**by M KIMURA · Cited by 9 — on ball motion in mills using the discrete element A ball mill is one kind of grinding machine, and it is a device in which media balls **

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1SUMITOMO KAGAKU2007 -IIThis paper is translated from R&D Report, ÒSUMITOMO KAGAKUÓ, vol. 2007-II. computational techniques that have accompaniedincreased performance in computers in recent years, it has become possible to analyze many fields using computer simulations. In the field of powder tech- nology, Cundall et al.1)have proposed a simulation method targeting the motion of particles, and it has had a great amount of success in application to the analysis of various phenomena. This method is called the discrete element method (DEM), and it is a method that tracks the motion of individual particles based on equations of motion. Research on ball motion in mills using the discrete element method has been proposed by Mishra et al.2)andYanagi et al.3)So far, reports on analyses simulat-ing three-dimensional analysis and complex liner shapes,4), 5)and research on mill power consump-tion 6), 7) have been published.However, most of them are concerned with the simu-lation of only the ballsÕ motion in a mill. In actual grind- ing, the balls are not alone in the mill, but rather are present with the solid materials. To have a more accu- rate simulation of the grinding behavior, we must also simulate the motion of the solid materials, but because the number of particles for the solid material is so large, it is impossible to track all of the particles includ- ed in the solid materials with current computer capabil- ities. Therefore, we must model the presence of the solid materials and introduce this into the simulation. Observing the ball motion in experiments, the surfacesIntroductionIn recent years the demands for functional inorganicmaterials have been expanding in a variety of fields, such as display materials, energy, automobiles and semiconductors. Since the performance of these inor- ganic materials greatly affects the performance of the products in the fields mentioned above, various compo- sitions and manufacturing conditions are explored to establish optimum performance. In the manufacturing of functional inorganic materials, ÒgrindingÓ can be cited as an important unit operation. Grinding opera- tions do not simply grind materials. They are used for the purpose of mixing, transporting, promoting physi- cal properties and heat transfer, preprocessing for recovery of valuable materials, expression of functions and the like.A ball mill is one kind of grinding machine, and it is adevice in which media balls and solid materials (the materials to be ground) are placed in a container. The materials are ground by moving the container. Because the structure of ball mills is simple and it is easy to operate, and so they are widely used.However, designing these devices and selecting con-ditions depend in many ways on empirical knowledge, and they have not been sufficiently systematized. Therefore, to scale-up these devices is not always easy, and collecting data requires a lot of effort and cost.On the other hand, with the recent improvements in Design Method of Ball Mill byDiscrete Element Method The grinding rate of gibbsite in tumbling and rocking ball mills using fins was well correlated with the spe- cific impact energy of the balls calculated from Discrete Element Method simulation. This relationship was successfully used for the scale-up of a rocking ball mill, and the optimum design and operating conditions for the rocking ball mill could be estimated by the specific impact energy of the balls cal- culated by a computer simulation.Sumitomo Chemical Co., Ltd.Process & Production Technology CenterMakio KIMURAMasayuki NARUMITomonari KOBAYASHI

