Powder Technology 268 (2014) 373–376

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Powder Technology

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Ceramic ball wear prediction in tumbling mills as a grinding mediaselection tool

Ismael E. Rivera Madrid a, Beatriz Álvarez Rodríguez b, Oswaldo Bustamante a,Oscar Jaime Restrepo Baena a, Juan M. Menéndez-Aguado b,⁎a CIMEX, Facultad de Minas, Universidad Nacional de Colombia, Medellín, Spainb Escuela Politécnica de Mieres, Universidad de Oviedo, Spain

⁎ Corresponding author at: Escuela Politécnica deMieres/n 33600 - Mieres - Spain. Tel.: +34 985458033; fax:

E-mail address: maguado@uniovi.es (J.M. Menéndez-A

http://dx.doi.org/10.1016/j.powtec.2014.08.0560032-5910/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 11 September 2013Received in revised form 19 July 2014Accepted 21 August 2014Available online 28 August 2014

Keywords:Ceramic grindingPopulation balance modelBall mill wear

This article presents the results of applying amedia selectionmethodology based on a population balance modelto a ceramic ball mill; the experimental data were obtained through the marked ball test in a white-cementgrinding pilot plant. Three types of balls from different suppliers (I, II and III) were used in order to study thewear occurring in each type of ball. Tests were carried out for 576 h and the law of wear for all three types ofballs was found to be zero order. These results were introduced into the model to obtain the ball charge of themill at steady state and the alumina consumption by wear.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The kinetics of grinding media wear is estimated based on theoriesput forward shortly before the second half of the Twentieth Century,as is evidenced in the work of Sepúlveda [1], which uses the theory oflinear wear to calculate specific rate constant wear. Other researchapproaches were found such as that of Albertin and Sinatora [2], inwhich ball wear showing linear behaviour in a laboratorymill was com-paredwith an industrial mill to calculate the rate of ball wear caused byvarious minerals. Albertin and Lucia de Moraes [3] also conducted fur-ther research on coal grinding for four different ball types and all fourpresented a linear behaviour in their wear. However, all tests werelimited to steel balls.

Plant results have shown that ball diameter is an operating variablein the grinding process that directly affects mill efficiency. In fact,according to Austin et al. [4] and Gupta et al. [5], it is known that thereis a ball size that maximizes the breaking rate of a given feed particlesize; in this respect, the ball size ratio and particle size were initiallymodelled by Bond [6], who, using a criterion based on the characteriza-tion of the mill feed size distribution, developed equations to select theball sizes at the start-up operation, but provide uswith little informationabout ball recharge into the mill.

From this perspective it is important to study ball recharge andgrinding media wear. From the point of view of kinetic equations

s, c/Gonzalo Gutierrez Quiros,+34 985458182.guado).

governing steel ball wear and their applications in population balancemodels, the contributions of Menacho and Concha [7,8] constitute agood advance; this is proven by the goodness of fit between theoreticaland experimental results. An introduction to population balancemodelsand their applications, with a wide variety of references can be found inJakobsen [9]; in this paper some of the equations proposed byMenachoand Concha are applied to ceramic ball wear behaviour prediction.

2. Experimental equipment and materials

Experiments were conducted using a laboratory mill (0.19 m indiameter) filled with alumina balls with a filling factor J = 45%, andoperated at 75% critical speed. Single-sized feed of quartz was used with100% fractional filling (U = 1), with a bulk density of 2.601 g/cm3 and asample net weight of 2.2 kg.

The diameter, weight and density of the three ball types, as well asthe volume calculated from these variables, can be seen in Table 1.

3. Mathematical model of ball wear in grinding mills

3.1. Kinetic ball wear

Themathematical model corresponds to a population balance equa-tion. Particularly if g (d) = α, we get

∂N d; tð Þ∂t þ α

∂N d; tð Þ∂d ¼ φI

XK

R¼1

mI0 dð Þδ d−dRð Þ: ð1Þ

Table 1Ceramic ball diameter calculation.

Ball type Weight (g) Density (g/cm3) Volume (cm3) Diameter (cm)

I 130.70 3.73 35.04 4.06II 116.40 3.72 31.30 3.91III 113.70 3.54 32.02 3.94

Table 2Kinetic equations.

Equation R2

Type I y = −0.0064x + 41.027 0.84867Type II y = −0.0051x + 39.428 0.61691Type III y = −0.0079x + 39.118 0.89461

374 I.E. Rivera Madrid et al. / Powder Technology 268 (2014) 373–376

Whose solution by the method of characteristics (Salsa [10]) is

N d; tð Þ ¼ N0 d−αtð Þ þ φI

α

XK

R¼1

mI0 dRð Þ U d−dRð Þ−U d−αt−dRð Þf g: ð2Þ

At steady state

NSS dð Þ ¼ φI

α

XK1

R¼1

d−βmI0 dRð Þ U d−dRð Þ−1½ � ð3Þ

where

N (d, t) is the number of balls with diameter d and time t in the ballcharge

ϕ1 is the total number of balls at the mill inletm0

I (dR) is the relative number frequency for the balls of size d at theinlet flow

δ is the Dirac delta functiond1 is the size of the largest ball entering the milld0 is the size of the balls leaving the millK1 is the number of ball size classes in the charge.

