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Modeling ore degradation during handling using continuum damage mechanics
International Journal of Mineral Processing 112–113 (2012) 1–6
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International Journal of Mineral Processing journal homepage: www.elsevier.com/locate/ijminpro
Modeling ore degradation during handling using continuum damage mechanics☆ Luís Marcelo Tavares ⁎, Rodrigo M. de Carvalho Department of Metallurgical and Materials Engineering, COPPE, Universidade Federal do Rio de Janeiro – UFRJ. Cx. Postal 68505, CEP 21941-972, Rio de Janeiro, RJ, Brazil
a r t i c l e
i n f o
Article history: Received 22 February 2010 Received in revised form 29 June 2010 Accepted 10 July 2010 Available online 16 April 2012 Keywords: Degradation Handling Modeling Particle breakage
a b s t r a c t Degradation of steelmaking materials during handling causes a significant economic impact, given the higher prices achieved by lump ore in comparison to finer products. It is relevant to model the amenability of ores to degradation, since it may be used to simulate the response of ores to different sequences of handling and transportation events from mine to port. The paper presents a mechanistic model to describe degradation caused by transfers and drops of particulate materials. The model is based on damage mechanics and the distribution of fracture energies of the original particles, using data collected in impact load cell and tumbling tests. This model, which is capable of predicting the proportion of particles broken and the entire size distribution resulting from any sequence of drops, has been validated at a preliminary level using data from drop tests of a Brazilian iron ore. © 2010 Elsevier Inc. All rights reserved.
1. Introduction Ores undergo significant degradation during mining as a result of blasting, mechanical handling by shovels and crushing in order to produce a size distribution that is capable of meeting specifications that are set by customers and the market. Such degradation is particularly relevant in the case of steelmaking materials. Degradation of lump ores, such as coal, iron and manganese ores into fines is often undesirable because lumps have premium prices when compared to fines. However, the relationship between the proportion of lump ore and fines that is produced in a given processing plant is codetermined by the ore's mechanical strength and the nature of the crushing and handling processes (Dukino et al., 1995). In addition to that, between the mine and the end user, the lump material is subject to a number of mechanical actions which cause degradation. These include at the mine site: crushing and screening, conveying, stockpiling and rail wagon loading; and at the port: rail wagon unloading, conveying, drops at transfer points, screening, stockpiling, reclaiming and ship loading (Sahoo, 2007). A number of experimental studies have been conducted to investigate ore degradation during handling. Some of the conclusions of these studies may be summarized as: • Degradation during handling results predominantly from impact from drops and, to a minor extent, also from abrasion due to gravity flow through bins (Fagerberg and Sandberg, 1973); ☆ A publishers' error resulted in this article appearing in the wrong issue. The article is reprinted here for the reader's convenience and for the continuity of the special issue. For citation purposes, please use the original publication details; International Journal of Mineral Processing, 101(1-4), pp. 21-27. **DOI of original item: doi:10.1016/j.minpro.2010.07.008. DOI of original article: 10.1016/j.minpro.2010.07.008. ⁎ Corresponding author. E-mail address: [email protected] (L.M. Tavares). 0301-7516/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.minpro.2010.07.011
• Degradation varies with particle size, with coarser particles suffering a greater degradation effect than finer ones (Fagerberg and Sandberg, 1973; Waters et al., 1987); • Greater fines generation is associated to handling weaker ores than tougher ones (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Sahoo, 2007); • Impact surface influences the level of degradation upon impact, and a cushioning effect occurs as the percentage of initial fines increases (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Sahoo, 2007); • Norgate et al. (1986) stated that a drop height of large magnitude will cause more degradation earlier in a handling system rather than later, because of the conditioning of the ore, which removes the weaker particles; • Degradation is significantly reduced by using drop heights below about 1 to 3 m (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Waters and Mikka, 1989). Prediction of the amenability of ores to degradation is relevant, since it can be used to investigate the response of ores to different sequences of handling and transportation events from mine to port. This can be a useful tool to assess the effectiveness of measures taken to minimize degradation during handling. A number of attempts have been made in the last couple of decades to describe quantitatively this phenomenon (Norgate et al., 1986; Teo et al., 1990; Dukino et al., 2000; Weedon and Wilson, 2000; Baxter et al., 2004; Sahoo, 2007), without any particular model finding wide acceptance or being able to describe in great detail the response of the material during the handling processes. The paper presents a simulation procedure that describes degradation caused by transfers and drops of the particulate material. It is based on data collected from drop weight, impact load cell tests and tumbling tests, a model from damage mechanics and the distribution of strengths in the original material.
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L.M. Tavares, R.M. de Carvalho / International Journal of Mineral Processing 112–113 (2012) 1–6
2. Model description When a particle is dropped during a transfer, it may break or not. Whenever the particle does not break catastrophically (body breakage), its surface will be abraded (surface breakage) and it may accumulate crack-like damage, so that it may eventually break as the result of an impact of a comparatively low magnitude in a future drop. Modeling weakening that the particle undergoes after repeated impacts is of major importance to describe degradation during handling, and a model has been proposed to describe it using elements from damage mechanics (Tavares and King, 2002; Tavares, 2009). The model is based on the recognition that the load-deformation response which results from impact of a spherical particle can be described as a combination of continuum damage mechanics (Kachanov, 1958) and Hertz contact theory (Tavares and King, 2005), giving
F¼
d1=2 ~ 3=2 kα 3
ð1Þ
where F is the load, d is the particle size, α is the local deformation and k~ ¼ kð1−DÞ
ð2Þ
is the stiffness of the particle during the stressing event. k is the stiffness of the particle prior to stressing, given by k = Y / (1 − μ 2), where Y is the modulus of elasticity and μ is the Poisson's ratio. The damage variable D may be described by the power law relationship D¼
α αc
γ
ð3Þ
which is valid during compression of the particle. αc is the deformation associated to fracture of the particle. By definition, damage is irreversible (Kachanov, 1958), so that, if during compression the particle does not fracture, then during restitution D = D*, where D* is the maximum value that the damage variable reached during the loading part of the cycle. Assuming that both the particle orientation and the model parameters αc and γ remain constant, Eqs. (1)–(3) can be used to predict the force–deformation curves resulting from repeatedly loading of a particle up to a given strain energy level, until its final failure (Fig. 1). Unfortunately, this procedure cannot be easily used in practice because it requires the knowledge of several material constants, including αc and k, besides the damage accumulation coefficient γ. These are material-specific constants that are not straight-forward to estimate directly from experiments (Tavares and King, 2005). Typically, only the distribution of fracture energies (probability distribution of body breakage) of the original material and the stressing energies in each event may be known, using, for instance, the impact load cell device (Tavares and King, 1998). A convenient procedure that allows simulating multiple loading events using a limited amount of data has been proposed and is presented elsewhere (Tavares and King, 2002; Tavares and Carvalho, 2009). It assumes that the constitutive equation (Eq. (1)) remains valid throughout the several loading events, so that the distribution of particle fracture energies after the nth loading is given by F nþ1 ðEÞ ¼ 2D ¼
F n ½E=ð1−DÞ−F n ðEk Þ 1−F n ðEk Þ
2γð1−DÞ Ek ð2γ−5Dþ5Þ E
ð4Þ
2γ 5
ð5Þ
where Fn(Ek) is the proportion of particles broken at the nth drop from a specific impact energy of Ek. Eq. (5) should be solved using an efficient numerical procedure, since D is implicit.
