International Journal of Mineral Processing 126 (2014) 49–54

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International Journal of Mineral Processing

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Rapid ore breakage parameter estimation from a laboratory crushing test

Fernando N. de Magalhães, Luís Marcelo Tavares ⁎Department of Metallurgical and Materials Engineering, Universidade Federal do Rio de Janeiro, COPPE/UFRJ, Cx. Postal 68505, CEP 21941-972, Rio de Janeiro, RJ, Brazil

⁎ Corresponding author. Tel.: +55 21 2562 8538.E-mail address: tavares@ufrj.br (L.M. Tavares).

0301-7516/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.minpro.2013.11.007

a b s t r a c t

a r t i c l e i n f o

Article history:Received 2 August 2013Received in revised form 7 November 2013Accepted 17 November 2013Available online 26 November 2013

Keywords:Breakage parametersCrushingPowerParticle breakageGeometallurgy

Dropweight testers have beenwidely and successfully used to characterize ore breakage parameters for simulationof crushers andmills. In some cases, limitations of time, sample availability, and even cost, can prevent the conduc-tion of these tests when preparing samples for chemical analysis and/or grindability and flotation/concentrationtesting. The present work demonstrates how a laboratory cone crusher equipped with a power meter, in conjunc-tion withWhiten–Awachie crusher model, has been used to quickly estimate the A * b breakage parameters. Withan average absolute error of 36% when applied to a variety of materials, it is demonstrated that themethod is not apriori restricted to the particular crusher used, neither to the crushing conditions employed. It is discussed that themethod, when part of a variability study, can be used as a convenient tool for geometallurgical mapping of ore de-posits regarding ore breakage response.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

A number of questions should be answered when selecting the ap-propriate crusher to an application, which include crusher capacity,power consumption, product size distribution and wear rate of liners,amongst others (Bearman et al., 1997; Tavares and da Silveira, 2008).In estimating the product size and the corresponding specific energyconsumption, a number of options exist, although it is not uncommonto solely rely on data provided by crusher manufacturers or to usedata from other plants that processes similar ores.

Throughout the life of a mine, crushers may be fed with ores withhighly variable crushing responses, so that characterizing the amenabil-ity of ores contained in different parts of a deposit to crushing is worth-while. This requires, however, proper crushability tests as well asmathematical expressions that can use characterization data to predictequipment performance.

Briefly, two main indices have been used to access the response ofores to crushing (Tavares, 2007; Tavares and da Silveira, 2008). Thecrushing work index (CWi), estimated on the basis of the twin-pendulum test, has been proposed by Fred Bond and accumulates along track record in predicting the performance of crushing operations,although with several cases of poor performance. On the other hand,the A ∗ b breakage parameters, obtained using the drop weight test(Napier-Munn et al., 1996) and,more recently, the rotary breakage tester(Shi et al., 2009), has been used with reasonable success in conjunctionwithWhiten's model of compression crushers (Whiten, 1972) to predictthe performance of full-scale machines.

ghts reserved.

The A ∗ b breakage parameters are calculated on the basis of impacttests on individual particles contained in five standard size classes, rang-ing from 63–53 mm down to 16.0–13.2 mm. In several instances,however, sample availability and even cost represent challenges toconducting the test, so that it may not be included in the suite of teststhat are carried out when processing drill core samples, especiallywhen preparing samples for assaying. Although alternative tests havebeen proposed that use more limited sample volumes (Chieregati andDelboni, 2002; Morrell, 2004) there are instances that not even thesetests are used. In all cases, however, laboratory crushers are invariablyused to prepare the sample for grindability or concentration tests or, atleast, for assaying or mineralogical analyses. Indeed, some researchers(Kojovic et al., 2010) have identified that the response of the samples,normally diamonddrill cores, to this crushing operation could provide in-formation on breakage response of ores to size reduction, and proposedan additional comminution index, the Ci.