In this work, we used a ‘zero-order’wear rate as the kinetic equationfor g (d) [g (d) = constant], as will be shown in Fig. 1.

3.2. Ball addition

In industrial practice, ball-filling level is set at steady state. Then,considering the law of wear, input and output ball size, daily ball addi-tion can be calculated using the following formula:

φ ¼ 6α 4−βð ÞwB

πρXk

R¼1

mI0 dRð Þ d4−β

0 −d4−βR

n o

1CCCCA

ð4Þ

38,00

38,50

39,00

39,50

40,00

40,50

41,00

41,50

42,00

0 1 2 3 4 5 6 7 8 9 10 1

Bal

l dia

met

er (

mm

)

Fig. 1. Kinetic

where:

φ daily ball addition (number of balls/day)α specific wear rate (cm/day)β wear rate orderwB mass hold-up of balls in mill (kg)ρ ball density (kg/cm3)dR mill input ball diameter (cm)d0 mill output ball diameter (cm).

In this case, the following considerations are assumed:

β ¼ 0 wear kinetics is zero orderð Þ

Xk

R¼1

mI0 dRð Þ ¼ 1 recharge is made with only one ball sizeð Þ:

Then, Eq. (4) can be expressed as follows:

φ ¼ 24αwB

πρ d40−d4R� � : ð5Þ

3.3. Consumption by wear

This is defined by:

CD ¼ 4wBα

Xk

R¼1

mI0 dRð Þ d3R−d30

n o

Xk

R¼1

mI0 dRð Þ d4o−d4R

n o ð6Þ

1 12 13 14 15 16 17 18 19 20 21 22 23 24

Cycle

Type I

Type II

Type III

ball wear.

4,0

5,0

6,0

7,0

8,0

2,001,751,501,251,000,750,500,200,150,10

Discharge Grill - do(cm)

Flo

w(b

all/h

) Type I

Type II

Type III

Fig. 2. Flow vs discharge grill.

375I.E. Rivera Madrid et al. / Powder Technology 268 (2014) 373–376

in our case ∑k

R¼1mI

0 dRð Þ ¼ 1, so we have,

CD ¼ 4wBαd3R−d30

n o

d40−d4R� � : ð7Þ

3.4. Consumption by purge

This is defined by:

CP ¼ 4wBα

Xk

R¼1

mI0 dRð Þd30

Xk

R¼1

mI0 dRð Þ d40−d4R

n o : ð8Þ

As in the previous case, ∑k

R¼1mI

0 dRð Þ ¼ 1, then Eq. (8) becomes:

CP ¼ 4wBαd30

d40−d4R� � : ð9Þ

Table 3Consumption of ceramic balls by size and type.

d0 = 2.00 cm d0 = 1.75 cm

C (kg/h) CD CP CT CD CP CT

Type I 0.601 0.079 0.679 0.611 0.051 0.66Type II 0.515 0.077 0.592 0.525 0.050 0.57Type III 0.672 0.104 0.776 0.686 0.067 0.75

d0 = 1.00 cm d0 = 0.75 cm

C (kg/h) CD CP CT CD CP CT

Type I 0.634 0.009 0.643 0.638 0.004 0.64Type II 0.547 0.009 0.556 0.550 0.004 0.55Type III 0.714 0.012 0.726 0.719 0.005 0.72

d0 = 0.15 cm

C (kg/h) CD CP CT

Type I 0.641 0.000 0.64Type II 0.553 0.000 0.55Type III 0.723 0.000 0.72

3.5. Total consumption

The alumina consumption from the ball charge in an industrial mill,defined as the total mass CT of alumina lost, comes from two contribu-tions, the wear consumption (CD) and the purge consumption (CP).Therefore, the total alumina consumption (CT) must be:

CT ¼ CD þ CP : ð10Þ

4. Characterization procedure

The charge consisting of ceramic balls and mineral was grinded for24 cycles, each cycle lasting 24 h. After each cycle, themill was stopped,a sample of twelve balls was taken andwearwasmeasured for each ballto later calculate the arithmetic mean. The aim was to study ball sizevariations throughout the cycles. Traditionally, mill media wear rate isevaluated by measuring the media level in the mill, or by removingmedia charge and weighing it after a certain number of hours. As men-tioned above, the mill was stopped after 24 h (one cycle) with a total of24 cycles for each ceramic ball type.

