Ž .Powder Technology 108 2000 116–121www.elsevier.comrlocaterpowtec

Impact energy spectra of tumbling mills

Raj K. Rajamani ), Poly Songfack, B.K. Mishra 1

The Comminution Center UniÕersity of Utah, Salt Lake City, UT 84112, USA

Accepted 20 September 1999

Abstract

Tumbling mills using balls as grinding media are used in a variety of ways to produce fine powders. Mills in this category are ball,planetary and vibratory mills. The fineness of powder produced in such mills depends on the powder strength characteristics as well as theimpact energy of the tumbling balls. In order to understand the collision frequency and the collision intensity of the ball media, a

Ž .numerical simulation tool using the discrete element method DEM has been developed. The impact energy spectra of these mills can bemodified via suitable choice of operating conditions. The significance of the impact energy spectra of vibratory and planetary mills arediscussed. q 2000 Elsevier Science S.A. All rights reserved.

Keywords: Impact energy spectra; Tumbling mills; Fine powders; Discrete Element Method

1. Introduction

The size distribution of material ground in tumblingmills is determined by the strength characteristics of thematerial and the energy available for grinding. This energyis imparted to the particles as hundreds and thousands ofcollisions of varying intensity and frequency. Thus, themill may be visualized as a device generating a character-istic impact energy spectrum which depends on the set ofdesign and operating conditions. However, all impacts maynot lead to material breakage at all. Some impacts are ofexcessive intensity that only a part of the energy is utilizedin breakage, the rest is lost in overcoming interparticlefriction. Impacts of low energy intensity may not causeany breakage. Therefore, the study of impact energy spec-trum of a given mill ultimately leads to the selection ofoptimal operating conditions for a desired degree of fine-ness of the product.

There has been an increasing demand for ultrafinepowders of specified size and shape dispersion, ranging intheir application from minerals to modern materials. Inminerals processing operations, particles in the range of200 mm and below are often desired for complete libera-tion of valuable material from the gangue. In magneticmaterial for recording, it is required not only to produce

) Corresponding author.1 On leave from Indian Institute of Technology, Kanpur, India.

micron-level particles in a narrow size range but to haveparticles conform to a particular shape. In pigment produc-tion for paints, the pigment particle size must be below300 nm for high light reflectivity. Such stringent require-ments make the powder processing operation quite com-plex and often challenging.

There is a limit to the fineness of material that can beproduced by conventional milling such as ball milling witha tolerable energy efficiency. This has prompted the use ofother tumbling devices such as vibrating mills. The motionof vibrating mills is characterized by the circular oscilla-tion of a point on the mill shell at high frequency. They aregenerally of small diameter, 25 to 91 cm and operate athigh mill filling, up to 60% to 70% of mill volume, with

Ž .smaller diameter balls about 6 to 12 mm . This results inhigher energy efficiency for the production of micron-sizematerial.

Ultrafine grinding to the submicron level is possible inplanetary mills. The planetary motion of the mill as awhole by means of a gyrating shaft produces excessivelyhigh force field sufficient enough to grind the particles tomicron and submicron size range. In some situations, uponprolonged grinding the smaller particles acquire enoughsurface charge to agglomerate, thus limiting the grindabil-ity of the particles. Operation of planetary mill is con-trolled by the relative speeds of the mill shell and thegyration shaft.

A comparison of relevant data pertaining to these tum-bling mills is shown in Table 1. The essential features

0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00208-9

( )R.K. Rajamani et al.rPowder Technology 108 2000 116–121 117

Table 1Design and operational features of tumbling mills

Features Ball mill Vibration mill Planetary mill

Grinding range -200 mm 10–100 mm 1–10-mmMode of operation rotation oscillation gyrationMill diameter 0.9–5.5 m 0.25–0.91 m 0.10–0.25 mBall diameter 12–50 mm 6–12 mm 3–12 mmEnergy supplied 10–3000 kW 3–15 kW 1–8 kW

describing the configuration of these mills is shown in Fig.1. It is easy to realize from Fig. 1 and Table 1 thattumbling mills are similar in many respects, although therange of application varies. The similarity in these millslies in the fact that the grinding action is a result ofmulti-body collisions taking place within the mill. Thesecollisions are caused by the motion of the mill shell that inturn is driven by various mechanisms depending on thetype of mill.

