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Modelling and simulation of ball mill wear Article in Wear · February 1993 DOI: 10.1016/0043-1648(93)90435-O
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Wear, 160 (1993) 309-316
Modelling and simulation of ball mill wear P. Radziszewski” and S. Tarasiewiczb ‘Dkpartement des sciences appliqu&es, Universite’ du Qu&bec en Abitibi-TPmbcamingue, 42, Rue A4gr Rhdaume Est, C.P. 700, Rouyn-Norandn, Qut!. J9X 5E4 (Canada) ‘~~~~ent de g&ie mkanique, Universite’ Lava& Ste-Fey, Qw’. GlK 7P4 (Canada)
(Received May 27, 1992; revised and accepted July 23, 1992)
Abstract In the mineral processing industry, ball mills are used to reduce ore from one size distribution to another. Ball mill wear occurs as a result of the violent interactions within the ball charge. In the present article, a mathematical description of wear has been added to a ball charge motion model. Wear is associated with the comminution mechanisms found in the ball charge profile. It is assumed that ball mill wear occurs in each of three comminution zones, with adhesive wear found in the crushing and tumbling zones and abrasive wear arising in the grinding zone. Wear rates are proportional to the energy dissipated in these zones as predicted by the charge motion model. Three laboratory case studies are investigated followed by one full-scale example in a wet milling context. Simulation results are compared with experimental data from which real-system predictions are presented. The use of these results provides the possibility of optimizing ball mill performance while including the wear rate factor.
1. Introduction Ball mills play an important role in the mineral processing industry of today. Yet as such, they are difficult to study given the closed nature of the operation. Once operating, the ball mill becomes a black box where only input and output materials can be observed, as well as the power draw of the grinding system. As a result of the highly stochastic nature of the ball mill environment, most ball mill studies to date have been experimental. This had led to the development of empirical and phenomenological ball mill models. Models of this type have limited utility due to inadequate representation of the dynamic interaction and the effect of ball mill elements on ball mill operation and performance. In the field of wear, as observed in ballmills, a number of studies have been completed. Some studies present extensive experimental data [l-3], while others illustrate models useful to the understanding of wear phenomena [4-8]. The goal of this work is the development of a better understanding of ball mill wear as a function of mill element interaction. The following report will limit itself to the presentation of a wear model and to the model validation using published ball wear test data, and conclude with a real-system simulated case example.
0043-1618/93/$6.00
2. Background Particle breakage, in a ball mill, is achieved by the collision of balls, or by balls sliding past one another. Both these actions continually nip and break certain quantities of material. This type of breakage is the final event in a chain of energy transfo~ations which commences with rotational energy of the ballmill transformed into the ball charge movement (Fig. 1). The charge profile shows three zones that characterize the type of action produced there, namely; the grinding zone, which is described by ball layers sliding over one another grinding the material trapped between them; the tumbling zone, which is described by balls rolling over one another and breaking the material in lowenergy impact; the crushing zone, which is described by balls in flight re-entering the ball charge and crushing the material in hip-ener~ impact.
Fig. 1. Typical ball charge action [9].
0 1993 - Elsevier Sequoia. All rights reserved
J’. Radziszewskl.
310
S. Tarasiewicz / Ball mill weal
The ball charge form, as described in Fig. 1, is dependant on the energy transferred between the ball charge and mill liners. This transfer is a function of the friction characteristics of the charge sliding on the material. These characteristics can be modified by different mill liners (Fig. 2). Solving a force balance on a single ball in the ball mill environment (Fig. 3) allows the definition of the point of flight, 4, of the ball by [lo] y
4=@-/3+arcsin
;
cos(O-p)
[
1
mblg
Fig. 4. Ball lifter effect.
(1)
Fig. 5. Charge foot stability criterion
development.
This relationship can also describe the ball lifter effect (Fig. 4), where other balls can act as lifters [ll, 121. When applied to the ball charge, eqn. (1) describes two different cases: (i) the angular position of the center of mass of a layer of balls in slippage (I_L< 1.0) C$= arctan@.,) - p + arcsin
%-Abw2R
cos(arctan(p.,)
g
- p)
1
(ii) the angular point of flight for a ball (ILz 1.0)
Fig. 6. Charge discretization.