Fig. 1. Illustration of the effect of weakening due to accrual of damage in repeated impacts at a constant stressing energy.
L.M. Tavares, R.M. de Carvalho / International Journal of Mineral Processing 112–113 (2012) 1–6
The initial distribution of particle fracture energies can be generally well described using the log-normal distribution (Tavares and King, 1998), given by 2
0
ð6Þ
where the median fracture energy may be described by ð7Þ
where di is the representative size of particles contained in the ith class, σi2 is the variance of the distribution and E∞, do and ϕ are model parameters that should be fitted to experimental data. Results of experiments used to determine the parameter γ are shown in Fig. 2, where the fraction of particles that suffered body breakage (those that lost at least 10% of their original weight) is presented as a function of number of drops, along with model predictions. In order to simulate degradation during handling, it is necessary to predict the complete size distribution of the solids after the drops, not only the proportion of material broken. This is given from a mass balance of each size fraction after an individual drop of all the particles, which is given by wi;nþ1
i ¼ wi;n 1−F i;n ðeEk;n Þ ð1−κ i Þ i h h i i X þ wj;n F j;n ðeEk;n Þbij ðeEk;n Þ þ κ j 1−F j;n ðeEk;n Þ aij
j¼1
fered body breakage and reported to the size class i. The term h h i i i P wj;n κ j 1−F j;n ðeEk;n Þ aij gives the cumulative proportion of surface
j¼1
i
ϕ d E50i ¼ E∞ 1 þ o di
fraction of the particles that suffered neither body nor surface breakage h i i P during the nth drop, whereas the term wj;n F j;n ðeEk;n Þbij ðeEk;n Þ represents the proportion of material from sizes classes i and coarser that suf-
13
16 B ln E− ln E C7 F i;0 ðEÞ ¼ 41þ erf @ qffiffiffiffiffiffiffiffiffi 50i A5 2 2σ 2
3
h
ð8Þ
j¼1
where wi,n + 1 and wi,n are the weight fractions of the material contained in size class i after and before the nth drop, respectively. e is the fraction of the energy from each collision that is captured by an individual particle during a stressing event, κi is the surface breakage rate, aij is the surface breakage distribution function and bij is the energy-specific body breakage distribution function. The term h i 1−F i;n ðeEk;n Þ ð1−κ i Þ on the right-hand side of Eq. (8) represents the
breakage products that report to size class i. Fi,n(eEk,n) is the probability that a particle contained in size class i will suffer body breakage when it captures energy eEk,n from a drop. κi is the proportion of surface breakage products generated in each impact of particles contained in size class i, which is here assumed to be independent of drop height, given the lack of data that allows a proper modeling of the phenomenon. The cumulative surface breakage function may be given by (King, 2001) Ai;j ¼ 1 for di ≥dA λ di ¼ for di bdA dA
ð9Þ
where dA and λ are material parameters. The surface breakage distribution function is assumed to be independent of parent particle size, so that Ai,j = Aj, leading to aj = Aj − Aj − 1. Ek,n is the specific impact energy, which, considering free-fall, is given by Ek;n ¼ ghn
ð10Þ
where g is the acceleration due to gravity and hn is the drop height used in the nth impact. The fraction of the impact energy that is captured by a particle when it drops, hitting a target, depends both on the characteristics of the particles and of the target surface. It may be estimated on the basis of Hertz contact theory considering a perfectly elastic impact, which gives (Tavares, 2004) e¼
ksurface ksurface þ k
! ksteel þ k ksteel
ð11Þ
where ksteel is the stiffness of steel (about 230 GPa), which is the surface used to characterize material breakage properties in the impact load cell, k is the stiffness of the particle and ksurface is the stiffness of the target surface against which the particle hits. This parameter allows simulating impacts of particles against a steel plate, a rubber plate or against other particles. In the later case it becomes e = 0.5(ksteel + k) / ksteel. The body breakage function bij is calculated on the basis of the parameter t10, which may be estimated from (Tavares, 2009) " t 10j ¼ A 1− exp −
b′eEk;n E50bj
!# ð12Þ
where Ek,n is the specific impact energy and A and b′ are parameters that must be fit to single-particle breakage data. E50bj is the median specific fracture energy of the particles contained in size class j that suffered body breakage, which may be given by (Tavares, 2009) −1
E50bj ¼ F j;n
Fig. 2. Experimental (triangles) and predicted (small circles and lines) results showing the cumulative percentage broken after repeated drops from two heights of a Brazilian bauxite ore contained in size range of 45.0–37.5 mm against a steel plate (γ = 3.7). Error bars represent the 90% confidence intervals.
