Advanced Powder Technology 21 (2010) 623–629

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Advanced Powder Technology

journal homepage: www.elsevier .com/locate /apt

Original Research Paper

Scale-up methodology for tumbling ball mill based on impact energyof grinding balls using discrete element analysis

Tomohiro Iwasaki *, Tomoya Yabuuchi, Haruki Nakagawa, Satoru WatanoDepartment of Chemical Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 November 2009Received in revised form 21 April 2010Accepted 24 April 2010

Keywords:Discrete element modelBall-millingImpact energyScale-upDry grinding

0921-8831/$ - see front matter � 2010 The Society ofdoi:10.1016/j.apt.2010.04.008

* Corresponding author. Tel.: +81 72 254 9307; faxE-mail address: iwasaki@chemeng.osakafu-u.ac.jp

This paper provides a method to scale-up horizontal tumbling ball mills, i.e. to determine the dimensionsof the rotating drum and the drum rotational speed. In order to develop the scale-up methodology, themotion of grinding balls in tumbling ball mills with different drum diameters was calculated using thediscrete element method (DEM). The impact energy of grinding balls was numerically analyzed, andthe influence of drum dimensions and drum rotational speed on the impact energy was investigated. Itwas found that scale-up of the rotating drum should be carried out based on the mechanical energyinstantaneously applied to the powder and its cumulative amount. The former was evaluated in termsof the frequency distribution of the impact energy and the latter its cumulative amount over the elapsedmilling time, which could be controlled by the drum rotational speed and the milling time, respectively.Validity of the proposed scale-up methodology was evaluated through dry grinding experiments ofaluminum hydroxide powder, and the experimental results supported its usefulness in practicalapplications.� 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder

Technology Japan. All rights reserved.

1. Introduction

Tumbling ball mills have been widely used in many manufac-turing industries not only for size reduction (grinding) of particu-late materials but also for synthesis of functional compositeparticles by mechanical alloying [1] and mechanochemical treat-ment [2–5] because of its simple structure and easy operation.The properties of milled products such as size distribution andcrystallinity depend strongly on both amount and history of themechanical energy applied to the particles during the milling treat-ment. The mechanical energy generated in the ball mills is derivedfrom the impact energy of grinding balls [6].

As a tool for analyzing the impact energy, the discrete elementmodel (DEM) simulation of grinding balls in tumbling ball mills isvery useful [7,8]. The impact energy estimated by the DEM simula-tion could be related to the product characteristics. For example,Kano et al. [9] reported the correlation between the impact energyand the grinding rate constant in the grinding treatment of inor-ganic particulate material. In ball-milling treatments of powders,most of the impact energy may contribute to the powder grindingalthough a part of the impact energy dissipates due to ball–powderbed interactions such as compression and friction except when theamount of powder placed in the mill is too much. Thus, there is no

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: +81 72 254 9911.(T. Iwasaki).

doubt that the progress of grinding depends strongly on the impactenergy. This suggests that the impact energy is a controlling factorin the ball-milling treatments. Therefore, in order to prepare theproducts with controlled properties, it is very important to controlprecisely the impact energy. Furthermore, Herbst [10] suggestedthat the scale-up information of tumbling ball mills could be ob-tained from the power draw predicted from the impact energy.Accordingly, in case of scale-up of the rotating drum, the dimen-sions of drum and the operating conditions in larger-sized millsshould be determined based on the impact energy required for pre-paring the milled product with the same properties as in labora-tory-sized mills. Several researches for studying the effect of thedrum dimensions on the impact energy have been conducted[11,12]. However, the methodology providing guidelines of processdesign and optimization for the scale-up has not been establishedyet. Unfortunately, at present not only the scale-up but also thecontrol of characteristics of milled products has been carried outbased on knowledge from trial-and-error and experience.

In this study, for developing a scale-up methodology based onthe impact energy, the motion of grinding balls in the rotatingdrum was simulated using the three-dimensional DEM. Both theimpact energy instantaneously generated and its cumulativeamount over the milling time were numerically analyzed. Validityof the proposed method is confirmed experimentally through thegrinding treatment of aluminum hydroxide powder in ball millsof different dimensions.

ed by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

Nomenclature

C1 coefficient in Eq. (18) (J m�3.5 s�1)C2 coefficient in Eq. (19) (J m�3.5 s�1)D drum diameter (m)d ball diameter (m)E cumulative specific energy during milling treatment

