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1.College of Mechanical and Electrical Engineering，China Jiliang University，Hangzhou 310018，P.R.China；2.College of Mechanical Engineering，Zhejiang University of Technology，Hangzhou 310014，P.R.China

Abstract: Motion behavior of grinding balls plays a vital role in improving efficiency of particle crushing. A method of preparing micro-particles by changing ball-motion behavior in a flutter mill is proposed and multiple grinding experiments are conducted. Crushing performance parameters，such as breakage rate Si，production rates of fine particles Fi and Fi*，are studied in different motion conditions. From the results，a better crushing performance is attained in the coupled motion modes of rotating speed ratio of 85%，with a vibrating amplitude of 8 mm and a frequency of 12 Hz. In addition，the influence of ball-motion behavior on particle crushing performance is discussed.The ball-motion behaviors，such as the collision energy loss E，among grinding balls have some relationship with the particle crushing performance of Si. Therefore，this study not just provides an efficiency way of accumulating micro-particles，but also reveals how the ball-motion behavior influence particle crushing performance in the flutter mill.

Key words：flutter ball mill；ball-motion behavior；breakage rate；collision energy loss；micro-particle preparation

0 Introduction

Many modern technologies used today rely on the preparation of micro-particles，such as ceramic industries，chemical product，3D printing and so on. These industries require billions of tons of me?tallic ores，cement，minerals，and other solid parti?cles and ball milling is the most important mechani?cal treatment used to reduce the particle size［1］.Fine powders with the size of submicron are urgent?ly required in preparing high-quality products，but a large amount of electrical energy ranging from 5 kWh/t to 50 kWh/t is consumed in the milling processes due to the low energy utilization［2］. Effec?tive crushing behavior of ball-motion can bring fine powders by consuming only a little amount of ener?gy［3］. Many scholars pay much attention to improve and change the method of ball-motion behavior in order to accumulate micro-particles［4-6］. For exam?ple，the structure of mill cylinder has been altered by adding different stirring rods，paddles or liners to enhance grinding behavior of ball-motion like the stirred mill，two-way rotating ball mill［7-12］. Thus，the motion mode of mill has been changed by trans?forming mill motion or adding other motions，such as vibration，planetary，or flutters［13-17］.

There are a number of reports on the effect of motion behavior of grinding balls on the crushed performance of particles. Gupta et al.［18］studied the force condition and kinematic equation of a single grinding ball. Fan et al.［19］recommended a theoreti?cal model on the motion area of a group of grinding balls，supplementing the ball-motion function in a complex condition. Cleary and Powell et al.［20-21］dis?cussed on the influence of many factors including ro?tating speed ratio，filling rate，and balls’size on the alteration of ball-motion behavior. Some re?searchers concluded that different motion behaviors of grinding balls affect the milling process，crushing ways of impacting and grinding， and crushing zones［22］. A better motion behavior should involve a reasonable proportion of the two crushing ways to improve milling efficiency as well as accumulating micro-particles［23］. Thus，the division of motion re?gions for the grinding balls were carried out earlier and we analyzed the ball?motion behavior in differ?ent conditions by changing motion parameters in a flutter ball mill［24］. However， the influence of changing the ball-motion behavior on the crushing performance of ground particles in the flutter mill is still unclear；and motion behavior of balls is totally available by understanding the simulation software of discrete element method（DEM）［25］，the crush?ing conditions of ground particles in actual grinding experiments haven’t carried out till now. More?over，if the crushing performance of ground parti?cles have some relation with the ball-motion behav?ior which can be obtained from numerical simula?tion，time and cost of milling processes may be dra?matically reduced.

Therefore，ball-motion behavior can influence the crushing performance of ground particles obvi?ously，as is focused in this report. This paper de?scribes the way of micro-particle preparation in a flutter ball mill in different motion conditions.Then，numerical simulation of the grinding balls is conducted for analysis the ball-motion behavior to study how it influences particle crushing perfor?mance and the more efficiency way of accumulating micro-particles.