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2SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element Methodof the balls are in a state where they are covered withthe solid materials. It appears as if balls coated with the solid materials are moving. Therefore, Kano et al.8)have simulated changing the coefficients of friction of the balls and have discovered that the coefficients of friction on the ball motion are extremely effective. It has been reported that if a suitable coefficient of fric- tion is selected for each solid material from these results, the ball motion can be reproduced with good precision.In addition, the collision frequency of the balls, thekinetic energy of the balls, the contact force between balls, the trajectory of ball motion and the like can be obtained from the simulation of ball motion. This infor- mation is an important factor for controlling the changes in characteristics of the solid materials in the grinding process. Kano et al. 8)found that it was the impact ener-gy of the balls that has a large effect on the grinding.Rocking mills (ball mills where the pot is rockedwhile rotating) are used to prevent adhesion and solidi- fication of the solid materials on the pot during grind- ing, and to scale-up these has been investigated. How- ever, collection of data has required effort as men- tioned earlier. Therefore, we focused on the discrete element method, and carried out joint research with Professor Saito and Senior Assistant Professor Kano at Tohoku University for the purpose of establishing a method for scale-up using rocking ball mill simulations.In this paper, we will introduce the results of predic-tions of grinding phenomena in rocking mills with fin plates and rocking mills using a combination of dis- crete element method simulation analysis and experi- mental investigations as well as the scale-up method and the optimum design method of these mills.Simulations1.Discrete Element Method The discrete element method is a method that mod- els contact forces such as the elastic repulsive force and frictional force working between particles that are in contact with each other and performs a numerical analysis of the motion of the individual particles operat- ed on by the forces of contact based on the equations of motion for each. In a ball mill, the collisions between two balls or a ball and the mill wall are expressed by a Voigt model that expresses elastic spring (spring coef- ficient K) where elastic and inelastic properties of theobjects are introduced between the points of contact asshown in Fig. 1and the viscosity dashpot (dampingcoefficient ). In addition, a friction slider (coefficient of friction ) is introduced into the tangential compo- nent of the interactive force to express frictional inter- action attendant to ball contact.The forces of contact between balls are given by the following equation as the compressive forces in the normal direction (Fn) and the shear forces in the tan-gential direction (Ft).Here, uand are the relative displacement and therelative angular displacement, respectively, between the two particles we are focusing on, and k, , and rare the spring coefficient damping coefficient, coeffi- cient of friction and the radius of the balls.The spring coefficient Knin the normal direction isgiven by the following equation using the YoungÕs modulus Eof the ball and the mill wall, and PoissonÕs ratio and using the Hertz theory of elastic contact. The subscripts i, jand windicate the balls iand jandthe mill wall.(Eq. 3)(Eq. 4)(Eq. 5)(Eq. 6) (Eq. 7)(Eq. 1)(Eq. 2)Fig. 1Model of interactive forces between two balls(a) Compressive force(b) Shear force KnSliderKtunut

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3SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element MethodThe spring coefficient in the tangential direction Kscan be obtained based on the defining equation for theLame constant shown in (Eq. 8), which shows the rela- tionship between the shear ratio and YoungÕs modulusfor the substance.In a vibration equation with a single degree of free- dom having the elastic spring and viscosity dashpot,is when the damping is the fastest. Cundall1)has pro-posed considering the relationship in (Eq. 9) where the rebounding phenomenon that arises from collisions among the elements is made to dampen as quickly as possible with determined. Simulations in this paper employed the same method of determination.The physical constants used in these calculations are given in Table 1.In a discrete element method simulation of theinside of a ball mill, the coefficient of friction of the balls is the most important factor, and it has been reported that the spring coefficient and damping coef- ficient calculated from YoungÕs modulus and Pois-sonÕs ratio do not have a large effect on the ballmotion.9)There is correlation between the coefficientof friction and grinding, and with gibbsite powder, it has been reported to be roughly 0.3Ð0.8,8)so we used0.8 here.2.Impact Energy of Balls Using the discrete element method, it is possible toobtain the collision frequency of the balls , the kinetic energy of the balls the contact force between balls, the trajectories of motion and other information about the movements of the balls inside the ball mill either tem- porally or spatially. As mentioned above, Kano et al.8)found the specific impact energy of the balls ( EW)(Eq. 9)(Eq. 8)defined by (Eq. 10) has an especially large effect on thegrinding.where, Wis the weight of the solid materials, nthenumber of collisions, mthe mass of a ball and jtherelative velocity between two colliding balls or a ball colliding against the mill wall.ExperimentGibbsite powder (Al(OH)3, with an average particlediameter of 30 Ð 50 m, Sumitomo Chemical Co. Ltd., CHP-340S) was used for the solid material.Fig. 2shows a schematic diagram of the rockingmill (Aichi Electric Co., Ltd.) used in this study. The rocking mill has the axis of tumbling rotation of a typi- cal tumbling mill and an axis of rocking rotation per- pendicular to this. It is a grinding device that is capa- ble of rocking the pot while it is rotating. Three fins are provided on the inside of the mill, and we used two types of mill where the capacity of the pot was 60 L and 300 L.The mill was filled with 15 mm nylon coated ironballs, and the rotational speed Nof the mill was variedin a range from 40% Ð100% using the critical rotationalspeed Ncdefined by (Eq. 11) as a reference. The criti-cal rotational speed Ncis the limiting speed where theballs move together with the mill wall due to the cen- trifugal force.where, Dmis the inside diameter of the mill, and gisgravitational acceleration. Grinding was carried out for 180 minutes. At specific times during that period, the mill was stopped, and small volume samples were(Eq. 11)(Eq. 10)Table 1Physical properties for DEM simulationYoungÕs modulusPoissonÕs ratioFrictional coefficient Density of balls Time step3.5 1080.230.8345210.0Fig. 2Schematic diagram of the rocking ball mill 2020Fin