(J kg�1)Ei impact energy of contacting ball (J)Ei50 geometric mean of energy distribution (J)F contact force acting on ball (N)f ball filling ratio (�)g gravity acceleration (m s�2)I inertia moment of ball (kg m2)k average number of contact points per unit time (s�1)L drum depth (m)m mass of ball (kg)N drum rotational speed ratio to critical speed (�)n unit vector in normal direction at contacting point

(m)n0 ideal critical rotational speed of drum (s�1)P power (J s�1)r radius of ball (m)S ratio of true volume of powder to drum capacity (�)T torque caused by tangential contact force (N m)t milling time (s)v relative velocity of contacting balls (m s�1)

W loading mass of powder (kg)X position of ball (m)x scale-up index (�)Y Young’s modulus (Pa)y position in drum (m)

Greek lettersa constant depending on restitution coefficient (�)b ratio of drum depth to drum diameter (�)d overlap displacement between contacting balls (m)g damping coefficient (kg s�1)jn stiffness in normal direction (N m�3/2)jt stiffness in tangential direction (N m�1)l sliding friction coefficient (�)q true density of powder (kg m�3)r Poisson’s ratio (�)re geometric standard deviation of energy distribution (�)x angular velocity of ball (s�1)

Subscripts0 smaller-sized millb balln normal componentt tangential componentw wall

624 T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629

2. Model description

In order to analyze the impact energy of grinding balls, theirbehavior in the rotating drum under dry condition was simulatedusing the 3D DEM. This model describes the motion of each ballbased on Newton’s second law for individual ball, allowing forthe external forces acting on the ball [13]. The interaction betweenball and atmosphere gas was neglected because the contact forceacting on the balls at the collision is much stronger than the dragforce acting on the balls in the translational motion. The funda-mental equations of translational and rotational motions of a ballare expressed as follows:

md2Xdt2 ¼ F þmg ð1Þ

Idxdt¼ T ð2Þ

where m, X, I and x are mass, position, inertia moment and angularvelocity of a ball, respectively. t, g, F and T are time, gravity acceler-ation, contact force, and torque caused by the tangential contactforce, respectively. X and x are calculated by integrating Eqs. (1)and (2) with respect to time between t and t + Dt.

For estimating the contact force, the Hertz-Mindlin contactmodel was employed, which is generally used as a model whichcan predict the impact energy relatively accurately [14]. The con-tact forces of normal and tangential directions, Fn and Ft, were esti-mated using the following formulas:

Fn ¼ �jnd3=2n � gnvn � n

� �n ð3Þ

F t ¼ �jtdt � gtv t if jF tj 6 ljFnjð Þ ð4Þ

F t ¼ �ljFnjv t

jv tjif jF tj > ljFnjð Þ ð5Þ

v t ¼ v � ðv � nÞnþ rðxi þxjÞ � n ð6Þ

where d, v, j, g, l, r and n are overlap displacement between con-tacting balls or between ball and drum wall, relative velocity of con-

tacting ball, stiffness, damping coefficient, sliding frictioncoefficient, ball radius and unit vector of normal direction at a con-tact point, respectively. The subscripts n and t mean the compo-nents of normal and tangential directions at a contact point,respectively. The subscripts i and j indicate a number of contactingballs. j and g are determined from the following formulas [15].

Stiffness for ball-to-ball collision:

jn ¼ffiffiffiffiffi2rp

Yb

3 1� r2b

� � ð7Þ

jt ¼2ffiffiffiffiffi2rp

Yb

2ð1þ rbÞð2� rbÞd1=2

n ð8Þ

Stiffness for ball-to-drum wall collision:

jn ¼ð4=3Þ

ffiffiffirp

ð1� r2bÞ=Yb þ ð1� r2

wÞ=Ywð9Þ

jt ¼8ffiffiffirp

Yb

2ð1þ rbÞð2� rbÞd1=2

n ð10Þ

Damping coefficient:

gn ¼ gt ¼ aðmjnÞ1=2d1=4

n ð11Þ

where Y and r are Young’s modulus and Poisson’s ratio, respec-tively. The subscripts b and w indicate ball and drum wall, respec-tively. a is the constant depending on the restitution coefficient andwas determined to be 0.20 based on the experimental value of res-titution coefficient (=0.75) according to the method proposed byTsuji et al. [15].

The impact energy of a contacting ball, Ei, was defined as:

Ei ¼12

mjv j2 ð12Þ

v represents the relative velocity of the ball at the moment whenthe ball comes into contact with a ball or the drum wall, as illus-trated in Fig. 1. The impact energy generated within a certain timehas a distribution, depending on the size and operating conditions

Fig. 1. Illustration of collision of balls.