C#J = arctan&)
- p
+ arcsin $ [
cos(arctan(p,)
- /3) I
(3)
where p = tan arctan&) [
Fig. 2. Typical ball mill liners [9].
- p+ T 2
1
(4)
and Ao is the slippage speed relative to the mill center [ll-131. A ball in the ball mill may follow one of two general trajectories depending on whether it is in the charge or in flight. The trajectory of the ball in the charge is circular, while for the ball in flight the trajectory is parabolic. Before this description of ball charge motion can be solved, the charge foot stability must be determined by a moment calculation (Fig. 5). (5)
Fig. 3. Point of flight for a single ball [lo].
The solution of eqns. (2)-(5) is aided by a discretization of the charge at rest (Fig. 6) [ll, 141. Thus each charge element can be defined in time and space by its center on mass. Using the principal of momentum conservation, it is assumed that during flight, ball charge element mass center trajectories are not affected by other similar trajectories in flight elements.
P, Rm%smvski, S. Tarasiewicz I Balf mill wear
Having thus defined ball charge motion, it now becomes possible to define and calculate the following energies:
311
of wear as a function of the energy rate consumed in wear. The mass rate of wear for these models is developed by multiplying the metal density with the volume rate of wear, resulting in the following equations [ 171: adhesive wear
abrasive wear (I31
With these energies, it is possible to solve the following energy balance [ll, 151 which describes the energy distributed in the ball charge of Fig. 1. -Lsumed
=Egrinding
+
ztumbling
+
Ecrushing
+ timarerial
(9)
Together, these relationships provide the possibility of simulating ball charge movement in a ball mill. The results of this simulation is a function of mill size, ball charge volume, lifter profile and mill rotation speed. These relationships also allow the possibility of determining the amount of energy consumed and distributed in the ball charge as a function of the variables mentioned.
It follows that if a rate of energy is dissipated in a given comminution zone of the ball mill (Fig. l), then the associated wear in that zone can be described by eqns. (12) and (13). Depending on the wear phenomena in that particular zone as well as the energy rate distributed there as determined from eqns. (6)-(8), the associated wear rate can be determined. Assuming that in the grinding zone wear is primarily abrasive, and in the tumbling and crushing zone wear is prima~ly adhesive, the ball mill wear model becomes a function of the energy rates calculated using the ball charge motion model. These wear rates are functions of ball mill physical and operating parameters. The model can now be written as
3. Wear model Even though there are many wear mechanisms (adhesive, abrasive, corrosive, surface fatigue) occurring in a ball mill, only adhesive and abrasive wear is considered in this development. These wear mechanisms can be modeled using [16]: adhesive wear
v= PFX 3Hr
L
s
(10)
&t/tot = &wb + fkfw
(15)
are wear rates for mill lifters, where: fistfw, f?I,*, fisytoc ball charge and total wear (kg s- ‘); &,, & are energy rates distributed in grinding on mill lifters and in the ball charge (kW); .&,, 8,,, are energy rates distributed in the crushing and grinding zones (kW). The liner wear rate can be determined from realsystem data using the following relationship:
abrasive wear
(17) where: V is the volume of metal worn (m’); F,_ is the load force (N); x, is the sliding distance (m); EI, is metal hardness (N m-“); P is the probability of adhesion [l]; 8 (abrasion factor) is the angular representation of a conix abrasive grain (“). After taking the product of load FL and sliding distance x,, the total energy, E, used to remove worn metal of volume V can be determined. The time derivative of the resulting equation gives the volume rate
where: P, is the average liner fraction worn [l]; A is the original liner profile area (m’); L, is mill interior length (m); n is the number of ball mill lifters [l]; T is the lifter life span (s).
