Ball wear and ball size distributions in tumbling ball mills_Austin

Powder Technology, 41 (1965) 279 - 286

Ball

Wear and Ball Size Distributions in Tumbling Ball Mills

The Pennsylunnrn State University, University Park, PA (US A_) R. R. KLIMPEL

The Dow Chemical Company, Midhnd. MI (USA.)

(Received Octaber 17.1983; ,n revised form May 15, 1984)

SUMMARY

The theory of the calculation of the size drstributzon of the equdtbrrum mixture of balls in a ball mtll is developed The differen- teal equatron is solved for wear laws of the form

wear rate per bail = r2+ A

where r IS ball radrus A = 0 gives the Bond wear law and A = 1 giues the Dams wear law. Methods of determining A are illustrated- Experimental data are presented which show that A = 0 for some cases of wet mulling, A = I for the two mills reported by Dauis, and A = 2 for a case of wet milling. The reason for this wide divergence is not known-

INTRODUCTION

The rates of ball wear m a ball null are of unportance for three major reasons. Fust, one of the major unsolved problems in the optunrzation of ball mrll design IS the choice of the ball size m the mill. To construct a design simulation model for a ball mfl [l, 2, 31 which can be used to predict optimum ball mixture, it is necessary to know not only the effect on breakage of different ball mixtures, but also the equilibrium ball nuxture in the mill. For wet ball mills, the equilibnum size distribution of balls in the miU is a function of the make-up balls added to the mill and the rate of wear of the balls. Second, economic studies of gnnding processes [S] show that steel loss of media and liners during grindmg is a substantial haction of the total cost of gnndmg. Third, in order for a mill to produce at a steady optimum rate, it is desirable to start a new mill charge with a ball size

oo32-5910/85i.s3.30

distribution close to the equrhbrium mixture of balls produced by natural wear, with addi- tion of make-up balls to give a correct eqmhb- rium ball mix.

Partial treatments of the mathematics of ball wear were given by Davis [4] and Bond [ 53. This paper extends their treatments and grves several examples of wear laws and ball srze distnbutions determined from plant data.

ABRASION TESTS

The abrasiveness of a particular material is often determined by some form of an em- prncal abrasion test [S, 71 Drfferent manufacturers have developed their own tests, and some users also have developed tests specific to their particular needs. A discussion and comparison of such tests IS beyond the scope of this paper, especrally smce there is little mformation on tests on the same material in different abrasion testers. A typical test rs that developed by Bond [5] from an ongmal test reported by P. Crush. TO quote [ 5]-

“A flat paddle 3” X 1” X l/4”, of SAE 4325 chrome-nrckel-molybdenum steel

hardened to 500 Bnneh, is inserted for one inch into a rotor 4 5 Inches in diameter, which rotates on a horizontal shaft at 632 through fallmg ore particles_ Two square inches of paddle surface are exposed to abrasion, and the paddle tip, with a radius of 4.25 in., has a linear speed of 1410 feet per minute sufficient for a good rmpact blow.

The rotor is enclosed by a concentric drum 12 inches in drameter and 4.5 inches deep, which rotates at 70 rpm, or 90% of cntical speed, m the same direction as the paddle

2SO

The inner crrcumference of the drum 1s lined with perforated steel plate to furnish a rough surface for continuously elevating the ore particles and showenng them through the path of the rotating paddle. In operation, screened particles passing 3/4 inch square and retained on l/2 mch square are used as feed Four hundred grams of 314” X 112” feed are placed in the drum, the end cover is attached, and abrasion is continued for 15 mmutes, then the drum is emptied, another 400 grams are added, and the abrasion continued. In each complete test four 400 gram samples are each

abraded for 15 minutes Thus the paddle IS abraded for a total of one hour, after which it is weighed to the tenth of a milligram. The loss of weight in grams is the abrasion indes A, of the material_”

Based on averages of large numbers of tests compared with collected plant experience, Bond gave the following average wear loss formcll-_, presumably for typical steels for ball dnd liners:

Wet ball mills Balls.

kg/kWh = 0_16(A, - 0.015)1-3

Lmers- (I)

kg/kWh = 0_012(A, - 0 015)’ 3 ₍₂₎ Dry ball mills (grate discharge)

Balls.

kg/kWh = 0_023A,‘-’ ₍₃₎

Liners:

kg/kWh = 0.0023Aio-5 ₍₄₎

Table 1 gives average abrasion indices for a number of materials [ 7]_

This type of test has several drawbacks. The typical appearance of balls from a dry grindmg ball mill shows surface scratches, indicating wear by abrasion [ 5]_ Balls from wet grinding operations are smoother but pitted, indicating the role of corrosion in metal loss There 1s little doubt that micro- surfaces formed by abrasion under mechamcal stress are highly reactive until the chemical bonds at the surface have been stabilized by reaction with the grinding fluid [8]_ It is expected, therefore, that metal wear rates in wet grinding would be highly variable com- pared with a dry abrasion test, depending on the corrosive (electrochemical) properties of the system [ 91 Since no standard deviations were reported for eqns. (1) - (4), it 1s not