1 F j;n ðEk;n Þ : 2
ð13Þ
The entire single-particle breakage function is then calculated on the basis of interpolating data on t10 versus the various tns using incomplete beta functions. After each impact, the distribution of mass-specific fracture energies of the particles contained in each size class i should be modified, given that some particles contained in this size class may have been broken and others that did not break may have become weaker,
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whereas fragments resulting from breakage of coarser particles may have reported to this size. The distribution of fracture energies of particles contained in size class i is then given by "
# h i i P F i;nþ1 ðEÞwi;n 1−F i;n ðEk;n Þ ð1−κ i Þ þ F i;0 ðEÞ wj;n F j;n ðEk;n Þbij þ 1−F j;n ðEk;n Þ κ j ai j¼1
F i;n þ 1 ðEÞ ¼
wi;nþ1
ð14Þ where the distribution of fracture energies of the particles that were damaged (Fi,* n + 1) is given by rewriting Eqs. (4) and (5), so that F i;nþ1 ðEÞ
D¼
" # F i;n ½E=ð1−DÞ−F i;n ðeEk;n Þ ¼ 1−F i;n ðeEk;n Þ
2γð1−DÞ eEk;n ð2γ−5D þ 5Þ E
2γ: 5
ð15Þ
ð16Þ
In Eq. (14) it is assumed that, when particles contained in size class i suffer body breakage, fragments contained in size class j resulting from this event present the same distribution of particle fracture energies of the original material contained in class j, which is given by Eq. (6). As presented, the model is fully predictive and does not require fitting parameters to calculate the response from multiple drops. It relies on data from carefully conducted experiments in the impact load cell and also from a single drop test at different numbers of drops, besides a tumbling test. These data are obtained as follows. 3. Materials and methods A sample of iron ore (itabirite) from Brazil was collected. Sample preparation consisted of sieving particles into several narrow size ranges. These size fractions were subjected to single-particle impact-breakage tests in the impact load cell and tumbling tests in order to determine the material-specific body breakage parameters required for simulation, and then to drop tests for model validation. The impact load cell consists of a long rod onto which solid-state strain gauges are attached (Tavares and King, 1998). The experiment consisted of impacting individual particles with a free-falling drop weight and recording the force–time profile. From the force–time profile, the specific particle fracture energy E of individual particles and the particle stiffness values k were calculated. From testing several particles contained in a given size range the distribution parameters were determined from Eq. (6). From testing particles contained in different size ranges the parameters in Eq. (7) were estimated and results are presented in Fig. 3 for the iron ore sample. In addition, particles contained in different size ranges were also impacted at variable energy inputs and their fragment size distribution determined, so that parameters in Eq. (12) were estimated, as well as the relationship between the various t10 and tn values (Fig. 4) describing body breakage. Repeated impacts at a constant impact energy were conducted in the impact load cell for particles contained in a narrow size range and the damage accumulation parameter γ was determined from Eqs. (4) to (5). Details on the experimental procedure and equipment may be found elsewhere (Tavares and King, 1998; Tavares, 2009). Surface breakage parameters (κ, λ and dA from Eqs. (8) to (9)) were determined from measuring the product size distribution from tumbling loads of 3 kg contained in size range of 53.0–37.5 mm in a mill measuring 30 × 30 cm operating at 53 rpm in the absence of grinding media. Given the mill diameter and load, it is assumed that each particle will hit the mill shell 863 times during 10 min of the test. In the present paper the surface breakage rate was estimated considering first-order kinetics, assuming that it was independent of
Fig. 3. Comparison between particle fracture energies of a Brazilian iron ore (itabirite) as a function of particle size. Symbols represent the median values (E50), bars represent the 5 and 95 percentiles of the distribution and the solid line the fitted equation (Eq. (7)).
time, particle size and impact energy. The validity of these assumptions is being investigated in the authors' laboratory. A summary of model parameters of the iron ore studied, determined from single-particle impact-breakage and tumbling tests, is shown in Table 1. Drop tests used for model validation were conducted by dropping, individually and by hand, batches of 30 particles contained in size range 125 × 63 mm against a hard metal plate from heights of 2 and 4 m. Only the coarsest fragment after impacting each particle, which was retained in the 63 mm sieve, was subjected to additional impacts, reaching a total of 50 drops. After each impact, fragments were collected and sieved. 4. Results and discussion The effect of number of drops has been investigated from drop tests of the iron ore sample. Fig. 5 compares the cumulative size distributions predicted from the model to those obtained from experiments, demonstrating a good agreement between them. The effect of drop height on the proportion of fines generated has been additionally investigated (Fig. 6) and the results also show a reasonably good fit to the data, although the model seems to overpredict results for more than 15 impacts. This may be due to limitations associated to
Fig. 4. Relationship between t10 and the various tns from single-particle drop weight tests on iron ore particles and model fit to the incomplete beta function (lines).
L.M. Tavares, R.M. de Carvalho / International Journal of Mineral Processing 112–113 (2012) 1–6
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Table 1 Summary of iron ore characteristics determined experimentally and used in the simulations. Sample
Body breakage Eqs. ((9), (12), (13))
Surface breakage Eq. (10)
E∞ do ϕ σ k A b′ (J/kg) (mm) (–) (–) (GPa) (%) (–) Iron ore 16.8
20.1
0.84 1.01 12.5
Eq. (5) Eqs. (6, 7) γ (–)
44.2 0.0288 4.9
κ (%)
λ dA (–) (mm)
0.015 0.31 0.25
assumptions in the model, such as of invariability of surface breakage rates with impact energy, but also could be related to experimental error, given the relatively limited number of particles tested (30). Fig. 6 also shows the influence of impact surface on the proportion of fines generated. Simulations for impacts from a drop height of 2 m are presented for the cases in which individual particles collide against a hard steel plate (ksurface = ksteel = 230 GPa in Eq. 11), against a bed of particles – such as what happens when the material drops on a surface containing other particles (ksurface = k), for instance a stockpile – or against a rubber surface, such as a conveyor belt or a rubbercovered chute (ksurface = 1.3 GPa). Results are consistent with data from Sahoo (2007), which demonstrated that by changing from steel to an ore surface the proportion of fines generated after repeated drops of an Australian iron ore reduced in about half, as long as the proportion of fines in the bed was kept below at about 10%, in order to prevent the effect of cushioning. Fig. 6 shows that the reduction in fines production associated to the use of a rubber surface was even more significant and is consistent with empirical evidence in materials handling systems. It has been recognized by a number of authors (Fagerberg and Sandberg, 1973; Waters et al., 1987; Teo et al., 1990; Sahoo et al., 2002) that subjecting particles to repeated drops results in the phenomenon called stabilization. This happens because the weakest particles subjected to handling are disintegrated more rapidly, so that the remaining material becomes, on average, tougher than the original ore. This is illustrated in Fig. 7 which shows simulations on the variation of the median particle fracture energy (E50) for particles contained in the original size as a function of number of impacts. It demonstrates that the material “gained strength” upon handling, with the greatest increase occurring when handling was more severe (greater drop heights). In practice, what happens is that as the weakest particles are broken, only the toughest particles are left in the sample, although they have also been weakened.
