Design Method of Ball Mill by Discrete Element Method Sumitomo Chemical Co., Ltd. Process & Production Technology Center Makio KIMURA Masayuki NARUMI Tomonari KOBAYASHI The grinding rate of gibbsite in tumbling and rocking ball mills using fins was well correlated with the specific 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 calculated by a computer simulation. Cundall 1) DEM : Discrete Element Method Mishra 2) 3) 4), 5) 6), 7) 27-
8) 8) 2 Fig. 1 K Voigt Fn Ft n Fig. 1 u ϕ K r KnHertz E i j w i j Knij = Kniw = i = j = w = u n Fn = Kn un + n 4 3 4 3 1 i 2 Ei 1 j 2 Ej 1 w 2 Ew Ks Eq. 8 Kn Ks = 2(1+ ) = 2 m K 1 K n (a) Compressive force i + j un t 1 ri i + w u t rirj ri + rj (b) Shear force { } (ut + rϕ) Ft = min Fn, Kt (ut + rϕ) + t t Slider Model of interactive forces between two balls (Eq. 1) (Eq. 2) (Eq. 3) (Eq. 4) (Eq. 5) (Eq. 6) (Eq. 7) (Eq. 8) (Eq. 9) Cundall 1) ϕ K t t 27-
Eq. 9 Table 1 2 2 Table 1 Young s modulus Poisson s ratio Frictional coefficient Density of balls Time step 9).3.8 8).8 8) Eq. 1 EW n 1 1 EW = m j 2 W j=1 2 Physical properties for DEM simulation [Pa] [ ] [ ] [kg/m 3 ] [ s] 3.5 1 8.23.8 3452 1. (Eq. 1) Wnm j Fig. 2 15mm NEq. 11 Nc41 Nc Nc = 6 g 2Dm (Eq. 11) Dm g 18 2 Table 2 Table 2 Pot diameter Pot depth Height of fin Swing speed Critical rotational speed Number of balls Weight of gibbsite Fin Schematic diagram of the rocking ball mill Mill configuration and experimental conditions [mm] [mm] [mm] [spm]* [rpm] [ ] [kg] * spm : frequency of swing per minute 6L mill 344 69 2 12 72 787 1.2 3L mill 59 1185 4 12 55 3912 51 Al (OH)3 3 5 m CHP-34S Fig. 2 6L 3L 8).17 27-
.3.8.15.8 6L Fig. 3 6LFig. 4 Specific impact energy [J/s/kg] 4 3 2.5Nc.7Nc 1.9Nc 1.Nc.2.4.6.8 1 Frictional coefficient [ ] Table 3 Table 4.4Nc 1.Nc.8 Fig. 3 Relation between specific impact energy and frictional coefficient of 6L tumbling mill without fin Table 3 Snapshots of the motion of balls in the tumbling mill ( and DEM simulation results) Specific impact energy [J/s/kg] 4 3 2 1.4Nc.7Nc.8Nc 1.Nc.4Nc.2.4.6.8 1 Frictional coefficient [ ].6Nc Fig. 4 Relation between specific impact energy and frictional coefficient of 6L tumbling mill with fin 1.Nc.8Nc.8Nc 1.Nc 27-
Table 4 Snapshots of the motion of balls in the rocking mill ( and DEM simulation results).4nc.6nc.8nc 1.Nc Fig. 5 6L Dt/D D Dt t Eq. 12 Dt Dl D Dl = exp( Kpt.5 ) (Eq. 12) Kp Dl Dl/D.135 6LDt/D Fig. 6 Fig. 7 Eq. 12 N/Nc.8Nc Normalized median diameter of the gibbsite [ ] Fig. 5 1..8.6.4.2. 2 4 6 8 1 12 Grinding time [sec].4nc.6nc.8nc Relation between normalized median diameter of the gibbsite and grinding time at 6L tumbling mill 1.Nc 1.Nc 27-
Normalized median diameter of the gibbsite [ ] 1..8.6.4.2.4Nc.6Nc.8Nc 1.Nc Specific impact energy [J/(s kg)] 4 6L tumbling mill 6L rocking mill 3 2 1.2.4.6.8 1 1.2 Fig. 6 Grinding rate [1/s]. 2 4 6 8 1 12.4.3.2.1 Grinding time [sec] Relation between normalized median diameter of the gibbsite and grinding time at 6L rocking mill 6L tumbling mill 6L rocking mill.2.4.6.8 1 1.2 Fig. 8 N/Nc [ ] Relation between specific impact energy and relative rotational speed Fig. 7 Fig. 8 Fig. 9 Fig. 7 N/Nc [ ] Relation between grinding rate and relative rotational speed Grinding rate [1/s].5.4.3.2.1 6L rocking mill 3L rocking mill 6L tumbling mill 3L tumbling mill Fig. 8 EWN/Nc.8Nc 1.Nc 1.Nc Fig. 9 1 2 3 4 5 Specific impact energy [J/kg/s] Relation between grinding rate and specific impact energy 3L 3L 27-
Eq. 12 Fig.9 1 2 3 4 53 6L 1 Fig. 1.8Nc Table 5 1.2kg Specific impact energy [J/(s kg)] 4 3 2 1 Fig. 1 Table 5 1 2 3 4 Ball diameter [mm] Relation between ball diameter and specific impact energy Calculation conditions Ball diameter 1mm 15mm 2mm 3mm Number of balls 2656 787 332 984 2 Fig. 11.6Nc.8Nc Specific impact energy [J/(s kg)] 4 3 2 1 Fig. 11 1 2 3 4 Height of fin [mm].6nc.8nc Relation between the height of fin and specific impact energy 27-
.8Nc5mm 1) P.A. Cundall and O.D.L. Strack, Geotechnique, 29, 47 (1979). 2) B. K. Mishra and R. K. Rajamani, KONA, 8, 92 (199). 3) H. Ryu, H. Hashimoto, F. Saito and R.Watanabe, Shigen-to-Sozai, 18, 549 (1992). 4) P. W. Cleary, Minerals Engineering, 11, 161 (1998). 5) R. K. Rajamani, B. K. Mishra, R. Venugopal and A.Datta, Powder Technology, 19, 15 (2). 6) A.Datta, B. K. Mishra, and R. K. Rajamani, Canadian 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 (21). 8) J. Kano, N. Chujo and F. Saito, Advanced Powder Technology, 8, 39 (1997). 9),,, (1998), p.74. PROFILE Makio KIMURA Tomonari KOBAYASHI Masayuki NARUMI 27-