4. Laboratory ball wear simulation In order to validate the wear model, a series of simulation tests were completed using published ex-
312
I? Radziszewski,
S. Tarasiewicz
perimental data [18, 191. The objective of these simulation experiments was to determine the applicability of the wear model to reproduce laboratory data as a function of ball and material hardness as well as grinding (wet/dry) conditions. Liner wear in these experiments is considered to be negligible. Table 1 gives the laboratory physical specifications. Figure 7 illustrates the simulated ball charge profile for dry grinding conditions. Table 2 shows the energy calculated in grinding for three simulated grinding conditions. After a back-calculation (Table 3) from dry grinding conditions with quartize, wear model parameters for abrasive grinding wear were determined. By keeping these parameters constant for changing grinding conditions, it was possible to compare laboratory and simulated wearrates as a function of material and grinding conditions (Fig. 8). At this point, it should be noted that adhesive wear was considered to be negligible in the laboratory test data used because the calculated energy level in tumbling and crushing (zones associated with adhesive wear) were quite low or non-existent. TABLE
1. Laboratory
specifications
6S.11
Fig. 7. Simulated
TABLE
ball charge profile.
2. Grinding
Grinding condition
DV Wet (water) Organic liquid
condition/grinding Friction
coefficient
Static
Dynamic
0.2 0.12 0.09
0.18 0.09 0.05
TABLE 3. Abrasion quartize)
parameter
back-calculation
(dry grinding. .-
Metal
Mild steel High carbon low alloy steel Austenite
Wear rate (g h-‘)
Hardness
Density (kg m-“)
Abrasion factor e 0
2.73 1.85
127* 505*
7800*** 7soo***
5.32 14.08
2.24
370**
7800***
12.24
z
mm-‘)
*[18]; **[20]; ***[,,I.
[18, 191 0.229 (m) 0.203 (m) =45% 70% of critical 1.15 (kg) 126 steel; 136 cast 9.0 (kg)
Ball mill length Ball mill diameter Mill filling Rotation speed Mineral charge Ball charge Ball mass
I Ball mill wear
energy Calculated grinding energy (W)
1.5 3.2 3.39
Fig. 8. Simulated
and real wear data.
Figure 8 shows that there is relatively good correspondence between laboratory data and simulated wear rates for changing grinding conditions. Further, it can be observed that the abrasion factor, 8, is hardnessdependent (Table 3). This is substantiated using data from ref. 18 for various metals. Plotting abrasion factors against metal hardness for the grinding of quartz illustrates this point (Fig. 9). The abrasion factor dependence on material hardness was obtained using wear data for taconite (3.5-4 Mohs scale) and mild steel balls [19]. The abrasion factor back-calculated for the wet grinding case gives 8= 1.2. Keeping in mind the development of the abrasive wear model [16], it can be hypothesized that the abrasion factor, 0, is a material-metal interface parameter. For a given metal and decreasing material hardness, 8 decreases and would possibly show shallower plowingwear phenomena, while for a given material hardness and increasing metal hardness, 8 increases, which possibly illustrates finer micro-cutting phenomena.
313
Fig. 11. Simulated
Fig. 9. Abrasion
factor as a function
TABLE 4. Ball mill specifications Length Inside diameter Operating speed Lifters No. of fifuxs Lifter profiie area Ball size Ball density Bali charge voIume Ball hardness Materiaf density (pyrochlore ore) Input tonnage Ball mill liner life span Charge and liner wear rate Operating condition
ball charge profire.
of ball hardness.
[22]
4.27 m
3.06 m (with lifters) 16.9 i-pm Shiplap (see Fig_ $1) 25 0.0325 mz 0.05 m 7800 kg me3 30% of mill voluwre 220 kg mm-2 2900 kg me3 17.5 t h-’ = 2 years (= 51iS worn) =0.04 kg per tonne production wet &=0.12; rk;=1).09f
Fig. 12. Ener@
rate distrz&ution.
ball charge; the smaller circles describe the charge elements in flight. To obtain the solution to eqn. (9), 5.325 s of real time were simulated, giving the energy rate values shown in Fig. 12. Examining the simulated energy rate results, the following eq~~t~~ns are obtained: &;,=I0
kW
&=6O kW I.$,,== 130 kW
Fig. 10. Original and iinal Iiftcr profiles
[23].