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4SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element Methodcollected. The diameter of the gibbsite powder wasmeasured using a Master Sizer 2000 (Sysmex Corpora- tion). Details of the experimental conditions are given in Table 2.Results and Discussion1.Effects of Fins on Ball Motion in the Mill To predict ball mill grinding behavior using the dis- crete element method, we must first be able to repro- duce the ballsÕ motion in the ball mill using simula- tions.Kano et al.8)have made it clear that it is possible toreproduce the ball motion when the solid material is also present in a tumbling ball mill without fins by pro- viding a suitable coefficient of friction for the balls and carrying out simulations. That suitable coefficient of friction is 0.17 for the balls alone, and roughly in the range of 0.3Ð0.8 when gibbsite is coexistent.In this paper, we carried out simulations varyingthe coefficient of friction for the balls in a range of0.15Ð0.8, and by comparing EWobtained as a result,we confirmed the effects of the coefficient of friction on ball motion. The relationship between the coefficient of friction andEWfor a 60 L tumbling mill without fins isshown in Fig. 3, and that for a 60 L tumbling mill withfins is shown in Fig. 4.We can see that when the coefficient of frictionchanges without fins, there is also a large change in EW. On the other hand, when the tumbling rotationalspeed is 1.0 Ncwhen there are fins, there is a tendencyfor EWto change with changes in the coefficient of fric-tion, but when the tumbling rotational speed is 0.8 Ncor less, EWhas substantially the same value. With themill that does not have fins, the height of the balls lift- ed up by the mill wall increases with a larger coefficient of friction, so even at the same tumbling rotational speed, different behavior appears. However, we can assume that when the mill is provided with fins, the balls are lifted up by the fins, so the effect of the coeffi- cient of friction becomes extremely small. Thus, we can assume that with a tumbling mill that has fins, the ball motion in the mill is substantially the same regard- less of the presence or absence of the material being ground and with a tumbling rotational speed smaller than the critical speed of rotation.Therefore, we carried out observational experimentswhere the motion of the balls in the mill with fins was made visible and compared it with the simulation results. Table 3gives the results for a tumbling mill,and Table 4gives the results for a rocking mill.In these experiments, a clear acrylic plate was used for the mill cover to make visual observations possible. When the ball motion had sufficiently stabilized after Fig. 3Relation between specific impact energy and frictional coefficient of 60L tumbling mill without fin010 203040Frictional coefficient [Ð]Specific impact energy [J/s/kg] 0.5Nc0.7Nc0.9Nc1.0Nc00.20.40.60.81 Fig. 4Relation between specific impact energy and frictional coefficient of 60L tumbling mill with fin01020304000.2 0.40.60.81 Frictional coefficient [Ð]Specific impact energy [J/s/kg] 0.4Nc0.7Nc0.8Nc1.0NcTable 2Mill configuration and experimental condi- tionsPot diameterPot depth Height of fin Swing speed Critical rotational speed Number of balls Weight of gibbsite[mm][mm] [mm] [spm]* [rpm] [Ð][kg]34469020 12 72787010.260L mill590118540 12 553912051300L mill* spm : frequency of swing per minute

