed of particles the mass of particles actually impacted is not known and the pressure is neither measured nor evenly spread over the impacted mass. Only the energy input is known, so the breakage has to be correlated with energy and not grinding pressure. The method of analysis presented here enables the impacted mass to be estimated, but the resulting mean grinding pressure still cannot be calculated because the mean deformation of this mass is not measured. Thus, the particle data set used here is from a test that has the advantage over single particle impact tests that each impact involves many particles, but the test has the disadvantages mentioned. The major disadvantages, that the impacted mass is not known and the pressure not uniformly distributed, could be removed if the drop weight test used a vertical cylindrical weight falling into a cylindrical container of slightly greater diameter. The form of Eq. (8) may be applicable to many impact conditions, although it is also probable that not all materials will fit this simple form. For a given value of Ā, the value of C is a direct measure of the ‘‘strength’’ of unit size (1  0.707 mm) since a higher value of C means that a higher E is necessary to cause the same fraction of breakage. H. Cho, L.G. Austin / Powder Technology 132 (2003) 161–166 165 If the median ‘‘strength’’ is defined as the specific energy where 50% of the particles are broken, E50, then the median strength of the standard size is a function of both C and Ā, as shown in the following: From Eq. (3), forces can be readily described (feature three) knowing the velocity and geometry of the hammers. It was found that the high number of high energy impacts as feed material was converted to product required the use of damage mechanics [4] to allow for particle weakening (feature four). 0:5 ¼ ĀlnðE50i =Ki Þ Nomenclature A A measured constant in Eq. (1), material-and sizedependent (g) Ā A/mo, a material constant, assumed independent of particle size (– ) B A measured constant in Eq. (1), material- and sizedependent (g) C A measured constant in Eq. (7) (J/g) E Specific impact energy (J/g) Emini The minimum specific impact energy to give significant breakage of material of size i (J/g) Emaxi The specific impact energy that breaks all feed material of size i (J/g) E50 The specific impact energy required to produce 50% breakage of a given particle size (J/g) The value of E50 for the material of size 1  0.707 E50o mm (J/g) i An integer index to a M2 sieve size interval of upper size xi ( –) i* Value of i for the size interval where mo has been estimated (– ) I Kinetic energy of the impacting drop weight (J) Ki A material constant for size i defined by Eq. (2) (J/g) M Mass of particles broken out of the feed size interval per impact (g) m Exponent in Eq. (7) (– ) Mass of particles actually impacted by the falling mo weight (g) m̄ Fraction broken of actually impacted particles, m/ mo ( –) n Integer index of the ‘‘sink’’ interval (– ) The upper sieve size of the M2 sieve interval xi indexed by i (mm) ð9aÞ It follows E50i ¼ Ki expð0:5=ĀÞ ð9bÞ From Eq. (7), Ki = C for the standard size because xi = xo. Therefore, E50o ¼ Cexpð0:5=ĀÞ ð9cÞ The mean strength of the standard size is higher for larger C and smaller Ā. The relative width of the distribution, defined as the spread of E/E50, is dependent on Ā Emin Ki 1 ¼ ¼ E50 Ki expð0:5=ĀÞ expð1=2ĀÞ ð10aÞ Emax Ki expð1=ĀÞ ¼ expð1=2ĀÞ ¼ E50 Ki expð0:5=ĀÞ ð10bÞ where a smaller Ā gives a wider relative spread. Combining Eqs. (7), (9a) and (9b) gives E50i ¼ ðxi =xo Þm ¼ ðxo =xi Þm E50o ð10cÞ Therefore, a higher value of m gives a higher relative increase of strength as particle size decreases. The analysis of the apparent breakage distribution functions from the data (the second required feature) has been presented separately [14]. The basic concept is that weaker particles struck with a high impact energy will fracture and the primary fragments will then fracture again because the dropping weight will still be moving, until all the kinetic energy of the weight has been converted to strain energy and hence to multiple fractures. Obviously, in order to do the energy and breakage balances required to deduce true primary breakage distributions it was necessary to use Eq. (8). It was concluded that a single true primary breakage distribution function of dimensionally normalized form could explain the product size distributions obtained at higher specific impact energies. Eq. (8) has been used [15] to construct a simulation model for high-speed hammer milling. In this case, the particles are not struck between a moving object and an anvil but are struck by the moving object while in suspension. However, the basic concepts are the same and this type of mill is amenable to analysis because the range of impact References [1] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction; Ball Milling, SME – AIME, New York, NY, 1984. [2] L.G. Austin, O. Trass, Handbook of powder science and technology, in: M.E. Fayed, L. Otten (Eds.), Size Reduction of Solids, 2nd ed., Chapman & Hall, New York, NY, 1997, pp. 586 – 634, Chap. 12. [3] R.P. King, A User’s Guide to MODSIM, Comminution Center, University of Utah, Salt Lake City, UT. [4] L.M., Tavares, Microscale investigation of particle breakage applied to the study of thermal and mechanical predamage, PhD thesis, Dept. of Metallurgical Engineering, University of Utah, Salt Lake City, UT, 1997. [5] H. Cho, L.G. Austin, J.Y. 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Schönert, Zem-Kalk-Gips 32 (1979) 1 – 9. [14] L.G. Austin, Powder Technol. 126 (2002) 85 – 90. [15] L.G. Austin, A preliminary simulation model for high speed hammer milling, Powder Technol. (2002) (submitted for publication).