5, Real-system simulative After v~~i~~g the wear model using laboratory test data, a first attempt was made to simulate wear in an industriaf ball mill ~n~~o~rnent. A b~~k~alculatjo~ determined wear model parameters for the known operating case. From this fixed point, mill speed and charge volume were varied in order to determine their effect on ball mill wear, The specifications of the ball mill are found in Table 4 and Fig. 10. The charge form, as simulated for these operating conditions, is shown in Fig. 11. There, the ball charge is defined by the circled centers of mass of the discretized
with I$ = I,8 kW being calculated separately” Equations (17), (14), (16) and (15), in the order of their use, give the following values as a ~~~t~on of the real-system wear rate of I)fSlitot=0.00194kg s-‘:
P= 0.~74 After adjusting the model to the ball mill data, the effects of ball mill rotation speed (Fig. 13) and bail mill filling (Fig. 14) on the respective energy rate distributions were simulated. As shown, the energy rates for the two cases increase with increasing rotation speed and increasing ball milf filling.
04--‘”
GO
/
600
t 65 0
70 0
75 0
800
Rcm.T,GN
SPEED
LX Wcrl
Fig. 13. Energy rates as a function
of rotation
speed. 800
ROTATION I%
t
Fig. 15. Wear rates as a function
speed.
not appreciably increase ball mill wear. In fact, the simulation results show that total wear decreases due to lower liner wear rates, while ball charge wear increases. This may be explained by the kinetics of the ball charge. Increasing mill rotation speed results in more balls in flight more frequently. This in turn lightens the active charge load which causes lower energy rates to be distributed in the grinding zone. Energy distributed on the mill liner is also decreased as a result. The ball charge wear, on the other hand, increases due to the higher levels of energy rates distributed in the impact zones of tumbling and crushing (see Fig. 14). Figure 16 shows that wear is dependant on the ball mill filling percentage. The wear rate increases with increasing ball mill filling percentage. Interestingly, the ratio of total wear rates to mill filling percentage indicates that the per unit percentage of ball charge volume decreases with increasing ball charge volume (Table 5).
21 0
200
16 0
12 0
80
A0
0.0 -I00
Fig. 14. Energy rates as a function
of rotation
SPEED
LJcrl
of ball mill filling.
The resulting effects on wear rates are shown in Fig. 15 as a function of rotation speed and in Fig. 16 as a function of ball mill filling. Examining the simulated results shows that for this case increasing the rotation speed of the ball mill does
6. Conclusion The ball mill is both a violent and a closed operating environment where the effects of ball mill parameters
P. Radziszewski, S. Tarasiewicz I Ball mill wear
315
models for wear effects on ball charge volume, ball size distribution and lifter profiles. Such models would allow subsequent simulation studies of non-stationary behavior in ball mill grinding.
References
14.0
12 0
IO 0
80
60
LO
v
O00
20
25
30
Fig. 16. Wear rate as a function
TABLE
35
I-4
of ball mill filling.
5. Ratio of total wear to mill filling
Mill filling, Pt (%)
Total wear rate, m,, (kg s-‘)
Ratio mJP, (kg s-’ %)
20 25 30 35 40
17.0 17.5 19.4 20.2 20.8
0.85 0.70 0.65 0.58 0.52
on mill wear are difficult to measure during mill operation. The use of a ball charge motion model permits the formulation and use of a wear model to predict mill wear. This wear model integrates the abrasive and adhesive wear mechanisms as a function of energy rates used in various comminution zones of the ball charge profile. Comparing the abrasive wear model results with published laboratory data shows relatively good correspondence. Further, it was observed that the abrasion factor was a parameter which may indicate a material-metal hardness relationship. Simulation of a realsystem wear context showed the effect of mill rotation speed and charge filling on charge and liner wear rates. These findings encourage further work in developing