The influence of drop height on degradation is investigated in greater detail in Fig. 8, which compares the amount of fines generated after impacts at different heights, all totaling 40 m (1 × 40 m, 2 × 20 m, 4 × 10 m, …). These simulations are conducted in analogy to experiments carried out by Norgate et al. (1986) and Sahoo (2007). Model predictions are compared to experimental estimates for selected drop heights, showing a reasonable agreement between them. The figure shows that degradation can be reduced significantly by using lower drops, typically below about 1 m. This is consistent with observations by Waters and Mikka (1989), Norgate et al. (1986) and Sahoo et al. (2002). Fig. 8 also suggests that there is an optimum drop height below which abrasion becomes significant and degradation increases again. This increase, however, may be an artifact of the assumption of independence of abrasion breakage rate upon drop height considered in the model. Since no data in the literature seem to support this observation, simulations are illustrated in Fig. 8 with dotted lines. Further, some of the researchers cited recognized that drop heights above about 3 m are generally very detrimental to iron ores, which is also evident in the increased degradation shown in the figure. The same general trend is also shown in Fig. 8 for the case in which the total drop height is 20 m.
Fig. 5. Experimental (symbols) and predicted (lines) results showing the size distributions after repeated drops of iron ore particles contained in size range 125–63 mm against a steel plate from a height of 4 m.
Fig. 7. Simulations showing the variation of the median specific fracture energies of particles contained in the range 106 × 90 mm as a function of drop height and impact surface (simulations from Fig. 6) of 125 × 63 mm iron ore particles.
Fig. 6. Experimental (symbols) and predicted (lines) results showing the proportion of fines (− 6.4 mm material) after repeated single impacts at two drop heights of iron ore particles contained in size range 125–63 mm against a steel surface. Predictions are also presented for drops from 2 m height against other surfaces.
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by changing the order of drops of different magnitudes, the proportion of fines changes, with the greatest degradation occurring from impacts at high magnitude earlier in the handling process rather than later. This model, which can be used to predict the response of different ore types within a mineral deposit to handling, requires fitting a number of material-specific parameters. However, these parameters can be fitted to data collected from controlled testing on single particles using the impact load cell and tumbling tests. Further, these same parameters are not system-specific, so that they can be used not only to simulate degradation due to handling, but also comminution in crushers and mills using state-of-the-art mechanistic models under development in the authors' laboratory. Acknowledgements
Fig. 8. Simulations showing the proportion of fines (− 6.4 mm material) after repeated impacts of iron ore (125 × 63 mm) against a steel surface for different drop heights for total drop heights of 20 and 40 m. Empty symbols represent experimental data and filled symbols model predictions.
Some authors (Norgate et al., 1986) have recognized that degradation also depends on the sequence of impact events. They observed that a drop height of a given value will cause more degradation earlier in the handling system rather than later because of conditioning of the ore, removing the weaker particles, although the difference reported was relatively small (1–2%) for the sequence of drops investigated by them. Simulations have been conducted by dropping lumps of iron ore in a steel surface following two different sequences: in the first sequence, particles were first dropped once from 20 m and then 9 times from a 2 m height, yielding 10.8% of −6.3 mm material (fines); in the second case the sequence of impacts was reversed, yielding 9.8% of fines. These results confirm observations by Norgate et al. (1986), even quantitatively (1% difference), demonstrating that the model is able to account for the main physical phenomena involved in degradation during handling.
5. Conclusions A fundamental model of degradation of ores as a result of drops and transfers has been developed. It is based on principles of continuum damage mechanics and measurements of single-particle fracture energies. The model has been validated using data from repeated drops of 125–63 mm iron ore lumps against a steel plate in the laboratory, with reasonably good agreement. The model was then used to simulate the influence of impact height, number of drops and impact surface on the proportion of fines (− 6.4 mm material) generated. It also predicted the phenomenon of stabilization, by which the unbroken material becomes progressively tougher, on average, as a result of repeated impacts. Finally, the model was also able to demonstrate that
The authors would like to acknowledge the financial support from the Brazilian Council of Research, CNPq. The authors also thank Vale S.A. for providing the samples used for testing. References Baxter, J., Abu-Nahar, A., Tüzün, U., 2004. The breakage matrix approach to inadvertent particulate degradation: dealing with intra-mixture interactions. Powder Technol. 143–144, 174–178. Dukino, R.D., Swain, M.V., Loo, C.E., Bristow, N., England, B.M., 1995. Fracture behaviour of three Australian iron ores. Trans. Intn. Min. Metall. C 104, C11–C19. Dukino, R.D., Swain, M.V., Loo, C.E., 2000. A simple contact and fracture mechanics approach to tumble drum breakage. Int. J. Miner. Process. 59, 175–183. Fagerberg, F., Sandberg, N., 1973. Degradation of lump ores in transport. Proc. 2nd Int. Symp. on Transportation and Handling of Minerals, Rotterdam 2, 128–156. Kachanov, L.M., 1958. Time of the rupture process under creep conditions (in Russian). Izv. Akad. Nauk AN SSSR 8, 26–31. King, R.P., 2001. Modeling and Simulation of Mineral Processing Systems. ButterworthHeinemann. Norgate, T.E., Tompsitt, D.F., Batterham, R.J., 1986. Computer simulation of the degradation of lump ore during transportation and handling. Proc., 2nd Int. Conf. on Bulk Materials Storage, Handling and Transportation, Wollongong, pp. 20–25. Sahoo, R., 2007. Degradation of steelmaking materials during handling. Powder Technol. 176, 77–87. Sahoo, R.K., Weedon, D.M., Roach, D., 2002. Experimental study of several factors effecting Gladstone Port Authority's lump degradation. Bulk Solids Handling 22, 356–361. Tavares, L.M., 2004. Optimum routes in particle breakage by impact. Powder Technol. 142, 81–91. Tavares, L.M., 2009. Analysis of particle fracture by repeated impacts using damage mechanics. Powder Technol. 190, 327–339. Tavares, L.M., Carvalho, R.M., 2009. Modeling breakage rates of coarse particles in ball mills. Miner. Eng. 22, 650–659. Tavares, L.M., King, R.P., 1998. Single-particle fracture under impact loading. Int. J. Miner. Process. 54, 1–28. Tavares, L.M., King, R.P., 2002. Modeling of particle breakage by repeated impacts using continuum damage mechanics. Powder Technol. 123, 138–146. Tavares, L.M., King, R.P., 2005. Continuum damage modeling of particle fracture. ZKG Int. 58, 49–58. Teo, C.S., Waters, A.G., Nikol, S.K., 1990. Quantification of the breakage of lump material during handling operations. Int. J. Miner. Process. 30, 159–184. Waters, A.G., Mikka, R.A., 1989. Segregation of fines of lump iron ore due to vibration on a conveyor belt. Proc. 3rd Int. Conf. on Bulk Materials Storage, Handling and Transportation, Newcastle, pp. 89–93. Waters, A.G., Vince, A., Teo, C.S., 1987. A technique for determining the resistance of shatter of lump materials: Proc. of CHEMECA, Melbourne, vol. 2, pp. 107.1–107.8. Weedon, D.M., Wilson, F., 2000. Modelling iron ore degradation using a twin pendulum breakage device. Int. J. Miner. Process. 59, 195–213.