Evidently, a number of measures of breakage response of ores havesome degree of correlation with the A ∗ b breakage parameters. For in-stance, Kojovic et al. (2010) show that it is correlated to the Ci. Napier-Munn et al. (1996) show the correlation between A ∗ b and the Bondball mill work index, whereas Tavares and da Silveira (2008) demon-strates that it is correlated with the Bond crushing work index (CWi)and even the Los Angeles index, more commonly used to characterizethe amenability of a rock to be used in pavements. However, empiricalcorrelations with no physical background carry the risk of only beingvalid for the cases forwhich theywere developed, besides requiring strictadherence to the standard used in the test.

The present paper demonstrates that the laboratory crushing opera-tion that is used to prepare drill core samples for grindability or concen-tration tests, or even assaying, can be, after proper calibration, used toestimate the breakage indices A ∗ b of ores.

50 F.N. de Magalhães, L.M. Tavares / International Journal of Mineral Processing 126 (2014) 49–54

2. Background

Comminution in compression crushers can be described by the well-known model from W. Whiten (Whiten, 1972; Napier-Munn et al.,1996). The mass balance equations can be defined in matrix (Napier-Munn et al., 1996) or in algebraic form (King, 2001). King (2001) showedthat the size distribution in the crusher product pi can be calculated from

pi ¼ 1−cið Þxi ð1Þ

where the various xi can be calculated sequentially from the coarsest sizeclass in the feed (i = 1) using the expression

xi ¼1

1−cibiip fi þ

Xi−1

j¼1

c jx jbij

0@

1A ð2Þ

which requires the knowledge of the size distribution of the feed pif, the

classification function ci and the breakage or appearance function bij,which corresponds to the fraction of particles contained in class j that re-port to size class i after a single breakage event. ci is a functionwhose pa-rameters are machine-dependent. A useful model for the classificationfunction is given by (Whiten, 1972)

ci ¼1 for dpi≥d2

1−dpi−d2d1−d2

� �nford1bdpibd2

0 for dpi≤d1

8>><>>: ð3Þ

where dpi is the representative size of particles contained in class i and d1,d2 and n are model parameters that must be fit from experimental data.Whereas n is usually constant at 2.3, d1 and d2 are normally influencedby operating variables of the crusher,mainly varyingwith crusher setting,thoughput, feedrate and eccentric throw (Napier-Munn et al., 1996). d1 isthe size belowwhich all particles escape breakage, whereas d2 is the sizeabove which all particles are broken inside the machine.

The breakage function in density form bij is calculated from the cu-mulative breakage function by

bij ¼ B Di−1;dpj� �

−B Di;dpj� �

andbjj ¼ 1−B Dj;dpj

� � ð4Þ

where Di is the ith screen size.The power model relates the actual power drawn by the crusher to

the power required by the laboratory drop weight tester to achievethe same size reduction (Morrel et al., 1992). It is of the form

Pc ¼ SPd þ Pn ð5Þ

where Pc is the actual power drawn by the crusher under load, Pd is thecalculated power in thedropweight tester, Pn is the power drawnby thecrusher under no load, often also regarded as a fitting parameter, and Sis a dimensionless factor for a particular crusher, estimated by regres-sion (Napier-Munn et al., 1996).

The drop weight equivalent power Pd is given by

Pd ¼ WXNi¼1

Ecst10i ci xi ð6Þ

where N is the number of size intervals and W is the throughput. Pd istherefore the total energy required to reduce the crusher feed to theproduct size distribution, as if all the size reduction took place in thedrop weight tester at a specific impact energy Ecs, given in kWh/t.

Values of S for industrial-scale secondary and tertiary cone andgyratory crushers have been found to vary typically from 1.2 to 1.55(Napier-Munn et al., 1996). The inverse of this number is considered

the efficiency of the crusher, which typically ranges from 70 to 80%(Napier-Munn et al., 1996).

3. Experimental

3.1. Materials

Samples of over 25 materials have been collected for testing, whichinclude several samples of bauxites, copper ores, limestones, acidrocks, coals and an iron ore. They have been prepared for testing bysieving in the appropriate size fractions. Sampleswere dried tomoisturecontents below 0.1% prior to all tests. More details on the sample char-acteristics can be found elsewhere (Tavares and Carvalho, 2007).