The purpose of this paper is to show that the impactenergy spectrum could be calculated. Then the differentmills are compared in terms of the spectra. The usefulnessof certain impacts and the energy wasted in some otherimpacts are also discussed.

The modeling of the motion and collision of balls in amulti-body system is definitely not an easy task. In thepast, most works have treated the ball charge as a contin-uum with specific properties. These over-simplified as-sumptions have only led to an empirical treatment of thecharge dynamics. However, with the use of the discrete

Ž .element method DEM , the mechanics of charge motion istackled without any of the simplifying assumptions. In thispaper, DEM is used to model the motion of the balls.Results of numerical simulations are presented to charac-terize the en masse motion of the balls within the mill. Aunique aspect of DEM is that it is possible to quantify theenergy involved in individual collisions which leads toimpact energy spectrum.

2. Discrete element method

w xTumbling mills are well suited for analysis by DEM 1 .It is primarily devised to simulate any problem dealingwith discretely interacting bodies. Ball mills, vibratingmills and planetary mills invariably deal with interacting

Ž .bodies grinding balls set in motion by the rotation,oscillation, or gyration of the mill shell, respectively. Inessence, these devices are very similar, and the basic DEMscheme has been applied with necessary modifications foreach type of mill.

Discrete element models are used to analyze dynamicŽsystems. Tumbling mills problems analysis of media mo-

.tion in particular are mainly dynamic problems wheretime dependent ball motion is of interest. The individualballs are modeled as rigid element having deformable

contacts. The details of the numerical algorithm and modelw xparameter values is described in Mishra and Rajamani 2 .

The equation of motion for each ball is written as:

d2 x d xM qC qK xsp 1Ž .

d t d t

where M, C and K are the mass, damping, and stiffnessmatrix, respectively; p is the applied force vector and x is

Ž .the position vector. Eq. 1 describes the motion of asystem of balls where the internal restoring forces are dueto a pair of springs acting at contact in both normal andshear directions. In addition, a pair of dashpots at contact

Ž .Fig. 1. Schematic representation of a planetary mill top and a vibratoryŽ .mill bottom .

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point damp the forces. Coefficient of friction plays a roleat the contact where the absolute value of the force in theshear direction is never allowed to increase beyond alimiting value given by the product of normal force andcoefficient of friction.

Ž .The motion of balls is computed by integrating Eq. 1by a central difference method where the incrementaldisplacements are computed at every time step as givenbelow.

d x 1 1y1tq sa x ty qa M p tq1 yk tŽ . Ž .1 2 xž / ž /d t 2 2

2Ž .

d x 1D x tq1 sD t tq 3Ž . Ž .ž /d t 2

Ž . Žwhere D t is the time of integration; a s 2yaD t r 2q1. Ž . Ž .aD t ; and a s 2D t r 2qaD t .2

In the above equations, a is a constant for massproportional damping as

Csa M. 4Ž .The central difference scheme as used in DEM is condi-tionally stable. The limit on the time step for integration istherefore set at:

2D tF . 5Ž .

vmax

The highest natural frequency of the system, v , ismax

estimated from the mass of the smallest ball and thehighest contact stiffness.

In summary, the discrete element analysis starts with amathematical representation of each element or entity com-prising the physical system. In a two dimensional model ofa simple system such as tumbling mills, these elements areline-type or disc-type or both. More complex systems aremodeled using superquadric elements to study spatial be-havior of particulate systems in three dimensions. A propercontact detection scheme is implemented depending on theshape of these elements. Knowing the contacts for anygiven element and their relative displacements, the forcesacting at each contact are calculated using a contact defor-mational equation. These forces are incorporated into thetranslational and rotational equations of motion. The majorcomputational tasks of DEM in each time step are as

Ž . Ž .follows: a sum all forces on balls and update position, bŽ .addrdelete contact between particles, and c compute

contact forces from contact properties.Contact detection takes the bulk of the computation

time. For this reason, the motion of balls is monitoredwithin a subregion of the system. Contact detection istriggered when the displacement of a ball exceeds a speci-fied value. The details of the contact search procedure andother pertinent details of the algorithm is best described in

w xCundall and Strack 3 .