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5SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element Methodstarting up the mill, we filmed the ball motion with a video camera under conditions with a tumbling speed of rotation of 0.4 NcÐ1.0 Nc. When the gibbsite powderwas introduced, it became difficult to visualize the ball behavior inside the mill because the gibbsite powder adhered to the clear acrylic plate, so we have only shown the results when the mill was filled with balls only. On the other hand, the results for a simulation carried out with a coefficient of friction of 0.8 are given. The ball motion at all of the conditions in these results agrees well with the simulation results, and we were able to confirm that the ball behavior could be repro- duced in simulations.2.Relationship between Grinding Rate and Tum- bling Rotational Speed (60 L Tumbling and Rocking Mills)Fig. 5shows the normalized median diameter(Dt/D0) of the gibbsite powder when using the 60 Ltumbling mill as a function of grinding period time depending on the tumbling rotational speed. Here, D0is the initial median particle diameter and Dtis themedian particle diameter after tseconds of grinding.Table 4Snapshots of the motion of balls in the rocking mill (Experiment and DEM simulation results)0.4NcExperimentSimulation 0.6NcExperimentSimulation 0.8NcExperimentSimulation 1.0NcExperimentSimulation Table 3Snapshots of the motion of balls in the tumbling mill (Experiment and DEM sim- ulation results)0.4Nc0.6Nc0.8Nc1.0NcExperimentSimulation

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6SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element MethodFig. 7shows the relationship between Kpdefined in(Eq. 12) and the relative tumbling rotational speed (N/Nc). We can see that at a tumbling rotational speed of 0.8 Nc, the tumbling mill and the rocking mill have a Kpof substantially the same value. In addition, Kpin therocking mill increases with an increase in the tumbling rotational speed, and an extremely large value was exhibited at 0.8 Nc. A trend where it dropped rapidlywhen the tumbling rotational speed was 1.0 Nccould beseen. The reason for this can be assumed to be that the potential energy of the balls that are lifted up increases with an increase in the tumbling rotational speed, and Kpincreases along with this. Furthermore, when thetumbling rotational speed reaches 1.0 Nc, the centrifu-gal force becomes large, but the balls adhere to the wall surface and make a state where they rotate togeth- er, so the collisions between the balls are reduced dra- matically, and Kpsuddenly drops off.3.Relationship between Impact Energy of Balls and Tumbling Rotational Speed (60 L Tumbling and Rocking Mills)Fig. 8shows the relationship between the specificimpact energy of the balls (EW) calculated from thesimulation results and the relative tumbling rotational speed (N/Nc). With the tumbling mill, there was anextremely large value in the neighborhood of 0.8 Nc,and with the rocking mill, there was a trend that exhib- ited an extremely large value in the neighborhood of 1.0 Nc. The reason for this can be assumed to be that, as with the relationship between Kpand N/Nc, thepotential energy of the balls increases with an increase in the tumbling rotational speed, and because of this,We can see that the normalized median diameter isreduced exponentially with time. This process can be approximated by (Eq. 12) so as to be shown as a solid line.where, Kpis defined as the grinding rate. In addition,Dldenotes the median diameter at the grinding limit,and Dl/D0= 0.135 from the experimental.In the same manner, Fig. 6shows the normalizedmedian diameter (Dt/D0) of the gibbsite powder whenusing the 60 L rocking mill as a function of the grinding period time depending on the tumbling rotational speed.(Eq. 12)= exp( ÐKpt0.5)D0 Ð DlDt Ð DlFig. 6Relation between normalized median di- ameter of the gibbsite and grinding time at 60L rocking mill0.00.2 0.4 0.6 0.8 1.0020004000600080001000012000 Grinding time [sec]0.4Nc0.6Nc0.8Nc1.0NcNormalized median diameter of the gibbsite [Ð]Fig. 7Relation between grinding rate and rela- tive rotational speed00.010.020.030.0400.20.40.60.811.2 N/Nc [Ð]Grinding rate [1/s]60L tumbling mill60L rocking millFig. 5Relation between normalized median di- ameter of the gibbsite and grinding time at 60L tumbling mill0.00.2 0.4 0.6 0.8 1.0020004000600080001000012000 Grinding time [sec]Normalized median diameter of the gibbsite [Ð]0.4Nc0.6Nc0.8Nc