3.2. Drop weight tester

In the impact load cell (Tavares, 2007), such as in any drop weighttester, the energy-size reduction relationship can be analyzed byimpacting single particles at variable input energies. In the test, stan-dardized at the Julius Kruttschnitt Mineral Research Centre (Napier-Munn et al., 1996), drop weights from about 3 to 50 kg are used to de-termine the impact breakage characteristics of particles contained infive size ranges, namely 63.0–53.0 mm, 47.5–37.5 mm, 31.5–26.5 mm,22.4–19.0 mm and 16.0–13.2 mm. Following preparation by sieving oflots contained in these five classes, the mean weight of each set of par-ticles to be broken is calculated. Based on the required specific input en-ergy for each test, which is variable from 0.1 to 2.5 kWh/t, the heightfrom which the drop weight is to be released is determined assumingfree-fall conditions. The test results in a breakage index, t10, which is re-lated to the specific input energy as (Napier-Munn et al., 1996)

t10 ¼ A 1− exp −bEcsð Þ½ � ð7Þ

where t10 is the percentage of breakage product that passes 1/10th ofthe initial particle size; Ecs is the specific input energy (kWh/t) as calcu-lated from the input energy of the fallingweight and the averageweightof the impacted particles.

Parameter A in Eq. (7) is the maximum value of t10, i.e., the highestlevel of size reduction from a single impact event, typically varyingfrom 35 to 70%. The product A ∗ b, that is, the derivative of (7) atEcs = 0, can be used to compare the amenability of the ore to fragmen-tation by impact. A high value of A ∗ bmeans that the rock has a low re-sistance to impact breakage and vice versa. Materials used in thepresent study presented values of A ∗ b ranging from 21 (copper ore#1) to 633 (limestone #3).

3.3. Laboratory crusher

Crushing experimentswere conducted in a laboratory-scale shortheadcone crusher (Denver No. 12) equipped with a power meter (Fig. 1). Thecrusher has a stroke of 5 mm, a cone angle of 44° (in relation to horizon-tal) and a mantle length of 150 mm. The feed opening gap is 25 mm,which is particularly convenient, since it allows feeding 50 mmdiameter(2¨) split cores. A constant closed-side setting (CSS) of 7.6 mmwas usedin all experiments, whichwasmeasuredwith the aid of a lead spacer. Themotor runs at about 1725 rpm, transferring torque to the crusher bowlthat turns at a frequency of 616 ± 5 rpm, via a rubber belt.

Crushing experiments consisted of continuous feeding samples con-taining from 7 to 15 kg of material, depending on specific gravity, withsize in the range of 22.4–16.0 mm to the crusher and collecting timedsamples of the product. Choke-fed conditions were maintained duringthe tests, but reasonably limited particle interaction existed, given therelatively coarse feed size distribution. In order to ensure that, carewas taken to reject the material generated in the beginning and finalportions of the test, when non-choke conditions prevailed. Size analyzesof the crusher discharge were measured by wet-dry sieving. From the

Fig. 1. Laboratory cone crusher (Denver®No. 12), equippedwith a powermeter (top) andconnected to a laptop computer (left).

51F.N. de Magalhães, L.M. Tavares / International Journal of Mineral Processing 126 (2014) 49–54

sampling time and the sample weights, the throughput was estimatedfor each test. Timed samples for size analyses and for estimating capac-ity were taken in duplicate and average values were used in the compu-tations. Reduction ratios achieved in the tests were in the range of 3.5–4.5, which were well within those obtained in industrial cone crushers.

4. Results and discussion

Fig. 2 shows size distributions of the crusher product for selectedmaterials, demonstrating the variety of material-dependent results.Crusher capacities for the different materials were found to vary from0.5 to 1.9 t/h, whereas the average power of the crusher in operationvaried from 1.2 to 4.1 kW.