T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629 625

of mills. The impact energy per unit time, i.e. the power P, was de-fined as the total kinetic energy of contacting balls per unit time andcalculated for the balls determined by the average number k of con-tact points per unit time [9].

P ¼Xk

Ei ¼Xk 1

2mjvj2 ð13Þ

In other words, Eq. (13) means that the kinetic energy of balls justbefore contacting (i.e., at |d| = 0) is summed up for all the contactpoints within 1 s but the kinetic energy during contacting (i.e.,within |d| > 0) is not calculated. Accordingly, the impact energy thusobtained corresponds to the maximum kinetic energy of balls at thecollision, i.e. the maximum mechanical energy which the balls cangive to the powder per unit time.

In case of scale-up of the drum, it is required that the productsobtained in larger-sized mills have the same properties (e.g., thesize distribution) as in a smaller-sized (laboratory) mill. In orderto achieve that, the amount of the impact energy applied to thepowder per unit mass, i.e. the specific energy, should be the samein both larger-sized and smaller-sized mills. It is obvious that thespecific energy depends on both the power and the milling time.Therefore, for calculating the power the impact energy distributionshould be analyzed, leading to determination of the operating con-ditions in larger-sized mills.

3. Experimental

In the experiments for evaluating validity of the proposedmethod, three commercially available drums with different sizes

Fig. 2. Milling drums used in experiments.

(TM-1, TM-2 and TM-3) made of stainless steel were used, asshown in Fig. 2. The dimensions of drums are summarized inTable 1. The calculation of the impact energy was performed forthe drums with the same sizes as the experimental ones. TM-4 inTable 1 is a virtual drum used only in the calculation. As grindingballs, zirconia balls with a diameter of 3 mm were used regardlessof the drum size, and the filling ratio of balls to the drum capacitywas 40% in all the drums. The powder sample was an aluminumhydroxide powder (C-31, Sumitomo Chemical Co., Ltd.) with amedian size of 51.0 lm. Proper amounts of aluminum hydroxidepowder and grinding balls were placed in the drum, and the grind-ing treatment was performed at predetermined rotational speeds.The loading amount of powder corresponded to 50% of the totalvoid volume of which the void was formed among the grindingballs under static condition. After a proper grinding time elapsed,all the powder was removed from the drum and mixed well in aplastic bag. A part of the sampled powder was dispersed in anaqueous solution containing a small amount of anti-aggregationagent (sodium dodecyl sulfate) under ultrasonic irradiation, andthen the size distribution was measured with a laser scatteringparticle size analyzer (Microtrac FRA, Nikkiso Co., Ltd.).

4. Results and discussion

4.1. Effects of drum diameter and drum rotational speed on impactenergy

The parameters used in the calculation are given in Table 2.Physical properties of the materials of drum and ball correspondto those of stainless steel and zirconia, respectively. The drum rota-tional speed was expressed as the ratio N to the ideal critical rota-tional speed n0 defined as:

n0 ¼1

2p

ffiffiffiffiffiffiffiffiffi2jgj

D

rð14Þ

where D is the drum diameter. The sliding friction coefficient usedin the calculation was determined based on the critical rotationalspeed measured experimentally under dry condition [16]. Fig. 3shows the calculation results of frequency distribution of the im-pact energy Ei in the mills of TM-1 to TM-4 under constant rota-tional speed ratios N. The impact energy distribution wasexpressed similar to a representation of particle size distributionwith a logarithmic abscissa [17] in terms of a frequency distributionfunction F(ln Ei) defined by:Z 1

0Fðln EiÞdðln EiÞ ¼ 1 ð15Þ

Fig. 4 indicates change in the median value of the energy distribu-tion with N. The energy distribution was found to shift to larger en-ergy range as N increased in all the mills since the motion of ballsbecame more vigorous [18]. The energy distributions in TM-2,TM-3 and TM-4 at the same N almost agreed with each other. Thisresult implies that the motion of balls was similar at the same Nregardless of the drum diameter because the Froude number Fr ofassembly of balls, defined by Eq. (16), was the same, resulting in asimilar flow condition of balls in the drums.

Table 1Drum dimensions.

TM-1 TM-2 TM-3 TM-4

Volume (ml) 115 510 2650 4710Internal diameter (mm) 54 90 150 200Internal depth (mm) 50 80 150 150Ratio of depth to diameter (�) 0.93 0.89 1 0.75Critical rotational speed (ideal) (s�1) 3.03 2.35 1.82 1.58

Table 2Simulation parameters.