3. Numerical results

In the past, characterization of charge motion insidetumbling mills has received considerable attention in orderto predict power draw, breakage of particles, and hence,mill performance. Here, several numerical experiments aredone to analyze the complex charge dynamics of tumblingmills. All of the results reported here are done with a twodimensional version of DEM algorithm. Two types ofmills are considered: vibratory and planetary mills.

In vibratory mill, the mill frequency and the vibrationamplitude are generally varied to achieve the proper circu-lation of the charge. Fig. 2 shows the charge profile in avibratory mill. In this figure, a 91-cm diameter mill issimulated to show the variation of charge profile withincreasing vibration frequency. The mill is loaded with

Ž .2-cm diameter balls, up to 65% ball load 1068 balls . Thevibration amplitude is maintained at 10 mm. The threesnapshots show the mill profile at steady state, for threevibration frequencies: 1000, 1500 and 2000 rpm. As thefrequency of oscillation or the angular speed of the wallabout its center of eccentricity is changed, the profile ofthe charge changes, resulting in steeper slope of ballsurface at higher frequency.

Fig. 2. Charge profile in a vibratory mill.

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Fig. 3. Impact energy distribution in vibratory mill.

The computer animation of ball charge profile in vibrat-ing mills shows that, in the radial plane of the mill,grinding balls circulate in the reverse direction of the millshell oscillation. The circulation of ball charge acceleratesas the vibration frequency increases. The circulation ofballs in the radial plane of the mill is very importantbecause it helps the transportation of ground particles from

w xthe bottom of the mill to the grinding zone 4 .Fine grinding in vibration mills takes place due to its

inherent design that increases the number of collisionsavailable per unit volume of mill. The high surface areaŽ .per unit volume of the grinding media leads to anincrease in the number of collisions. Fig. 3 shows theimpact energy distribution inside the mill. It is seen thatimpacts of 10 to 15 mJ number in tens of millions persecond while in the 100 mJ range, the impacts are only inthe thousands per second. The highest energy impacts, inthis case upwards of 225 mJ, are in the thousands persecond. The intensity of impacts in vibratory mill is astrong function of amplitude of vibration. For constantamplitude and frequency the impact distribution can alsobe varied by changing the ball load. When the ball load isincreased, the ball collision frequency increases but thehigh energy impacts relatively decrease. From a practicalstandpoint, in order to grind coarser and stronger particles Fig. 4. Charge profile in a planetary mill.

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lesser ball load should be used to generate more highenergy impacts. On the other hand, softer and smallerparticles must be ground with a higher ball load.

The motion in planetary mill as opposed to vibratorymills is even more complex. As shown in Fig. 1 the ballsexperience different accelerations depending on their loca-tion within the system. The acceleration field has a maxi-mum and minimum value at some location within the mill.The critical speed corresponds to a condition where theabsolute centrifugal acceleration due to the rotation of themill about its own axis is high enough for the balls to pressagainst the shell throughout a cycle. Based on this idea itcan be shown that the critical speed N is given by,c

1qRN s 6Ž .c 'GrD

where R is ratio of the angular speed of the mill to that ofthe gyrating shaft.