1

10

100

0.1 1

Pas

sin

g (

%)

Particle

Fig. 2. Size distributions of the laboratory crush

Data from three laboratory crushing tests were, initially, used to fit aconsistent set of parameters of the classification function (Eq. (3)), aswell as of the breakage function for each material, by comparing mea-sured product size distributions to those estimated using Eqs. (1)–(2).It is customary to describe the breakage or appearance function usingthe so-called t10 procedure and the different tmarkers, originally devel-oped by Narayanan andWhiten (1988). Alternatively, Whiten's crushermodel has also been used in conjunction to the expression (King, 1990)

B Di; dpj� �

¼ ϕDi

dpj

!γ

þ 1−ϕð Þ Di

dpj

!β

ð8Þ

where ϕ, γ and β are the model parameters to be fitted from data.Fig. 3 shows the classification function fitted to the data for the lab-

oratory crusher used in the present study. This has been assumed to beconstant for all materials tested under the standard conditions. Indeed,for a given crusher operating under a constant gap and choke-fed condi-tions and constant feed size, the parameters that describe the classifica-tion function are not expected to vary. Nevertheless, parameters d1 andd2 are likely to increase with the lamelarity of the particles, so thatchanges of these parameters with material might have allowed im-proved fitting of the data. However, this was not carried out, in orderto prevent masking the results, thus guaranteeing that the breakagefunction carries all material-specific responses from the test. The small-er value of d1 in comparison to the gap (Fig. 3) may be explained by thechoke-fed conditions, which result in the crusher selecting for breakagea fraction of the particles that are smaller than the closed side setting.

Fig. 4 presents the breakage functions for selected materials, whichshows significant material-dependent variations. Although the break-age function was not described using Narayanan and Whiten's proce-dure, the t10 value may be estimated from Eq. (8) with the expression

t10 ¼ φ10γ þ 1−φ

10βð9Þ

Napier-Munnet al. (1996) observed that t10 values for crushers typicallyvary from 5 to 25%. In the tests, these values ranged from 4.5% (copperore #4) to 25.0% (limestone #3).

Fig. 5 compares measured to fitted product size distributions, show-ing that Whiten's model in conjunction to the functional forms used todescribe the classification and breakage functions were capable of de-scribing the observed product size distributions with very good fidelity.

The following step consisted of fitting a unique value of the parame-ter S from Eq. (5). This was accomplished by minimizing the sum ofsquares of the differences between the net power measured in eachtest (Pc − Pn) and the product of different values of S and Pd. The latter

10 100

size (mm)

Feed

Limestone 1

Limestone 2

Limestone 3

Limestone 4

Limestone 5

Limestone 6

Limestone 7

Acid rock 1

Acid rock 2

Acid rock 3

Acid rock 4

er feed and product for selected materials.

Fig. 3. Fitted classification function (Eq. (3)), showing the fixed CSS used in the tests(d1 = 6.1 mm, d2 = 10.6 mm, n = 2.3).

1

10

100

0.1 1 10

Pas

sin

g (

%)

Particle size (mm)

Limestone 3

Acid rock 2

Acid rock 4

Fig. 5.Measured (symbols) andfitted (lines) size distributions of the crusher product fromselected tests.

52 F.N. de Magalhães, L.M. Tavares / International Journal of Mineral Processing 126 (2014) 49–54

was estimated on the basis of Eq. (6) using the fitted cixi results fromWhiten's model (Eqs. (1) and (2)) and a value of Ecs for each test and,estimated by rearranging Eq. (7), so that Ecs = − ln(1 − t10 / A′) / b′.A′ and b′ are identified as to distinguish from the A and b values fromthe standard dropweight test, since it is recognized that breakage char-acteristics of ores are often found to vary with particle size (Tavares,2007; Shi and Kojovic, 2007). As particles fed to the crusher in questionwere finer than 22.4 mm, the values of the breakage parameters werecalculated on the basis of particle breakage data of 22.4–19.0 mm and16.0–13.2 mm from the drop weight tests. As such, the parameters A′and b′ are used to identify the breakage response of the materials thatare consistent with these finer sizes. The optimal value of S that was ca-pable of fitting the data was 5.1, resulting in an energy efficiency of onlyabout 19.6% (100/S). Such value is consistent with the transmission oftorque from the motor to the cone shaft by the rubber belt, which isused in the laboratory crusher, which has much lower efficiency thanthose used in industrial crushers (Napier-Munn et al., 1996).