Density of ball 6000 kg/m3

Young’s modulus of ball 210 GPaYoung’s modulus of drum wall 193 GPaPoisson’s ratio of ball 0.32Poisson’s ratio of drum wall 0.3Coefficient of ball-to-ball restitution 0.95Coefficient of ball-to-drum wall restitution 0.89Sliding friction coefficient between balls 0.46Sliding friction coefficient between ball and drum wall 0.46Ball diameter 3–6 mmBall filling ratio to drum capacity (by volume) 0.2–0.6Rotational speed ratio to critical 0.3–0.9Time step 1.0 ls

Fig. 3. Impact energy distribution at various rotational speed ratios of N = (a) 0.3,(b) 0.5, (c) 0.7 and (d) 0.9 in different-sized drums under constant ball filling ratio off = 0.4 and ball diameter of d = 3 mm.

Fig. 4. Change in median value of energy distribution with rotational speed ratio N.

626 T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629

Fr ¼ Dn0NffiffiffiffiffiffiffiffiDjg

pj¼ Nffiffiffi

2p

pð16Þ

The calculated motion of balls in TM-1, TM-2 and TM-3 areillustrated in Fig. 5a–c, respectively, as an example. In addition,the experimental observation of the ball motion in TM-2 operatedunder the same conditions as those in the calculation (correspond-ing to Fig. 5b) is shown in Fig. 6 as a representative example. Thecalculated flow conditions of balls qualitatively matched with theexperiments under the operating conditions; this supports thevalidity of the assumptions in the model. As can be seen in Fig. 5,in larger-sized drums, balls were lifted up higher and surely hadhigher potential energy, leading to higher kinetic energy, eventhough at the same N. However, the higher potential energy didnot directly contribute to the increase in the impact energy. In or-der to investigate the cause, the average impact energy and num-ber of contact points of balls were analyzed. Fig. 7 shows theaverage impact energy per a single contact point and the number

of contact points per unit time within a region of0.5D � 0.25D � depth (=80, 150 and 150 mm for TM-2, TM-3 andTM-4, respectively), which was grayed in the cross section viewof a drum illustrated in Fig. 7a, as a function of the dimensionlessparameter y/D representing a position in the ball bed. The averageimpact energy on the surface of the ball bed where the lifted ballsfell down (y/D � 0.2) was higher than that in the inside and bottomof the ball bed. However, the number of contact points on the sur-face was remarkably small in comparison with that in the insideand bottom. Thus, the impact energy distribution was found tobe dominated by the impact energy generated in the inside andbottom of the ball bed, i.e. the shear flow field of balls, rather thanon the surface.

On the other hand, in TM-1 having the smallest diameter, theballs had relatively low impact energy as shown in Fig. 3 becausethe motion of balls was remarkably limited due to higher diameterratio of ball to drum. This resulted in the formation of insufficientcascade flow of balls, leading to notable difference of the energydistribution. Accordingly, it is needed to adjust the rotational speedof TM-1 so that the motion of balls in TM-1 is similar to those inthe larger mills. As an example, the energy distribution in TM-1was approximated to those at N = 0.7 in TM-2, TM-3 and TM-4 asfollows. The energy distributions were expressed approximatelyby a log-normal frequency distribution function defined as:

Fðln EiÞ ¼1

ln re �ffiffiffiffiffiffiffi2pp exp �ðln Ei � ln Ei50Þ2

2ðlnreÞ2

" #ð17Þ

where Ei50 and re are the geometric mean and the geometric stan-dard deviation of the energy distribution, respectively. Ei50 and re

calculated for the larger mills had similar values in the ranges ofEi50 = 2.13 ± 0.16 lJ and re = 4.94 ± 0.79, respectively. For TM-1,however, Ei50 was 1.32 lJ differing remarkably from those for thelarger mills, whereas re = 4.95 in TM-1 was almost the same. There-fore, N in TM-1 was increased so that Ei50 in TM-1 was almost thesame as those in the larger mills. By increasing N from 0.7 to 0.9in TM-1, the cascade flow developed and the shear flow was pro-moted as shown in Fig. 5d. Fig. 8 shows the energy distribution afteradjustment of N. As a result, the energy distribution in TM-1 couldsuccessfully approach those in the larger mills at N = 0.7. Conse-quently, in case of scale-up of the drum, it is important to expandthe shear flow field while maintaining the similarity of the regionof the shear flow field by using the rotational speed ratio as the con-trolling factor. It is well known that increase of the drum speed re-sults in development of the cascade flow of balls but the analysis of

Fig. 5. Simulated motion of grinding balls in different-sized drums: (a) TM-1 at N = 0.7, (b) TM-2 at N = 0.7, (c) TM-3 at N = 0.7 and (d) TM-1 at N = 0.9.