1 Experiments of Preparing Micro?particles

The flutter ball mill is a novel milling equip?ment combined with the two coupled motions of hor?izontal rotation and vertical vibration，which can change the ball?motion behavior and reduce the weak zone efficiently in a milling process［26-28］. Fig.1 shows a schematic diagram of a flutter ball mill［17］.There are two different motors to drive the mill cyl?inder and change the ball-motion behavior by adjust?ing three motion parameters including rotating speed ratio，vibrating amplitude，and frequency.

Fig.1 A schematic diagram of a flutter ball mill[17]

1.1 Experimental set?up and simulation model

Fig.2 shows the experimental mill set-up. It implements the change of motion behavior of balls caused by the varied motion parameters. The oper?ating conditions and physical characteristics are de?scribed in Table 1. The feed material in the mill is silica sand，provided from Zhejiang Dingchuang Sil?ica Sand Co.，and its initial particle size is 0.4—0.85 mm，as its chemical composition is shown in Table 2. The silica sand is comprised of SiO2（86%of the weight），Al2O3（1.2% of the weight），F(xiàn)e2O3（1.1% of the weight），NaO（1.3% of the weight），and other components（about 10% of the weight）.

Table 1 Experimental operating conditions and physical characteristics of the flutter mill

Table 2 Chemical composition of the used silica sand

Fig.2 Experimental set-up of the flutter ball mill

The vibrating motion of mill cylinder is welltraced by a sensor element in Fig.2. When a vibrat?ing amplitude of 4 mm is adjusted，the displacement of the point onZandYaxes are obtained after sev?eral cycles，as shown in Fig.3（a）. The maximum wavy distance on theZaxis is about 4 mm with an unavoidable wavy distance of about 1 mm on theYaxis，within the allowable error. The trajectory of the cylinder is shown in Fig.3（b），and its amplitude is almost the same with the selected value.

Fig.3 Testing results for the experimental set-up with the vibrating amplitude of 4 mm

The simulation process in EDEM contains three parts involving building a computational mod?el for the flutter mill，generating particles in the cyl?inder，and imposing coupled motions to the mill.The completed situational model is shown in Fig.4（a）. The parameters in simulation model are repre?sented in Table 3 and a tagged point on the shell is shown in Fig.4（b）. The trajectory of the tagged point is obtained after several cycles，as shown in Fig.4（c），which is close to the experimental set-up value.

Fig.4 Simulation model and the motion-path of the cylinder

Table 3 Simulation parameters in EDEM model

1.2 Experimental design

The following three coupling level motion pa?rameters were selected. Rotating speed ratio and vi?brating amplitude were chosen as 65%，75%，and 85% and 4，6，and 8 mm at the frequencies of 4，8，and 12 Hz，respectively［29］. Then，an effective optimal design scheme of uniformly orthogonal plan［30］was used to deduce the experimental group numbers. The experimental scheme and three pa?rameters in each group are displayed in Table 4.To compare with the coupled motion，the tenth group of rotating motion only is added as the con?trol group.

When the milling process ends，the specific surface area of ground particles is tested by an ASAP2020 Surface Area Instrument. Then，the particle size distribution is analyzed by a Malvern Laser Particle Size Analyzer.

2 Crushing Performance of Ground Particles

2.1 Particle size distribution

Fig.5 shows the ground particle size distribu?tion in different motion conditions. The curves is fit?ted by linear regression analysis，and the linear fit?ting equations match Rosin Rammler Bennet（RRB）equation completely，as shown in［31］

or

whereRis the cumulative mass fraction on a sieved mesh，x0the characteristic diameter that stands for the particle size，andnthe uniformity coefficient that stands for the width of particle size distribution.It should be noted that a smallnleads to a wider par?ticle size distribution［32-33］.

The RRB distribution function of the ground particles processed in each group is obtained based on Eq.（2）. The characteristic particle sizex0and uniformity coefficientnare summarized in Table 5.x0is decreased gradually with enhancing group numbers，except for group 10. It indicates that the ground particles in coupled motions are uni?formly refined because the ball-motion behaviors are preferred to accumulate micro-particles. How?ever，the irregular uniformity coefficient，n，indi?cates that in some groups，micro-particles began to aggregation in a granular manner［34］. There?fore，intensity of the coupled motions may signifi?cantly affect generation and accumulation of microparticles.