From the fitted parameters and the results from the tests it was thenpossible to estimate the breakage parameter b′. It is assumed that theparameter A′ in Eq. (7) could be set to an average value of 50% or 0.5,which is a typical value encountered when testing a variety of materials(Napier-Munn et al., 1996). As such, replacing Eqs. (5) and (6) in (7),after rearranging gives

b′pred ¼ −ln 1−t10=A′ð ÞSW

XNi¼1

ci xi

Pc−Pnð10Þ

where t10 is given by Eq. (9).A comparison of the breakage parameter values calculated on the

basis of data from the crushing tests (Eq. (10)) [(A′ ∗ b′)pred] and those

1

10

100

0.01 0.1 1

B(D

i; d

pj)

Relative size - Di /dpj

Limestone 3

Acid rock 2

Acid rock 4

Fig. 4. Fitted breakage functions (Eq. (8)) for selected materials.

obtained from the drop weight test conducted with particles containedin size classes 22.4–19.0 mmand 16.0–13.2 mm[A′ ∗ b′] is presented inFig. 6, which demonstrates the reasonable agreement between them. Inorder to use the proposed method to estimate the value of A ∗ b that isconsistent with the standard drop weight test, that is, the test that isconducted with particles contained in sizes from 63–53 mm to 16.0–13.2 mm, an additional relationship is required. This is shown in Fig. 7for the samples from the present work. The correlation between thetwo depends on material, in particular the material-specific particle-size effect on the strength of particulate materials (Tavares, 2007; Shiand Kojovic, 2007). If it is assumed that data for all materials analyzedin the present work can be described using the same correction factor,then

A � bð Þpred ¼ 1:38 A′ � b′ð Þpred ð11Þ

which means that values of A ∗ b of the standard drop weight test are38% larger than those found for the test conducted with particlescontained in sizes below 22.4 mm.

Evidently, this relationship is not universal, varying typically withore type (Tavares, 2007). As such, if one wishes to find a more accuraterepresentation for a particular ore deposit, then this relationship shouldbe established on the basis of a few standard drop weight tests, so thatthe relationship between the measured A ∗ b and the A′ ∗ b′ can be de-scribed with greater confidence. However, assuming that Eq. (11) pro-vides an average valid description of the size-scale effect on A ∗ b inthe present work, then a final comparison is possible between themea-sured values of A ∗ b from the standard drop weight test and values

10

100

1000

10 100 1000

(A'*

b' )

pre

d

A' *b'

Acid rocks

Copper ores

Limestones

Bauxites

Coals

Iron ore

Fig. 6. Comparison between measured A′ ∗ b′ values from the drop weight test with par-ticles contained in the ranges 22.4–19 mm and 16.0–13.2 mm, and (A′ ∗ b′)pred valueswhich are predicted using Eq. (10).

10

100

1000

10 100 1000

A*b

Acid rocks

Copper ores

Limestones

Bauxites

Coals

Iron ore

A' *b'

Fig. 7.Comparison betweenmeasured values of A′ ∗ b′ from the dropweight test conduct-ed on the basis of data from 16.0–13.2 mm and 22.4–19.0 mm size ranges and the stan-dard drop weight test (A ∗ b).

Table 1Estimates of breakage parameters for acid rock #2 as a function of crusher gap (CSS) andfeed size distribution (A*b = 36.2).

Feed CSS Capacity Specific net power (A ∗ b)pred Deviation

(mm) (mm) W (t/h) (kWh/t) (%)

26.0–12.5 9.2 1.91 0.23 40.0 1126.0–12.5 7.6 0.98 0.42 34.6 −522.4–16.0⁎ 7.6 0.95 0.26 54.4 5019.0–6.3 5.3 0.93 0.54 25.1 -31

⁎ Base case.