Fig. 6. Experimental observation of flow conditions of balls in TM-2 at N = 0.7.

Fig. 7. Changes of (a) average impact energy per single contact point and (b)number of contact points per unit time within region of 0.5D � 0.25D � drum depthas a function of position in ball bed under conditions of N = 0.7, f = 0.4, andd = 3 mm.

T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629 627

the impact energy clarifies the meaning of adjustment of the drumspeed in the scale-up.

4.2. Determination of milling time at scale-up

When the drum diameter, the ball diameter and the ball fillingratio are constant, the power P determined by Eq. (13) is propor-tional to the number of balls in the drum, depending on the drumdepth L [12]. Thus, the power was expressed in terms of P/L, andinfluence of the drum diameter D on P/L was numerically analyzed.Fig. 9 shows an example of the relationship between P/L and D.When the ball filling ratio f and the ball diameter d were constant,P/L was proportional to 2.5 power of D regardless of the rotationalspeed ratio N. The similar results were also obtained at other val-ues of f and d. The empirical value of exponent thus numericallydetermined was similar to the experimental values indicating thedrum diameter dependence of power consumption, reported byGow et al. [19], Coghill et al. [20], Hukki [21], and Hogg and Fuer-stenau [22]. Consequently, P was given as a function of D and thedepth-to-diameter ratio b (=L/D).

P ¼ C1bD3:5 ð18Þ

where C1 is the coefficient depending on the rotational speed.Therefore, dependence of C1 on the rotational speed ratio N wasinvestigated under various conditions of the ball filling ratio f andthe ball diameter d in TM-2. The results are shown in Fig. 10. It

was found that C1 was proportional to 1.3 power of N when f andd were constant.

C1 ¼ C2N1:3 ð19Þ

where C2 is the coefficient depending on f and d. Within the rangesof f = 0.4–0.6 and d = 3–6 mm, however, the C1–N correlation wasexpressed by a single exponential function; this means that C2 is al-most constant under the conditions of f and d.

Fig. 10. Relationship between coefficient C1 and rotational speed ratio N undervarious ball filling ratios and ball diameters in TM-2.

Fig. 8. Impact energy distribution at controlled rotational speed ratio N underconstant ball filling ratio of f = 0.4 and ball diameter of d = 3 mm.

Fig. 9. Change in power P with drum diameter D at various rotational speed ratios Nunder constant ball filling ratio of f = 0.4 and ball diameter of d = 3 mm.

628 T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629

The true volume of powder (=W/q, where W and q are the load-ing mass and true density of powder, respectively) does not changeduring the milling treatment. Thus, ratio S of the true volume ofpowder to the drum capacity, defined by:

S ¼ W=qðp=4ÞbD3 ð20Þ

is also constant. Accordingly, the drum diameter D is expressed asfollows:

D ¼ 4WpbqS

� �1=3

ð21Þ

The cumulative specific energy E for a given milling time t is ex-pressed as:

E ¼ PW

t ð22Þ

By substituting Eqs. (18)–(21) into Eq. (22), E is given by the follow-ing empirical formula.

E ¼ C24p

� �1:2

N1:3ðqSÞ�1:2 Wb

� �0:2

t ð23Þ

Eq. (23) can provide the milling time in a larger-sized mill by usingthe milling time in a smaller-sized mill assuming that the cumula-tive specific energy was the same.

The milling time t in a larger-sized mill is determined as fol-lows. A scale-up index is defined as the mass ratio x of powderin the larger-sized mill to in a smaller-sized mill. By making thevalues of E in the larger-sized and smaller-sized mills coincide witheach other, the following equation is obtained.