Table 5 RRB liner regression results for the ground particles

2.2 Production rate of fine particles

The production rate of fine particles，F(xiàn)i，in dif?ferent groups is inspected to study the crushing per?formance of particles in the flutter mill. The mass fraction of ground particles finer than the size ofxiat the timetis denoted asFi（t）. The production rate of particles with sizes finer than 325，650，and 1 600 mesh（F325，F(xiàn)650，F(xiàn)1600）［35］is selected as the desired size，as shown in Figs.6—8.

The values ofF325are increased rapidly with en?hancing the grinding time，as shown in Fig.6.Slopes of the fitting curves show the mean growthrate of the micro-particles. They are smaller in groups 1—4 than that in groups 5—9.And the same changes ofF650andF1600are obtained in Figs.7 and 8 except for their values. Finally，the values ofF325，F(xiàn)650，andF1600are compared in different groups after grinding for 120 min，as shown in Fig.9.

Fig.6 Variation of F325 of particles with a size finer than 325 mesh

From Fig.9，it can be envisaged that a little volatility，but with a steady increase is attained forF325，F(xiàn)650，andF1600with enhancing of the group numbers，except for group 10，in which a much smaller value ofF650andF1600is gained.

Fig.7 Variation of F650 of particles with a size finer than 650 mesh

Fig.8 Variation of F1 600 of particles with a size finer than 1 600 mesh

Fig.9 Comparison of F325, F650, and F1 600 in all groups

Considering the energy consumed by the flutter mill during milling process，the production rate of fine particles per unit power，F(xiàn)i*，can be defined as［18］

whereM，t，andPare the mass of feed particles，grinding time，and net power drawn by mill，respec?tively. Fig.10 provides the values ofFi*in the flutter mill and shows the energy consumption during mill?ing process.Fi*is changed significantly in different groups and has a sharp drop in the groups 2 and 4.

Fig.10 Values of Fi*in the flutter-motion conditions

2.3 Particles breakage rate

According to the first-order grinding dynamics，Mi（0）andMi（t）denote the mass fraction of particle size（i）before and after grinding fortmin，respective?ly. Actually，iis ranged from 0.4 mm to 0.65 mm，confirming approximately no change in size. The da?ta obtained from results in each group are fitted with the mass fraction of crushed particles，which is de?scribed according to［36］

whereSiis the breakage rate of the feed size（i）. As it is often the case，fori= 1，Eq.（4）can be rewrit?ten as［37］

The dependence of ln［Mi（t）/Mi（0）］withtalong with the corresponding fitting curves are shown in Fig.11. The distribution of curves in Fig.11（a）is almost similar and the corresponding slopes are little changed. However，more different curves are attained in Fig.11（b）. Based on Eq.（5），the value of specific breakage rate，S1，can be de?rived from fitting curves，as shown in Fig.12.S1is increased continually with group numbers until group 8，where reaches up to the maximum value of 0.12. These findings confirm that this motion condition has a strong ability to break the feed parti?cles.

Fig.11 First-order grinding dynamic plots for ten groups

Fig.12 Variation of particle breakage rate S1 in ten groups

3 Motion Behavior of Grinding Balls in Numerical Simulation Results

3.1 Mean contact force

Mean contact force，F(xiàn)mcf，is the mean value of stress when collision occurs between balls and wall.The parameter describes the impact and friction forc?es affecting ground particles.Fmcfamong balls in the collision process has the two components of normal direction and tangential component，as shown in Fig.13. The normal component，F(xiàn)n，can be deter?mined by viscous damping in a linear contact model as［38］

Fig.13 Viscous damping in a linear contact model[38]

The tangential component，F(xiàn)t，depends on the force of viscous damping and friction，as shown in［39］

In Eqs.（6—7），knandktare the normal and tangen?tial stiffness coefficients，respectively；d，c，andvthe overlap zone，the damping coefficients，and the relative velocities，respectively.μis the friction co?efficient，which equals to tanφinFmaxt=μFn=Fntanφ.