53F.N. de Magalhães, L.M. Tavares / International Journal of Mineral Processing 126 (2014) 49–54

predicted using the present work (A ∗ b)pred. These are presented inFig. 8, which shows the reasonable agreement between parameters es-timated from the standard and the rapid method of estimating A ∗ b.The average absolute error, calculated from the ratio of the differencebetween the measured and predicted values and the measured A ∗ bvalues, was 36%. However, for somematerials, notably those character-ized with low resistance to impact breakage (high A ∗ b values), themaximum absolute error was as high as 120%.

In summary, the application of the method proposed in the presentwork to estimate the ore-dependant parameter A ∗ bwould correspondto applying Eq. (10), followed by Eq. (11) on the basis of a previously cal-ibrated crusher. It is worthwhile noting that the method may be appliedto any laboratory cone or jaw crusher, provided that parameters from theclassification function are established, as well as the energy efficiency ofthe crusher estimated, which requires the calibration using data fromone or more materials with known A ∗ b breakage parameters.

Although in the present work the feed size distribution was main-tained constant for the differentmaterials, themethod allows, in princi-ple, varying the size distribution of the feed, since it explicitly accountsfor it in Eqs. (1) and (2). The onlywarning that exists is associated to theweak dependence of both d1 and d2 on feed coarseness (Napier-Munnet al., 1996). In addition to that, the method may also be used if crushergap is changed, since it can be accounted for by varying the parametersd1 and d2 (Eq. (3)) in proportion to the CSS. In order to illustrate that,

10

100

1000

10 100 1000

(A*b

) pre

d Acid rocks

Copper ores

Limestones

Bauxites

Coals

Iron ore

A*b

Fig. 8. Comparison between measured A ∗ b values from the standard drop weight testand (A ∗ b)pred predicted using the method proposed in the present work and Eq. (11)for correcting the data.

results are shown in Table 1, which compares the predicted to themea-sured value of A ∗ b for one particular sample, showing reasonableagreement.

The deviations found between the rapid method proposed in thepresentwork and the standardmethod of calculating theA ∗ b breakageparameters are such that it is not, by any means, proposed as an all-purpose replacement of the latter by back-calculation from laboratorycrushing data. The magnitude of the deviations found, however, qualifyit as a viable method for rapid estimation of the breakage parameters,when sample availability, cost and even time represent challengesthat could prevent a proper breakage characterization, which couldlead to loss of information that could be used to assess ore response tocomminution in a given portion of the deposit. As such, it is a good ex-ample of adding value to a laboratory operation that is often going tobe carried out when preparing samples for assaying and grindability/concentration tests.

Perhaps the greatest applicability of the method would be as part ofvariability studies when a large number of drill cores are generated, butare not often available for breakage characterization. In those cases, it isthe authors' experience that the ratio between the maximum and theminimum values of A ∗ b for individual mineral deposits are seldomsmaller than 2 and can be as high as 20. Such variability is greaterthan the deviations found in the present study, thus justifying its appli-cation in this context.

5. Summary and conclusions

A rapid and convenient method has been proposed to estimate theA ∗ b breakage parameters. The method is based on Whiten–Awachiecrusher model, applied to a laboratory crusher with properly calibratedvalues of the classification function and energy efficiency of the crusher.

The method has been able to predict the A ∗ b breakage parameterwith good confidence,with anaverage absolute error of 36%. It has dem-onstrated applicability to a wide range of materials, including bauxites,copper ores, limestones, acid rocks, coals and an iron ore, with values ofA ∗ b ranging from 21 to 633.

Although not meant to replace the standard drop weight or the rapidbreakage tests used to calculate the A ∗ b breakage parameter, the meth-od is a good example of adding value to a laboratory operation that isoften going to be carried out when preparing samples for assaying andgrindability tests. When used to process samples from variability tests,themethod then represents a convenient and low-cost option for provid-ing data for geometallurgical mapping of deposits.

Acknowledgments

The authors would like to acknowledge the financial support fromthe Brazilian Research Agency CNPq.

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