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International Journal of Mineral Processing journal homepage: www.elsevier.com/locate/ijminpro
Modeling ore degradation during handling using continuum damage mechanics☆ Luís Marcelo Tavares ⁎, Rodrigo M. de Carvalho Department of Metallurgical and Materials Engineering, COPPE, Universidade Federal do Rio de Janeiro – UFRJ. Cx. Postal 68505, CEP 21941-972, Rio de Janeiro, RJ, Brazil
a r t i c l e
i n f o
Article history: Received 22 February 2010 Received in revised form 29 June 2010 Accepted 10 July 2010 Available online 16 April 2012 Keywords: Degradation Handling Modeling Particle breakage
a b s t r a c t Degradation of steelmaking materials during handling causes a significant economic impact, given the higher prices achieved by lump ore in comparison to finer products. It is relevant to model the amenability of ores to degradation, since it may be used to simulate the response of ores to different sequences of handling and transportation events from mine to port. The paper presents a mechanistic model to describe degradation caused by transfers and drops of particulate materials. The model is based on damage mechanics and the distribution of fracture energies of the original particles, using data collected in impact load cell and tumbling tests. This model, which is capable of predicting the proportion of particles broken and the entire size distribution resulting from any sequence of drops, has been validated at a preliminary level using data from drop tests of a Brazilian iron ore. © 2010 Elsevier Inc. All rights reserved.
1. Introduction Ores undergo significant degradation during mining as a result of blasting, mechanical handling by shovels and crushing in order to produce a size distribution that is capable of meeting specifications that are set by customers and the market. Such degradation is particularly relevant in the case of steelmaking materials. Degradation of lump ores, such as coal, iron and manganese ores into fines is often undesirable because lumps have premium prices when compared to fines. However, the relationship between the proportion of lump ore and fines that is produced in a given processing plant is codetermined by the ore's mechanical strength and the nature of the crushing and handling processes (Dukino et al., 1995). In addition to that, between the mine and the end user, the lump material is subject to a number of mechanical actions which cause degradation. These include at the mine site: crushing and screening, conveying, stockpiling and rail wagon loading; and at the port: rail wagon unloading, conveying, drops at transfer points, screening, stockpiling, reclaiming and ship loading (Sahoo, 2007). A number of experimental studies have been conducted to investigate ore degradation during handling. Some of the conclusions of these studies may be summarized as: • Degradation during handling results predominantly from impact from drops and, to a minor extent, also from abrasion due to gravity flow through bins (Fagerberg and Sandberg, 1973); ☆ A publishers' error resulted in this article appearing in the wrong issue. The article is reprinted here for the reader's convenience and for the continuity of the special issue. For citation purposes, please use the original publication details; International Journal of Mineral Processing, 101(1-4), pp. 21-27. **DOI of original item: doi:10.1016/j.minpro.2010.07.008. DOI of original article: 10.1016/j.minpro.2010.07.008. ⁎ Corresponding author. E-mail address: [email protected] (L.M. Tavares). 0301-7516/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.minpro.2010.07.011
• Degradation varies with particle size, with coarser particles suffering a greater degradation effect than finer ones (Fagerberg and Sandberg, 1973; Waters et al., 1987); • Greater fines generation is associated to handling weaker ores than tougher ones (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Sahoo, 2007); • Impact surface influences the level of degradation upon impact, and a cushioning effect occurs as the percentage of initial fines increases (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Sahoo, 2007); • Norgate et al. (1986) stated that a drop height of large magnitude will cause more degradation earlier in a handling system rather than later, because of the conditioning of the ore, which removes the weaker particles; • Degradation is significantly reduced by using drop heights below about 1 to 3 m (Fagerberg and Sandberg, 1973; Norgate et al., 1986; Waters and Mikka, 1989). Prediction of the amenability of ores to degradation is relevant, since it can be used to investigate the response of ores to different sequences of handling and transportation events from mine to port. This can be a useful tool to assess the effectiveness of measures taken to minimize degradation during handling. A number of attempts have been made in the last couple of decades to describe quantitatively this phenomenon (Norgate et al., 1986; Teo et al., 1990; Dukino et al., 2000; Weedon and Wilson, 2000; Baxter et al., 2004; Sahoo, 2007), without any particular model finding wide acceptance or being able to describe in great detail the response of the material during the handling processes. The paper presents a simulation procedure that describes degradation caused by transfers and drops of the particulate material. It is based on data collected from drop weight, impact load cell tests and tumbling tests, a model from damage mechanics and the distribution of strengths in the original material.
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2. Model description When a particle is dropped during a transfer, it may break or not. Whenever the particle does not break catastrophically (body breakage), its surface will be abraded (surface breakage) and it may accumulate crack-like damage, so that it may eventually break as the result of an impact of a comparatively low magnitude in a future drop. Modeling weakening that the particle undergoes after repeated impacts is of major importance to describe degradation during handling, and a model has been proposed to describe it using elements from damage mechanics (Tavares and King, 2002; Tavares, 2009). The model is based on the recognition that the load-deformation response which results from impact of a spherical particle can be described as a combination of continuum damage mechanics (Kachanov, 1958) and Hertz contact theory (Tavares and King, 2005), giving
F¼
d1=2 ~ 3=2 kα 3
ð1Þ
where F is the load, d is the particle size, α is the local deformation and k~ ¼ kð1−DÞ
ð2Þ
is the stiffness of the particle during the stressing event. k is the stiffness of the particle prior to stressing, given by k = Y / (1 − μ 2), where Y is the modulus of elasticity and μ is the Poisson's ratio. The damage variable D may be described by the power law relationship D¼
α αc
γ
ð3Þ
which is valid during compression of the particle. αc is the deformation associated to fracture of the particle. By definition, damage is irreversible (Kachanov, 1958), so that, if during compression the particle does not fracture, then during restitution D = D*, where D* is the maximum value that the damage variable reached during the loading part of the cycle. Assuming that both the particle orientation and the model parameters αc and γ remain constant, Eqs. (1)–(3) can be used to predict the force–deformation curves resulting from repeatedly loading of a particle up to a given strain energy level, until its final failure (Fig. 1). Unfortunately, this procedure cannot be easily used in practice because it requires the knowledge of several material constants, including αc and k, besides the damage accumulation coefficient γ. These are material-specific constants that are not straight-forward to estimate directly from experiments (Tavares and King, 2005). Typically, only the distribution of fracture energies (probability distribution of body breakage) of the original material and the stressing energies in each event may be known, using, for instance, the impact load cell device (Tavares and King, 1998). A convenient procedure that allows simulating multiple loading events using a limited amount of data has been proposed and is presented elsewhere (Tavares and King, 2002; Tavares and Carvalho, 2009). It assumes that the constitutive equation (Eq. (1)) remains valid throughout the several loading events, so that the distribution of particle fracture energies after the nth loading is given by F nþ1 ðEÞ ¼ 2D ¼
F n ½E=ð1−DÞ−F n ðEk Þ 1−F n ðEk Þ
2γð1−DÞ Ek ð2γ−5Dþ5Þ E
ð4Þ
2γ 5
ð5Þ
where Fn(Ek) is the proportion of particles broken at the nth drop from a specific impact energy of Ek. Eq. (5) should be solved using an efficient numerical procedure, since D is implicit.