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8SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element Method(1)Effects of Ball Diameter Fig. 10shows the relationship between the balldiameter and EWat a tumbling rotational speed of 0.8Nc. The number of balls was set so as to have the sameweight of balls (Table 5), and the weight of gibbsitewas fixed at 10.2 kg.There was a tendency for EWto be larger with smallerdiameter balls, and it was estimated that Kpwould alsoshow the same tendency. We can assume that at condi- tions where the weight of the balls and the weight of gibbsite were the same, the number of balls increased as diameter of the balls became smaller and the number of collisions increased, so we had this kind of trend.(2)Effects of Fin Height Fig. 11shows the relationship between fin heightand EWunder the conditions of tumbling rotationalspeeds of 0.6 Ncand 0.8 Nc. We can see that EWbecomes somewhat higher with the installation of thefins. However, we can see that even if the height of the fins is increased, no large difference was found in the value for EW. In the case of 0.8 Nc, the trend was for EWto reach a maximum when the fin height was 5 mm.ConclusionWe have established a method where reproductionand prediction of the grinding phenomena in rocking mills are possible with good precision by combining simulation based on the discrete element method and experimental investigation. At rotational speeds below the critical rotational speed, a good correlative relation- ship is clearly established between the grinding rate and the specific impact energy of the balls regardless of the presence or absence of rocking and the size of the mill. If this relationship is used it is possible to scale-up with a minimum frequency of experiments and find the optimum operating conditions and design con- ditions. Moving forward, we would like to work on par- allel development for grinding equipment such as agi- tated bead mills and vibrating mills. If this can be achieved, it will be possible to select types of grinding equipment using this method, and we can assume that higher levels of grinding process development and increased speed will be achieved. Furthermore, since the same kind of approach can be applied to the mechanochemical reactions, we would also like to develop it in that direction.AcknowledgementsProfessor Fumio Saito and Senior Assistant Profes-sor Junya Kano of the Tohoku University Institute of Multidisciplinary Research for Advanced Materials gave us multifaceted guidance in this joint research. We would like to offer our thanks here.Fig. 10Relation between ball diameter and specif- ic impact energy0102030 40010203040 Ball diameter [mm]Specific impact energy [J/(sákg)]Table 5Calculation conditions10mm15mm 20mm 30mmBall diameter265607870 3320984Number of ballsFig. 11Relation between the height of fin and spe-cific impact energy01020 3040010203040 Height of fin [mm]Specific impact energy [J/(sákg)]0.6Nc0.8Nc

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9SUMITOMO KAGAKU2007 -IIDesign Method of Ball Mill by Discrete Element MethodReferences1)P.A. Cundall and O.D.L. Strack, Geotechnique, 29,47 (1979).2)B. K. Mishra and R. K. Rajamani, KONA, 8, 92(1990).3)H. Ryu, H. Hashimoto, F. Saito and R.Watanabe, Shigen-to-Sozai, 108, 549 (1992).4)P. W. Cleary, Minerals Engineering, 11, 1061(1998).5)R. K. Rajamani, B. K. Mishra, R. Venugopal and A.Datta, Powder Technology, 109, 105 (2000).6)A.Datta, B. K. Mishra, and R. K. Rajamani, Canadi-an Metallurgical Quarterly, 38, 133 (1999).7)M. A. van Nierop, G. Glover, A. L. Hinde and M. H. Moys, International Journal of Mineral Processing,61, 77 (2001).8)J. Kano, N. Chujo and F. Saito, Advanced PowderTechnology, 8, 39 (1997).9)The Society of Powder Technology, Japan: FuntaiSimulation Nyumon [Introduction to Granular Sim-ulation], p.74, Sangyo Tosho, (1998).PROFILEMakio KIMURASumitomo Chemical Co., Ltd.Process & Production Technology Center Senior Research AssociateMasayuki NARUMISumitomo Chemical Co., Ltd.Process & Production Technology Center Research AssociateTomonari KOBAYASHISumitomo Chemical Co., Ltd.Process & Production Technology Center Research Associate

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