Fig.14 showsFmcfin each group.FnandFtare a little changed in the groups 1，2，4，5，7 and 9；while they obviously increase in the groups 3，6，and 8 in which a large vibrating amplitude and fre?quency are performed. According to Eqs.（6—7），the result can be explained with the gradual increase of amplitude and frequency，leading to a larger space between balls，a shorter time of the single ball-motion，and decreasing the relative velocity；thus，F(xiàn)nandFtare increased. Moreover，the maxi?mum value ofFmcfin group 8 means that the ballmotion behavior can lead to a significant impact on the friction forces of the ground particles，providing the largest breakage rate ofSiin all groups（Fig.12）.

Fig.14 Mean contact force in ten groups

3.2 Collision energy loss

The collision energy loss among balls is origi?nated from damping and friction，which involves the two forms of normal and tangential energy losses.The normal energy loss can be obtained from Eqs.（6，8）.

The tangential energy loss can be derived from Eqs.（7，9）.

In Eqs.（8—9），Δtrefers to the time-step andkthe step. During a complete collision，the overlap zone，，is changed from zero to a maximum and，then，approached zero. So，the elastic strain energy becomes zero when the collision is finished，as shown in

The total collision energy loss during a com?plete collision is calculated by［40］

Fig.15 shows the collision energy loss，E，in each group. The variation ofEis similar to that ofFmcfin Fig.14，andFnis much larger thanFtin Fig.14. While，the normal energy loss，En，is much smaller than that of the tangential energy loss，Et，in Fig.15. It can be seen that althoughFnis larger，its contact frequency among balls is lower than that ofFt.Combined with the largest value of particle breakage rateSiin group 8 in Fig.12，the largest tangential contact force and tangential colli?sion energy loss are observed in group 8 as well，in?dicating that they may be effectively helpful to break the particles compared to the normal ones in the flutter mill.

Fig.15 Collision energy loss among balls in ten groups

4 Discussion

From the experimental and numerical results，it can be concluded that the ball-motion behavior in a rotating speed coupled with a high-intensity vibrat?ing is significantly effective to achieve micro-parti?cles owing to the more friction force，F(xiàn)t，and tan?gential collision energy，Et，imposed on the feed particles（as shown in Figs.9，12，14 and 15）. In?stead，the much lower values ofSiandFiin group 10 indicate that rotating motion only has been clear?ly undesirable to accumulate the desired particles fin?er than 325 mesh. Therefore，the coupled motion is necessary and helpful to change the motion behavior of grinding balls.

In addition，among the combined-motion condi?tions，group 8 shows a superior crushing perfor?mance for the feed particles，not only a larger break?age rate ofSibut also much smaller and more uni?form distribution of ground particles，as analyzed in Fig.5 and Table 5.

From Figs.12 and 15 it can be deduced that the curves ofEtorEn，andSichange similarly in for mer nine groups. Then，the points in Fig.16 are attempt?ed to express the relationship betweenEt（orEn）andSi. The points tagged by the squares are ignored because they are obtained in group 10. A linear func?tion is used to fit these points and a relational formu?la is expressed as

Fig.16 Linear dependence of Si to E

or

whereaandbare the constant coefficients with dif?ferent values for the flutter mill under various mo?tion conditions. According to Eqs.（13—14），Sican be predicted in some motion conditions and may be compared with the ball-motion behavior ofEnorEt.Therefore，high-cost，hard-operating，and environ?ment-polluting experiments can be extensively avoided for the flutter mill.

5 Conclusions

（1）The coupled motions can preferentially ac?cumulate micro-particles compared to the rotating motion only because of the more uniform distribu?tion-size，smaller size of ground particles，larger production rate of micro-particles，F(xiàn)i，and the big?ger particle breakage rate，Si，in the multiple grind?ing experiments.

（2）In the coupled motions，it is found that the main role of crushing particles is the tangential force and the tangential collision energy loss，which is dif?ferent from the traditional one. The optimal crush?ing performance is obtained in group 8，in which the rotating speed ratio，vibrating amplitude，and fre?quency are 85%，8 mm，and 12 Hz，respectively.Therefore，an efficiency way of accumulating microparticles is successfully achieved.

（3） How the ball-motion behavior influence particle crushing performance is discussed in the flut?ter mill by combining experimental and numerical analyses. A linear positive correlation is disclosed between particle breakage rateSiand collision ener?gy lossE.