Fig. 1. Illustration of the effect of weakening due to accrual of damage in repeated impacts at a constant stressing energy.
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The initial distribution of particle fracture energies can be generally well described using the log-normal distribution (Tavares and King, 1998), given by 2
0
ð6Þ
where the median fracture energy may be described by ð7Þ
where di is the representative size of particles contained in the ith class, σi2 is the variance of the distribution and E∞, do and ϕ are model parameters that should be fitted to experimental data. Results of experiments used to determine the parameter γ are shown in Fig. 2, where the fraction of particles that suffered body breakage (those that lost at least 10% of their original weight) is presented as a function of number of drops, along with model predictions. In order to simulate degradation during handling, it is necessary to predict the complete size distribution of the solids after the drops, not only the proportion of material broken. This is given from a mass balance of each size fraction after an individual drop of all the particles, which is given by wi;nþ1
i ¼ wi;n 1−F i;n ðeEk;n Þ ð1−κ i Þ i h h i i X þ wj;n F j;n ðeEk;n Þbij ðeEk;n Þ þ κ j 1−F j;n ðeEk;n Þ aij
j¼1
fered body breakage and reported to the size class i. The term h h i i i P wj;n κ j 1−F j;n ðeEk;n Þ aij gives the cumulative proportion of surface
j¼1
i
ϕ d E50i ¼ E∞ 1 þ o di
fraction of the particles that suffered neither body nor surface breakage h i i P during the nth drop, whereas the term wj;n F j;n ðeEk;n Þbij ðeEk;n Þ represents the proportion of material from sizes classes i and coarser that suf-
13
16 B ln E− ln E C7 F i;0 ðEÞ ¼ 41þ erf @ qffiffiffiffiffiffiffiffiffi 50i A5 2 2σ 2
3
h
ð8Þ
j¼1
where wi,n + 1 and wi,n are the weight fractions of the material contained in size class i after and before the nth drop, respectively. e is the fraction of the energy from each collision that is captured by an individual particle during a stressing event, κi is the surface breakage rate, aij is the surface breakage distribution function and bij is the energy-specific body breakage distribution function. The term h i 1−F i;n ðeEk;n Þ ð1−κ i Þ on the right-hand side of Eq. (8) represents the
breakage products that report to size class i. Fi,n(eEk,n) is the probability that a particle contained in size class i will suffer body breakage when it captures energy eEk,n from a drop. κi is the proportion of surface breakage products generated in each impact of particles contained in size class i, which is here assumed to be independent of drop height, given the lack of data that allows a proper modeling of the phenomenon. The cumulative surface breakage function may be given by (King, 2001) Ai;j ¼ 1 for di ≥dA λ di ¼ for di bdA dA
ð9Þ
where dA and λ are material parameters. The surface breakage distribution function is assumed to be independent of parent particle size, so that Ai,j = Aj, leading to aj = Aj − Aj − 1. Ek,n is the specific impact energy, which, considering free-fall, is given by Ek;n ¼ ghn
ð10Þ
where g is the acceleration due to gravity and hn is the drop height used in the nth impact. The fraction of the impact energy that is captured by a particle when it drops, hitting a target, depends both on the characteristics of the particles and of the target surface. It may be estimated on the basis of Hertz contact theory considering a perfectly elastic impact, which gives (Tavares, 2004) e¼
ksurface ksurface þ k
! ksteel þ k ksteel
ð11Þ
where ksteel is the stiffness of steel (about 230 GPa), which is the surface used to characterize material breakage properties in the impact load cell, k is the stiffness of the particle and ksurface is the stiffness of the target surface against which the particle hits. This parameter allows simulating impacts of particles against a steel plate, a rubber plate or against other particles. In the later case it becomes e = 0.5(ksteel + k) / ksteel. The body breakage function bij is calculated on the basis of the parameter t10, which may be estimated from (Tavares, 2009) " t 10j ¼ A 1− exp −
b′eEk;n E50bj
!# ð12Þ
where Ek,n is the specific impact energy and A and b′ are parameters that must be fit to single-particle breakage data. E50bj is the median specific fracture energy of the particles contained in size class j that suffered body breakage, which may be given by (Tavares, 2009) −1
E50bj ¼ F j;n
Fig. 2. Experimental (triangles) and predicted (small circles and lines) results showing the cumulative percentage broken after repeated drops from two heights of a Brazilian bauxite ore contained in size range of 45.0–37.5 mm against a steel plate (γ = 3.7). Error bars represent the 90% confidence intervals.