C24p

� �1:2

N1:30 ðqS0Þ�1:2 W0

b0

� �0:2

t0

¼ C24p

� �1:2

N1:3ðqSÞ�1:2 xW0

b

� �0:2

t ð24Þ

where the subscript 0 means the smaller-sized mill. From Eq. (24), tis given as:

t ¼ N0

N

� �1:3

xb0

b

� ��0:2 S0

S

� ��1:2

t0 ð25Þ

4.3. Verification of scale-up methodology

Validity of the proposed method for the scale-up was experi-mentally confirmed through the grinding experiments of the alu-minum hydroxide powder. As representative conditions of theball filling ratio and the ball diameter, f = 0.4 and d = 3 mm werechosen because C2 in Eq. (19) is independent of f and d withinthe ranges of f = 0.4–0.6 and d = 3–6 mm. Here, we deal with thescale-up issue of TM-1 to TM-2 and TM-3, in which TM-1 is oper-ated at the rotational speed ratio of N = 0.9 (=N0 in Eq. (25)) as anexample. First, N in TM-2 and TM-3 has to be determined; N wasdetermined to be 0.7 in TM-2 and TM-3 so that the energy distri-butions in TM-2 and TM-3 coincide with that in TM-1 at N = 0.9,as shown in Fig. 8. And then, the grinding time t in TM-2 andTM-3 are determined according to a given grinding time t0 inTM-1 by using Eq. (25) so that the specific energy E coincides witheach other. In addition, in order to confirm the effect of coincidenceof the energy distribution, the grinding experiments were carriedout at the same N (=0.7) where the energy distribution in TM-1 dif-fers from those in TM-2 and TM-3 as shown in Fig. 3c but the spe-cific energy coincides with each other. Table 3 summarizes theoperating conditions. Fig. 11 shows change in the size distributionof milled powder with the cumulative specific energy under theconditions thus determined. When the energy distribution wasnot adjusted, the size distribution in TM-1 differed from those inTM-2 and TM-3 even though the specific energy coincided witheach other. This indicates that only coincidence of the specific en-ergy is not enough for the scale-up. On the contrary, under the

Fig. 11. Change in size distribution of milled powder at cumulative impact energyof E = (a) 0.06, (b) 0.13, (c) 0.26 and (d) 0.51 MJ/kg in TM-1, TM-2 and TM-3 underconstant ball filling ratio of f = 0.4 and ball diameter of d = 3 mm.

Table 3Operating conditions in grinding experiments.

TM-1 TM-2 TM-3 TM-1

Ball filling ratio, f (�) 0.4 0.4 0.4 0.4Ball diameter, d [mm] 3 3 3 3Rotational speed ratio, N (�) 0.9 0.7 0.7 0.7Rotational speed of drum (s�1) 2.73 1.65 1.27 2.12Scale-up index, x (�) 1 4.4 23 1Volume ratio of powder to drum, S (�) 0.036 0.036 0.036 0.036Grinding time, t (min)

at E = 0.06 MJ/kg 4.7 5 3.9 6.5at E = 0.13 MJ/kg 9.4 10 7.7 12.9at E = 0.26 MJ/kg 18.9 20 15.5 25.8at E = 0.51 MJ/kg 37.7 40 31 51.6

T. Iwasaki et al. / Advanced Powder Technology 21 (2010) 623–629 629

same energy distribution, the size distribution in each mill almostcoincided with each other. The coincidence is satisfactory in prac-tical uses. This result supports validity of the proposed method andindicates that in the scale-up of tumbling ball mills coincidence ofnot only the specific energy but also the energy distribution (i.e.,similarity of the shear flow field of balls) in the mills with differentsizes is important. This method can be applied under other condi-tions within the ranges of f = 0.4–0.6 and d = 3–6 mm.

5. Conclusion

The impact energy of grinding balls in horizontal tumbling ballmills with different drum diameters was analyzed using the dis-crete element method. The scale-up methodology for tumbling ballmills based on the impact energy was proposed. It was found that

in case of the scale-up of the rotating drum it is important to ex-pand the shear flow field of grinding balls in the drum while main-taining the similarity of the region of the shear flow field. Whenboth the frequency distribution of the impact energy and its cumu-lative amount over the milling time coincided with each other indifferent-sized mills, the scale-up could be successfully achieved.Validity of the proposed method was confirmed experimentallythrough the dry grinding treatment of aluminum hydroxide pow-der. The experimental results supported that this method is usefulin practical applications. When the grinding balls with differentsizes are used and/or ball-lifters are fitted to the drum, this methodmay be also applied because the grinding mechanism in such casesis almost the same as that in the mills used in this work althoughthe impact energy distribution remarkably changes depending onthe ball motion. This scale-up methodology will be generalized inour future work by rearranging the obtained formulas in terms ofmeaningful dimensionless numbers such as Froude number andpower number.

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