TABLE 1

Abrasion Index averages [ 7 J

Material Specific gra-ity _*i

Dolomite 2.7 0.016 Shale 2.62 O-021 L.S. for cement 2.7 0 024 Llmestone 25 0 032 Cement cbnker 3 15 0 071 Magnesite 30 0 078 Heavy sulfides 3 56 0 128 Copper ore 2 95 0.147 Hematite 4 17 0 165 Magnetite 3.7 0.222 Gravel 2 68 0.283 Trap rock 2 80 0 364 Gramte 2 72 0 388 Taconite 3.37 O-624 Quartzlte 27 0.775 Alumina 3.9 0 891

possible to place error limits on their use_ In addition, the abrasion test does not give in- formation on the ball wear laws, so the abra- sion mdex cannot be used to predict the eqmlibrmm distnbution of ball sizes unless the wear law is known.

Another type of test involves the measure- ment of weight loss of a ball charge m a laboratory or pilot-scale mill, under condi- tions comparable with those expected in the full-scale mill. Under some circumstances, data from this type of test can be used to ob- tam the wear law, as discussed below

BALL WEAR AND BALL SIZE DISTRIBUTIONS:

THEORY

In order to solve the problem of choosing the best mixture of make-up balls to add to a mill, it is necessary to consider the process of wear and the establishment of a pseudo steady-state (equilibrium) mix of ball diam- eters in a miu. The treatment by Bond [lo] makes two major assumptions. (i) that the wear rate of a ball is proportional to its SLE- face area, and (ii) that bail makeup consists only of a single large size of ball The forrnula- tion gwen below extends this treatment to allow for other cases of wear laws and ball additions.

In ball wear, there is no problem m drstin- guishing between balls and the wear powder, so the wear products can he considered

simply as mass lost from the ball charge. Consider unit mass of balls in the mill, con- taining a total number of balls of NT, with a cumulative fractional number size distribution of N(r), r berg ball radius. Let nr be the number rate of addition of fresh balls per unit time, with a cumulative fractional number size distribution of n(r). Consider balls of size r to r + dr m the steady-state charge. A steady state number balance on this size interval is ‘number rate of balls entering by wear of size r + dr balls + number rate of addition of make-up balls of this size range = number rate of balls of size r wearing out of the interval’ or ‘number rate of balls wearmg out of the size intervaI (passing through size r) = number of rate of addition of make-up balls of all larger radii’

The number of balls wearmg out of the interval in time df mcludes all balls between r and r + dr where dr 1s defined by

-(4xr’p,) dr

= (rate of wear of each ball) dt = f(r) dt

where f(r) is mass per unit time. The number rate of balls weanng out is thus

=N

cWr)

f(r)

-~

T dr 47i+p,

(5)

Equating to the total number rate of addition of make-up of all larger sizes, n,[l - n(r)],

nT[l --n(r)] =NT

giving

cwr)

dr _NT f(r) (6)

This is the basic drfferential equation definmg the distnbutron of ball s=es N(r), with boundary conditions of N(r,,) = 0 and iWIll,, ) = 1, where r,,.,,,, is the minimum size

of ball which can exist m the mill and r,,,,, is the maxim urn size of baJl added. The equa- tion u-nplicitly defines the relation between ball addition rate nr and the wear rate. Exper- imental measurements of N(r), NT and nT for a known addition of balls n(r) enables f(r) to be calculated.

The difference

in

rate of conversion of mass to powder in the element 4 4 = nr[l --h(r)] -(r + &)3& --m3pP, 3 3

1

rate of conversion of mass to powder in the element

d.Mr)

= NTf@) 7

(8)

Equating eqns. (7) and (8) mves an alternative derivation of eqn. (6).

The relation between cumulative mass frac- tion M(r) in the charge and number fraction N(r) is

dM(r) = ~&&,~&~(r)

Also, mass and number fraction in the make- up are related by

where m(r) is the cumulative mass fraction of balls less than size r in the make-up and mr is the mass rate of make-up (per unit mass of balls). These convert eqn_ (6) to

w(r)

_f(r)

A convenient method of analysis is to assume that the variation of f(r) with r can be approximated by a power function r2+ 4

wezrate=

f(r)=Kpb4m2*P

or

f(r) =

where fI can be positive or negative. Since

We= ratI2 1s -d(47+&/3)/dt, Kpb4irr2+* =

292

dzstcnce per unit time (-dr/dt) is KT* (for A = 0 the wear distance per unit time = K). Equa-

tion (6a) becomes

AI(r) = (m,K/K) i [l - n(r)]r3-a dr (11)

If the make-up IS m defmite sizes of balls of ~1, r2, - -, rk, --- rm, and if the sizes are

ordered rmav =r,>r,...>r,> _ >r,>

rmm, and mk 1s the werght fraction of make-up of size rL, eqn. (9) becomes

K = znz,/rk3 ₍₉₄

I;