1 F j;n ðEk;n Þ : 2
ð13Þ
The entire single-particle breakage function is then calculated on the basis of interpolating data on t10 versus the various tns using incomplete beta functions. After each impact, the distribution of mass-specific fracture energies of the particles contained in each size class i should be modified, given that some particles contained in this size class may have been broken and others that did not break may have become weaker,
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whereas fragments resulting from breakage of coarser particles may have reported to this size. The distribution of fracture energies of particles contained in size class i is then given by "
# h i i P F i;nþ1 ðEÞwi;n 1−F i;n ðEk;n Þ ð1−κ i Þ þ F i;0 ðEÞ wj;n F j;n ðEk;n Þbij þ 1−F j;n ðEk;n Þ κ j ai j¼1
F i;n þ 1 ðEÞ ¼
wi;nþ1
ð14Þ where the distribution of fracture energies of the particles that were damaged (Fi,* n + 1) is given by rewriting Eqs. (4) and (5), so that F i;nþ1 ðEÞ
D¼
" # F i;n ½E=ð1−DÞ−F i;n ðeEk;n Þ ¼ 1−F i;n ðeEk;n Þ
2γð1−DÞ eEk;n ð2γ−5D þ 5Þ E
2γ: 5
ð15Þ
ð16Þ
In Eq. (14) it is assumed that, when particles contained in size class i suffer body breakage, fragments contained in size class j resulting from this event present the same distribution of particle fracture energies of the original material contained in class j, which is given by Eq. (6). As presented, the model is fully predictive and does not require fitting parameters to calculate the response from multiple drops. It relies on data from carefully conducted experiments in the impact load cell and also from a single drop test at different numbers of drops, besides a tumbling test. These data are obtained as follows. 3. Materials and methods A sample of iron ore (itabirite) from Brazil was collected. Sample preparation consisted of sieving particles into several narrow size ranges. These size fractions were subjected to single-particle impact-breakage tests in the impact load cell and tumbling tests in order to determine the material-specific body breakage parameters required for simulation, and then to drop tests for model validation. The impact load cell consists of a long rod onto which solid-state strain gauges are attached (Tavares and King, 1998). The experiment consisted of impacting individual particles with a free-falling drop weight and recording the force–time profile. From the force–time profile, the specific particle fracture energy E of individual particles and the particle stiffness values k were calculated. From testing several particles contained in a given size range the distribution parameters were determined from Eq. (6). From testing particles contained in different size ranges the parameters in Eq. (7) were estimated and results are presented in Fig. 3 for the iron ore sample. In addition, particles contained in different size ranges were also impacted at variable energy inputs and their fragment size distribution determined, so that parameters in Eq. (12) were estimated, as well as the relationship between the various t10 and tn values (Fig. 4) describing body breakage. Repeated impacts at a constant impact energy were conducted in the impact load cell for particles contained in a narrow size range and the damage accumulation parameter γ was determined from Eqs. (4) to (5). Details on the experimental procedure and equipment may be found elsewhere (Tavares and King, 1998; Tavares, 2009). Surface breakage parameters (κ, λ and dA from Eqs. (8) to (9)) were determined from measuring the product size distribution from tumbling loads of 3 kg contained in size range of 53.0–37.5 mm in a mill measuring 30 × 30 cm operating at 53 rpm in the absence of grinding media. Given the mill diameter and load, it is assumed that each particle will hit the mill shell 863 times during 10 min of the test. In the present paper the surface breakage rate was estimated considering first-order kinetics, assuming that it was independent of
Fig. 3. Comparison between particle fracture energies of a Brazilian iron ore (itabirite) as a function of particle size. Symbols represent the median values (E50), bars represent the 5 and 95 percentiles of the distribution and the solid line the fitted equation (Eq. (7)).
time, particle size and impact energy. The validity of these assumptions is being investigated in the authors' laboratory. A summary of model parameters of the iron ore studied, determined from single-particle impact-breakage and tumbling tests, is shown in Table 1. Drop tests used for model validation were conducted by dropping, individually and by hand, batches of 30 particles contained in size range 125 × 63 mm against a hard metal plate from heights of 2 and 4 m. Only the coarsest fragment after impacting each particle, which was retained in the 63 mm sieve, was subjected to additional impacts, reaching a total of 50 drops. After each impact, fragments were collected and sieved. 4. Results and discussion The effect of number of drops has been investigated from drop tests of the iron ore sample. Fig. 5 compares the cumulative size distributions predicted from the model to those obtained from experiments, demonstrating a good agreement between them. The effect of drop height on the proportion of fines generated has been additionally investigated (Fig. 6) and the results also show a reasonably good fit to the data, although the model seems to overpredict results for more than 15 impacts. This may be due to limitations associated to
Fig. 4. Relationship between t10 and the various tns from single-particle drop weight tests on iron ore particles and model fit to the incomplete beta function (lines).
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Table 1 Summary of iron ore characteristics determined experimentally and used in the simulations. Sample
Body breakage Eqs. ((9), (12), (13))
Surface breakage Eq. (10)
E∞ do ϕ σ k A b′ (J/kg) (mm) (–) (–) (GPa) (%) (–) Iron ore 16.8
20.1
0.84 1.01 12.5
Eq. (5) Eqs. (6, 7) γ (–)
44.2 0.0288 4.9
κ (%)
λ dA (–) (mm)
0.015 0.31 0.25
assumptions in the model, such as of invariability of surface breakage rates with impact energy, but also could be related to experimental error, given the relatively limited number of particles tested (30). Fig. 6 also shows the influence of impact surface on the proportion of fines generated. Simulations for impacts from a drop height of 2 m are presented for the cases in which individual particles collide against a hard steel plate (ksurface = ksteel = 230 GPa in Eq. 11), against a bed of particles – such as what happens when the material drops on a surface containing other particles (ksurface = k), for instance a stockpile – or against a rubber surface, such as a conveyor belt or a rubbercovered chute (ksurface = 1.3 GPa). Results are consistent with data from Sahoo (2007), which demonstrated that by changing from steel to an ore surface the proportion of fines generated after repeated drops of an Australian iron ore reduced in about half, as long as the proportion of fines in the bed was kept below at about 10%, in order to prevent the effect of cushioning. Fig. 6 shows that the reduction in fines production associated to the use of a rubber surface was even more significant and is consistent with empirical evidence in materials handling systems. It has been recognized by a number of authors (Fagerberg and Sandberg, 1973; Waters et al., 1987; Teo et al., 1990; Sahoo et al., 2002) that subjecting particles to repeated drops results in the phenomenon called stabilization. This happens because the weakest particles subjected to handling are disintegrated more rapidly, so that the remaining material becomes, on average, tougher than the original ore. This is illustrated in Fig. 7 which shows simulations on the variation of the median particle fracture energy (E50) for particles contained in the original size as a function of number of impacts. It demonstrates that the material “gained strength” upon handling, with the greatest increase occurring when handling was more severe (greater drop heights). In practice, what happens is that as the weakest particles are broken, only the toughest particles are left in the sample, although they have also been weakened.
The influence of drop height on degradation is investigated in greater detail in Fig. 8, which compares the amount of fines generated after impacts at different heights, all totaling 40 m (1 × 40 m, 2 × 20 m, 4 × 10 m, …). These simulations are conducted in analogy to experiments carried out by Norgate et al. (1986) and Sahoo (2007). Model predictions are compared to experimental estimates for selected drop heights, showing a reasonable agreement between them. The figure shows that degradation can be reduced significantly by using lower drops, typically below about 1 m. This is consistent with observations by Waters and Mikka (1989), Norgate et al. (1986) and Sahoo et al. (2002). Fig. 8 also suggests that there is an optimum drop height below which abrasion becomes significant and degradation increases again. This increase, however, may be an artifact of the assumption of independence of abrasion breakage rate upon drop height considered in the model. Since no data in the literature seem to support this observation, simulations are illustrated in Fig. 8 with dotted lines. Further, some of the researchers cited recognized that drop heights above about 3 m are generally very detrimental to iron ores, which is also evident in the increased degradation shown in the figure. The same general trend is also shown in Fig. 8 for the case in which the total drop height is 20 m.