A 10-cm diameter mill is simulated to study the motionof the charge during one revolution. This mill is loadedwith 400 balls of diameter 3 mm up to a 50% filling. Thegyration arm is 60 cm and it revolves at 315 rpm, generat-ing a centrifugal field of 33 times that of gravity. For thissystem it is found that R must lie within y3.513 to 1.513.The motion of the charge is known to vary with therelative speed of the mill in relation to the speed of thegyrating shaft. Fig. 4 shows three snapshots of chargemotion at 65%, 90% and 105% of critical speed for a millthat rests on a gyrating shaft of 60 cm. In these simulationsthe gyrating shaft speed is kept at 33 radrs while the millspeed is varied to match the critical speed. All thesesnapshots correspond to the same location of the mill withrespect to the center of gyration. The profile of the chargeis distinct in each of these snapshots. At 105% of criticalspeed complete centrifuging of the charge occurs. At lowerspeeds, the ball charge would break away from the mill

Fig. 5. Energy distribution in a planetary mill.

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walls and contribute to high energy impacts against thewall of the mill.

It is possible to compute the impact energy distributionw xin the planetary mills. Zhao 5 has shown the variation in

the maximum contact force with the mill operating param-eter. For 4-mm steel balls, the maximum contact forcemeasured was 7 N in a 72-mm diameter mill with 33%ball filling. This is in a mill supported by a 360-mmgyration arm rotating at 700 rpm, which results in revolu-tion to rotation gear ratio of 0.5. However, it is difficult, asZhao points out, to determine the entire spectrum ofimpact forces. Fig. 5 shows the DEM impact energydistribution in the planetary mill whose dimensions aregiven earlier. It is known that the charge motion and hencethe number and intensity of collisions inside the mill varieswith mill speed and filling. The effect of coefficient offriction is often neglected in these analyses. Consideringthe excessive amount of swapping of balls, surging andslippage in planetary mill, the effect of coefficient offriction must be properly understood. As seen from Fig. 5,at low coefficient of friction, all impacts are less than 50mJ and with the increase of coefficient of friction anenergy spectrum evolves. In other words, as the coefficientof friction increases, more number of higher energy im-pacts are observed. From a practical standpoint, as grind-ing progresses ultrafine particles tend to lubricate the millinterior, which reduces the coefficient of friction betweensliding layers of balls. This in turn may lead to decrease ingrinding rate.

4. Conclusions

Ø Tumbling mills such as conventional ball mills, vibratingand planetary mills are very similar in operation. Thesize distribution of material produced in these mills is

both a function of material strength and the impactenergy spectrum in the mill. In this paper, DEM is usedto calculate the ball charge motion and energy spectrumof vibrating and planetary mills.

Ø The numerical experiments are done to study the effectof vibration frequency on the ball charge profile andimpact energy spectrum of vibrating mills. These experi-ments can be readily extended to other parameters suchas mill filling, vibration amplitude, etc.

Ø The effect of gyration speed on the mill charge profileand energy spectrum of planetary mills is also studied bymeans of numerical experiments. The effect variablessuch as gyration arm length, mill filling, etc. can beinvestigated.

Ø DEM appears to be a very convenient tool to study theeffect of various design and operating parameters on theimpact energy spectrum of tumbling mills. Thus, thistool can be used to select the best set of parameters for adesired product fineness. More comprehensive work isbeing done to relate the impact energy spectrum to theoperating and design parameters with a view to optimiz-ing the process performance.

References

w x1 B.K. Mishra, R.K. Rajamani, Simulation of charge motion in ballŽ .mills: Part 2. Numerical simulations, Int. J. Miner. Process. 40 1994

187–197.w x2 B.K. Mishra, R.K. Rajamani, The discrete element method for the

Ž .simulation of ball mills, Appl. Math. Modell. 16 1992 598–604.w x3 P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular

Ž .assemblies, Geotechnique 29 1979 47–65.w x4 T. Yokoyama, K. Tamura, H. Usui, G. Jimbo, Simulation of ball

behavior in a vibration mill in relation with its grinding rate-effects offractional ball filling and liquid viscosity, Proc. of Particle Tech.Forum, Denver, USA, 1994, pp. 419–430.

w x5 Q. Zhao, Mechanism of fine grinding in a planetary mill, PhD Thesis,Nagoya University, Japan, 1989.