Grinding tests were carried out for 24 days (576 h) and small anddamaged media were taken from the grinding chamber and analysedafter each mill operation cycle.

d0 = 1.50 cm d0 = 1.25 cm

CD CP CT CD CP CT

3 0.621 0.032 0.653 0.628 0.018 0.6466 0.534 0.031 0.565 0.541 0.018 0.5593 0.697 0.042 0.739 0.707 0.024 0.731

d0 = 0.50 cm d0 = 0.20 cm

CD CP CT CD CP CT

2 0.640 0.001 0.641 0.641 0.000 0.6414 0.552 0.001 0.554 0.553 0.000 0.5534 0.722 0.002 0.723 0.723 0.000 0.723

d0 = 0.10 cm

CD CP CT

1 0.641 0.000 0.6413 0.553 0.000 0.5533 0.723 0.000 0.723

0,50

0,55

0,60

0,65

0,70

0,75

0,80

2,001,751,501,251,000,750,500,200,150,10

Discharge Grill - do(cm)

CT

(kg

/h)

Type I

Type II

Type III

Fig. 3. Total consumption vs discharge grill.

376 I.E. Rivera Madrid et al. / Powder Technology 268 (2014) 373–376

5. Results and discussion

Ball wear was minimal and very even. No broken or fractured ballswere detected in operation or in the test.

Fig. 1 shows the operational grinding time versus ceramic ball sizeand the results of uniform ball wear with slight variations in size canbe observed. The kinetic equations are shown in Table 2.

Results suggest that, for the three media shapes studied, kinetics fitzero order. This figure also illustrates the ball wear described by lineartime dependence. There are no signs of accelerated ball wear in thesecycles. The average wear between the first and last cycle is approxi-mately 1% in 576 operating hours (i.e. 0.41 mm diameter reduction).

In the case of Fig. 2, the necessary ball recharge flow for each balltype is plotted. It can be observed that in the case of ball types I and II,ball recharge flow estimated with this methodology is around 4–5balls per hour, while in the case of ball type III the addition must begreater, around 6–7 balls per hour. This figure also constitutes a toolfor evaluating different wear behaviour and can prove very useful inball type selection.

The results of ceramic ball consumption are shown in Table 3; inFig. 3 the results of total consumption with ball size for each ball typeare represented.

6. Conclusions

Considering that the worn balls in industrial mill charge constituteabout 15–40% [11] and that only marginal differences in ball size wereobserved, the effect of the presence of worn balls within themill chargeis negligibly small. Thus, the benefit that can be obtained does not justifythe cost of the removal operation from mills.

The most important conclusions to be drawn from this work are:

• The kinetic order is zero so consumption bywear is constant. This alsooccurs in the case of steel balls.

• The influence of wear consumption CD on total consumption CT is pre-dominant. No significant variation is observed in the purge wear, dueto the mechanism of discharge.

• From Fig. 2 it can be concluded that themill must be rechargedwith 4balls per hour in the case of ball types I and II, and 6 balls per hour forball type III.

• The methodology carried out in this study can be used to assist inceramic ball selection

NomenclatureN (d, t) number of balls with diameter d and time t in the ball chargeϕ1 total number of balls at the mill inletm0

I (dR) relative number frequency for the balls of size d at the inletflow

δ Dirac delta functiond1 size of the largest ball entering the milld0 size of the balls leaving the millK1 number of ball size classes in the chargeNSS (d) number of balls of size d in the steady stateg (d) wear kinetic ballsφ daily ball addition (number of balls/day)α Specific wear rate (cm/day)β Wear rate orderwB Mass hold-up of balls in mill (kg)ρ ball density (kg/cm3)dR mill input ball diameter (cm)d0 mill output ball diameter (cm).

References

[1] J. Sepúlveda, Methodologies for the evaluation of grindingmedia consumption ratesat full plant scale, Miner. Eng. 17 (2004) p1269.

[2] E. Albertin, A. Sinatora, Effect of carbide fraction and matrix microstructure on thewear of cast iron balls tested in a laboratory ball mill, Wear 250 (2001) 492–501.

[3] Eduardo Albertin, Sandra Lucia de Moraes, Maximizing wear resistance of balls forgrinding of coal, Wear 263 (2007) 43–47.

[4] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process engineering of size reduction: ball mill-ing, Ed: Society of Mining Engineers, New York, 1984.

[5] V.K. Gupta, H. Zouit, D. Hodouin, The effect of ball and mill diameters on grindingrate parameters in dry grinding operation, Powder Technol. 42 (1985) 199–208.

[6] F.C. Bond, New equation for calculating the work index from A–C closed circuit ballmill grindability test, Allis Chalmer Publication, 1960.

[7] J.M. Menacho, F. Concha, Mathematical model of ball wear in grindingmills: II. Gen-eral solution, Powder Technol. 52 (1987) 267–277.

[8] J.M. Menacho, F. Concha, Mathematical model of ball wear in grinding mills: I. Zero-order wear rate, Powder Technol. 47 (1986) 87–96.

[9] H.A. Jakobsen, The population balance equation, Chemical Reactor Modeling,Springer-Verlag, Berlin Heidelberg, 2008.

[10] S. Salsa, Partial differential equations in action, From Modelling to Theory,Universitext, Springer-Verlag, Italia, Milan, 2008.

[11] N.S. Lameck, M.H. Moys, Effects of media shape on milling kinetics, Miner. Eng. 19(2006) 1377–1379.