TECHNICAL NOTES 8 GRINDING R. P. King - …

1 Copyright R P king 2000 TECHNICAL NOTES 8 GRINDINGR. P. King8-2&ROOLVLRQ5 ROOLQJZLWKQLSSLQJF igure Different types of GRINDING action by thegrinding P EOL QJ6 KRXOGHURIWKHORDG*UDYLWDWLRQDOIRUFH)6 WDJQDQ W]RQHJ&HQWULIXJDOIRUFH)FJ,PSDFW] RQH7 RHRIWK HOR DG&DWDUDFWLQJ)FRVF igure Media motion in the tumbling GRINDING actionIndustrial GRINDING machines used in the mineralprocessing industries are mostly of the tumbling milltype. These mills exist in a variety of types - rod, ball,pebble autogenous and semi-autogenous. The grindingaction is induced by relative motion between theparticles of media - the rods, balls or pebbles. Thismotion can be characterized as collision with breakageinduced primarily by impact or as rolling with breakageinduced primarily by crushing and attrition.

2 Inautogenous GRINDING machines fracture of the mediaparticles also occurs by both impact (self breakage) andattrition. The relative motion of the media is determined by the tumbling action which in turn is quite strongly influenced by theliners and lifters that are always fixed inside the shell of the mill. Liners and lifters have two main protect the outer shell of the mill from wear - liners are renewable. prevent slipping between the medium and slurry charge in the mill and the mill shell. Slippage willconsume energy wastefully but more importantly it will reduce the ability of the mill shell to transmit energy tothe tumbling charge. This energy is required to cause GRINDING of the material in the mill. The shape anddimensions of the lifters control the tumbling action of the tumbling action is difficult to describe accurately but certain regions in the mill can be characterized in terms ofthe basic pattern of motion of material in the mill.

3 The motion of an individual ball in the charge iscomplicated in practice and it is not possible to calculatethe path taken by a particular particle during station ofthe charge. However the general pattern of the motionof the media can be simulated by discrete elementmethods which provide valuable information about thedynamic conditions inside the of the terms that are often used to describe themotion of the media in a tumbling mill are shown inFigure Critical speed of rotationThe force balance on a particle against the wall is given by 8-3 Centrifugal force outwardFc mp&2Dm2( )& is the angular velocity, mp is the mass of any particle (media or charge) in the mill and Dm is the diameter of the millinside the forceFg mpg( )The particle will remain against the wall if these two forces are in balance Fgcos ( )

4 Where is shown in Figure Thus a particle will separate from the wall at the point wherecos FcFg( )The critical speed of the mill, &c, is defined as the speed at which a single ball will just remain against the wall for a fullcycle. At the top of the cycle =0 andFc Fg( )mp&2cDm2 mpg( )&c 2gDm1/2( )The critical speed is usually expressed in terms of the number of revolutions per secondNc &c2 12 2gDm1/2 (2 )1/22 D1/2m ( )8-40J&HQWHURIJUDYLW\RIWKHFKDUJHG)U DFW LRQ-FFFigure Simplified calculation of the torque requiredto turn a mill. RIFULWLFDOVSHHG 3 RZHU Figure The effect of mill speed on the power drawnby a rotating liner profile and the stickiness of the pulp in the millcan have a significant effect on the actual criticalvelocity.

5 Mills usually operate in the range 65 - 82% of criticalbut values as high as 90% are sometimes crucial parameter that defines the performance of amill is the energy consumption. The power supplied tothe mill is used primarily to lift the load (medium andcharge). Additional power is required to keep the Power drawn by ball, semi-autogenous andautogenous mills A simplified picture of the mill load is shown in Ad this can be used to establish the essential featuresof a model for mill torque required to turn the mill is given byTorque T Mcgdc Tf( )Where Mc is the total mass of the charge in the mill and Tf is the torque required to overcome 2 NT( )For mills of different diameter but running at the samefraction of critical speed and having the same fractionalfillingNet power 2 NMcdc.

6 NcMcdc .. ( )The exponent on Dm has been variously reported tohave values as low as and as high as 10203040506070% loading in the mill34567 Loading factor K1 Grate discharge DryGrate discharge WetOverflow discharge WetFigure Effect of mill filling on power draft for ballmills. The data is taken from Rexnord Process MachineryReference Manual, Rexnord Process Machinery Division,Milwaukee, 1976 The effect of varying mill speed on the power drawn by the mill is shown graphically in Figure speed of rotation of the mill influences the power draft through two effects: the value of N and the shift in thecenter of gravity with speed. The center of gravity first starts to shift to the left as the speed of rotation increases butas critical speed is reached the center of gravity moves towards the center of the mill as more and more of the materialis held against the shell throughout the cycle.

7 Since critical speed is larger at smaller radii the centrifuging layer getsthicker and thicker as the speed increases until the entire charge is centrifuging and the net power draw is effect of mill charge is primarily through the shifting of the center of gravity and the mass of the charge. As thecharge increases the center of gravity moves inward. The power draft is more or less symmetrical about the 50% simple equation for calculating net power draft isP ( )Kl is the loading factor which can be obtained fromFigures for the popular mill types. 3c is the millspeed measured as a fraction of the critical reliable models for the prediction of the powerdrawn by ball, semi-autogenous and fully autogenousmills have been developed by Morrell and by Austin.

8 (Morrell, S. Power draw of wet tumbling mills and itsrelationship to charge dynamics - Part 2: An empiricalapproach to modeling of mill power draw. Trans. Metall. (Sect C:Mineral Processing ExtrMetall.) 105, January-April 1996 ppC54-C62. AustinLG A mill power equation for SAG mills. Minerals andMetallurgical Processing. Feb 1990 power No load power Net power drawn by the charge( )The net power is calculated fromNet power ! ( )In equation , D is the diameter inside the mill liners and Le is the effective length of the mill including the conicalends. !c is the specific gravity of the charge and . and / are factors that account for the fractional filling and the speedof rotation respectively. K is a calibration constant that varies with the type of discharge.)

9 For overflow mills K = for grate mills K = This difference is ascribed to the presence of a pool of slurry that is present on the bottomof overflow-discharge mills but not to the same extent in grate-discharge mills. This pool is situated more or lesssymmetrically with respect to the axis of the mill and therefore does not draw significant power. Austin recommendsK = for overflow semi-autogenous mills. A value of K = makes Austin s formula agree with Morrell s dataas shown in Figure Geometry of a mill with cylindrical ends. Alldimensions are inside liners. Lc = centerline length. L =belly length. Dm = mill diameter. Dt = no-load power accounts for all frictional and mechanical losses in the drive system of the mill and can be calculatedfromNo load power [3c( L)] ( )Ld is the mean length of the conical ends and is calculated as half the difference between the center-line length of themill and the length of the cylindrical geometry of a mill with conical ends is shown inFigure The total volume inside the mill is given byVm 4D2mL1 2(Lc L)L1 (Dt/Dm)31 Dt/Dm( )

10 The density of the charge must account for all of thematerial in the mill including the media which may besteel balls in a ball mill, or large lumps of ore in anautogenous mill or a mixture in a semi-autogenous mill,as well as the slurry that makes up the operating Jt be the fraction of the mill volume that is occupiedby the total charge, Jb the fraction of the mill volume thatis occupied by steel balls and E the voidage of the ballsand media. U is the fraction of the voidage that is filledby slurry. 3v is the volume fraction of solids in the slurry. Let VB be the volume of steel balls in the mill, VMed be thevolume of autogenous media and VS the volume of Jb(1 E)VmVS JtUEVmVMed (Jt Jb)(1 E)Vm( )The charge density is calculated from!