1 I. Iwasaki, R. L. Pozzo, K. A. Natarajan, K. Adam and J. N. Orlich, Nature of corrosion and abrasive wear in ball mill grinding, Znt. J. Miner. Process., 22 (1988) 345-360. 2 J. J. Moore, R. Perez, A. Gangopadhyay and J. K. Eggert, Factors affecting wear in tumbling mills: influence of composition and microstructure, Znt. J. Miner. Process., 22 (1988) 313-343. 3 C. H. Pitt, Y. M. Chang, M. E. Wadsworth and D. Kotlvar, Laboratory abrasion and electrochemical test methods as a means of determining mechanism and rates of corrosion and wear in ball mills, Znt. J. Miner. Process., 22 (1988) 361-380. 4 A. G. Tangena and P. J. M. Wijnhoven, Finite element calculations on the influence of surface roughness on friction, Wear, 103 (1985) 345-354. A. A. Torrance, A three-dimensional cutting criterion for abrasion, Wear, 123 (1988) 87-96. A. Zmitrowicz, A thermodynamical model of contact, friction and wear. I. Governing equations, Wear, I14 (1987) 135-168. A. Zmitrowicz, A thermodynamical model of contact, friction and wear. II. Constituative equations for materials and linearized theories, Wear, 114 (1987) 169-197. 8 A. Zmitrowicz, A thermodynamical model of contact, friction and wear. III. Constituative equations for friction, wear and frictional heat, Wear, 114 (1987) 199-221. 9 E. G. Kelly and D. J. Spottiswood, Introduction to Mineral Processing, Wiley, Toronto, 1982, pp. 140, 141. 10 R. E. McIvor, Effect of speed and liner configuration on ballmill performance, Miner. Eng., (June 1983) 617-622. 11 P. Radziszewski, Modtlisation de la comminution en fonction du concassage, du cascadage et du broyage dans un broyeur a boulets, These de maitrise, UniversitC Laval, 1986. 12 S. Tarasiewicz and P. Radziszewski, Ballmill simulation, Part I. A kinetic model for ballcharge motion, Trans. Sot. Computer Simulation, (1989) 61-73. 13 G. C. Lowrinson, Crushingand Grinding, Butterworth, London, 1974, p. 131. 14 S. Tarasiewicz and P. Radziszewski, Ballmill simulation, Part II. A numerical solution to the ballcharge model, Trans. Sot. Computer Simulation, (1989) 75-88. 15 S. Tarasiewicz and P. Radziszewski, Ballmill simulation. Part III. Energy and the breakage event, Trans. Sac. Computer Simulation, (1989) 291-306. 16 E. Rabinowicz, Friction and Wear of Materials, Wiley, Toronto, 1965, pp. 136-169. du pro&de de 17 P. Radziszewski, Etude de l’automatisation la comminution dans un broyeur a boulets, These de Doctorat, Universitd Laval, 1991. 18 A. K. Gangopadhyay and J. J. Moore, The role of abrasion and corrosion in grinding media wear, Wear, 104 (1985) 49-64. 19 I. Iwasaki, S. C. Riemer, J. N. Orlich and K A. Natarajan, Corrosive and abrasive wear in ore grinding, Wear, 103 (1985) 253-267. 20 P. C. Carman, Chemical Constitution and Properties of EngineeringMateriaZ.s,Edward Arnold, London, 1949, pp. 139-176. 21 Z. D. Jastrzebski, The Nature and ZVoperties of Engineering MatertaB, 2nd edn., Wiley, Toronto, 1976, pp. 612, 613.
P. Radziszewski, S. Tarasiewicz I Bail mill wear
316
22 Niobec data, Niobec, St-Honor& Quebec, Canada, 1986. 23 D. J. Dunn, Optimizing ball mill liners for performance and economy, Sot. Min. Eng., (Dec. 1976) 32-34.
Etumbling? g mbi
eff1 li Appendix A: Nomenclature
rci
R
db
E. consumed . E. crushing> E. CT egrinding,
4, Ematerial
Egrc
ball diameter (m) energy rate consumed energy rate distributed energy rate distributed energy rate distributed grinding (kW) energy rate used to
(kW)
vi
by the mill (kW) in crushing (kW) in grinding (kW) on liner through
0 ASPpk 43 @i
lift mill material WO
&ml
energy rate distributed in tumbling (kW) gravity (9.81 m SC’) ball mass (kg) discretized element mass (kg) normal force (N) radial position of mass center (m) Mill radius (m) element velocity (m s-r) Arctan ( pL,) static and dynamic friction factors [l] angular position of the ball in mill (radians) angular velocity of the ball relative to the mill center (radians s-l) angular velocity of the mill (radians s-l)