Fig. 5. Experimental (symbols) and predicted (lines) results showing the size distributions after repeated drops of iron ore particles contained in size range 125–63 mm against a steel plate from a height of 4 m.
Fig. 7. Simulations showing the variation of the median specific fracture energies of particles contained in the range 106 × 90 mm as a function of drop height and impact surface (simulations from Fig. 6) of 125 × 63 mm iron ore particles.
Fig. 6. Experimental (symbols) and predicted (lines) results showing the proportion of fines (− 6.4 mm material) after repeated single impacts at two drop heights of iron ore particles contained in size range 125–63 mm against a steel surface. Predictions are also presented for drops from 2 m height against other surfaces.
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by changing the order of drops of different magnitudes, the proportion of fines changes, with the greatest degradation occurring from impacts at high magnitude earlier in the handling process rather than later. This model, which can be used to predict the response of different ore types within a mineral deposit to handling, requires fitting a number of material-specific parameters. However, these parameters can be fitted to data collected from controlled testing on single particles using the impact load cell and tumbling tests. Further, these same parameters are not system-specific, so that they can be used not only to simulate degradation due to handling, but also comminution in crushers and mills using state-of-the-art mechanistic models under development in the authors' laboratory. Acknowledgements
Fig. 8. Simulations showing the proportion of fines (− 6.4 mm material) after repeated impacts of iron ore (125 × 63 mm) against a steel surface for different drop heights for total drop heights of 20 and 40 m. Empty symbols represent experimental data and filled symbols model predictions.
Some authors (Norgate et al., 1986) have recognized that degradation also depends on the sequence of impact events. They observed that a drop height of a given value will cause more degradation earlier in the handling system rather than later because of conditioning of the ore, removing the weaker particles, although the difference reported was relatively small (1–2%) for the sequence of drops investigated by them. Simulations have been conducted by dropping lumps of iron ore in a steel surface following two different sequences: in the first sequence, particles were first dropped once from 20 m and then 9 times from a 2 m height, yielding 10.8% of −6.3 mm material (fines); in the second case the sequence of impacts was reversed, yielding 9.8% of fines. These results confirm observations by Norgate et al. (1986), even quantitatively (1% difference), demonstrating that the model is able to account for the main physical phenomena involved in degradation during handling.
5. Conclusions A fundamental model of degradation of ores as a result of drops and transfers has been developed. It is based on principles of continuum damage mechanics and measurements of single-particle fracture energies. The model has been validated using data from repeated drops of 125–63 mm iron ore lumps against a steel plate in the laboratory, with reasonably good agreement. The model was then used to simulate the influence of impact height, number of drops and impact surface on the proportion of fines (− 6.4 mm material) generated. It also predicted the phenomenon of stabilization, by which the unbroken material becomes progressively tougher, on average, as a result of repeated impacts. Finally, the model was also able to demonstrate that
The authors would like to acknowledge the financial support from the Brazilian Council of Research, CNPq. The authors also thank Vale S.A. for providing the samples used for testing. References Baxter, J., Abu-Nahar, A., Tüzün, U., 2004. The breakage matrix approach to inadvertent particulate degradation: dealing with intra-mixture interactions. Powder Technol. 143–144, 174–178. Dukino, R.D., Swain, M.V., Loo, C.E., Bristow, N., England, B.M., 1995. Fracture behaviour of three Australian iron ores. Trans. Intn. Min. Metall. C 104, C11–C19. Dukino, R.D., Swain, M.V., Loo, C.E., 2000. A simple contact and fracture mechanics approach to tumble drum breakage. Int. J. Miner. Process. 59, 175–183. Fagerberg, F., Sandberg, N., 1973. Degradation of lump ores in transport. Proc. 2nd Int. Symp. on Transportation and Handling of Minerals, Rotterdam 2, 128–156. Kachanov, L.M., 1958. Time of the rupture process under creep conditions (in Russian). Izv. Akad. Nauk AN SSSR 8, 26–31. King, R.P., 2001. Modeling and Simulation of Mineral Processing Systems. ButterworthHeinemann. Norgate, T.E., Tompsitt, D.F., Batterham, R.J., 1986. Computer simulation of the degradation of lump ore during transportation and handling. Proc., 2nd Int. Conf. on Bulk Materials Storage, Handling and Transportation, Wollongong, pp. 20–25. Sahoo, R., 2007. Degradation of steelmaking materials during handling. Powder Technol. 176, 77–87. Sahoo, R.K., Weedon, D.M., Roach, D., 2002. Experimental study of several factors effecting Gladstone Port Authority's lump degradation. Bulk Solids Handling 22, 356–361. Tavares, L.M., 2004. Optimum routes in particle breakage by impact. Powder Technol. 142, 81–91. Tavares, L.M., 2009. Analysis of particle fracture by repeated impacts using damage mechanics. Powder Technol. 190, 327–339. Tavares, L.M., Carvalho, R.M., 2009. Modeling breakage rates of coarse particles in ball mills. Miner. Eng. 22, 650–659. Tavares, L.M., King, R.P., 1998. Single-particle fracture under impact loading. Int. J. Miner. Process. 54, 1–28. Tavares, L.M., King, R.P., 2002. Modeling of particle breakage by repeated impacts using continuum damage mechanics. Powder Technol. 123, 138–146. Tavares, L.M., King, R.P., 2005. Continuum damage modeling of particle fracture. ZKG Int. 58, 49–58. Teo, C.S., Waters, A.G., Nikol, S.K., 1990. Quantification of the breakage of lump material during handling operations. Int. J. Miner. Process. 30, 159–184. Waters, A.G., Mikka, R.A., 1989. Segregation of fines of lump iron ore due to vibration on a conveyor belt. Proc. 3rd Int. Conf. on Bulk Materials Storage, Handling and Transportation, Newcastle, pp. 89–93. Waters, A.G., Vince, A., Teo, C.S., 1987. A technique for determining the resistance of shatter of lump materials: Proc. of CHEMECA, Melbourne, vol. 2, pp. 107.1–107.8. Weedon, D.M., Wilson, F., 2000. Modelling iron ore degradation using a twin pendulum breakage device. Int. J. Miner. Process. 59, 195–213.