Slmrlarly, 12~ IS the number fraction of balls of size rk, where nh = (mk/rk3)/K Then

n(r) = I:+n

m--l +nk, _ +nk r,<r<r,_,

r > rm Equation (11) is then readily integrated to

+ (rm-14--a-rm4-A)(1 -rzn,)

-F _{___ (r} 4--1_

rk 3-A 1

x (1 -n, + -- +nk-,)]

rk+LGr<rh m> k> 1 (12)

and mT follows from eqns. (9a) and (12) using M(r,) = 1. In general, the choice of make-up ball size dlstrlbutmn n(r) to give the approach to a desired M(r) is a trial-and-error calcula- tion whrch requ~es a knowledge of K and A

For a szngle sEe of make-up ball n(r) = 0 for r < rmar and

M(r) = [ ,2:),1 (r4-- - rmm4-A)

Since M(r,,,) = 1, K = 1/r,,,r3,

(4 - A)IL~~,<’ (4 - A)Kr,,,A-l

mT = =

rmav 4-l_ rmln 4-A 1 - (rm~/rma-r)4-A

and (13) da-*-d 4-A M(d) = d mu-, 4-A-d 4-A ma-x mm (14)

where d = 2r. For the Bond assumption that wear IS proportional to the surface area, A = 0. For some reason, Bond approximated eqn. (14) by M(d) = (d/d,,,)3 ‘. Note that mr is the fractzon of ball charge replaced per unit time If (rmJrmax )4-A is taken as neghglble compared vvlth 1, the value of mT is the minimum wear rate: physrcally this means that the fraction of mT due to balls passmg out through the discharge or ball-retammg grate, (rmln/rmax)3, 1s neghglble.

For make-up of balls consisting of two ball sizes dl and dz (d, = d,,,), of mass fraction m, and m2, eqn (14) becomes

1 da-A -d,,4-- K,d,4-A + (1 - Kl)d,m4-A d ,,Gd<d, M(d) = (144 K,d4-* + (1 - K1)d14-A-d,m4-P + (1 - K,)d,4-A -d,,4-A where

K-1&+

(z)(z)‘]

and thus iies between 0 and 1.0

BALL SIZE DISTRIBUTION AND WEAR TESTS

There are three major types of test which can be used to estimate the ball size distribu- tron and wear law for ball mills. The most drrect 1s dumping of the mill charge and siz- mg of the balls Allred with mill records of darly make-up, the wear law and wear rates can be obtamed. The second type of test mvolves markmg of a number of balls of a given size in some way, and stopping the mrll after a period of grinding to find the marked balls, which are then weighed and calipered. Thus 1s repeated to determme the change in ball dimension as a function of g-rindmg time. The third method mvolves dumping of the mill charge after some suitable period of grinding of the startmg ball charge, without make up, and weighing of the various ball size fractions. Since the startmg charge 1s made up

of discrete sizes, not a continuum, separation into the various hactions is strqhtforward and the weight loss of each size can be deter- mmed. Examples of the three methods are given below

EXPERIMENTAL DATA

Davis [4] gave data from a Hardmge conical wet mill and a 6 ft dia. by 8 ft dry ball mill which indicated A = 1, correspondmg to a wear law of ‘wear rate 0~ ball weight, d 3r_ Using method 2, Lorenzetti [ll] has found that eqn. (10) unth A = 0 is a good assump- tion for wet milling and he gives K values rang- mg from 3.8 pm/h for a relatively soft iron ore to 15 4 pm/h for an extremely hard copper ore, for Armco Moly-Cop balls.

On the other hand we have analyzed data from 2 wet industrial ball mill where make-up was of a single size of ball Table 2 gives the milling conditions. Figure 1 shows the cumulative weight fraction of balls versus ball diameter, as measured by emptying the mill contents and counting and sizing balls using calipers Two different srzes of make-up balls were tested. It appears that A = 2 in this case, smce the data agreed reasonably well with the form of eqn. (14) with A = 2, that is, wear rate in mm/h a r4_ The values of K obtained from eqn. (13) using this value of A were (7 6) (10m6) mm-’ h-i for the equdibrium ball charge ongmating from 100 mm make-up balls and (12.3) (10m6) mm-’ h-i for that originating from 75 mm make-up balls. It should be noted that if the smaller balls formed by wear were softer because of the loss of a hardened outer layer then they would wear faster, giving A < 0.

TABLE 2

Ball wear data on 4 3 m dia. by 5 m long wet over- flow mrll grmdmg abrasive inorganic material

Type of ball steel, specific

gravity 8 5,600 Brine11 hardness Diameter of make-up ball, mm 100 75

Media weight, t 110 110

Daily addition, t 2.0 2.7

Daily throughput, t 1600 1480

Steel per ton throughput, kg/t 1 25 1.82

Steel loss, kg/kWh 0 068 0.098

ocz

001 I I I I ,llll

10 10 40 m 100

aall owmerer mm

Fig. 1. Cumulative ball size distribution at steady state (see Table 2)_

In this test it appeared that the wear of larger balls was much faster wrth respect to smaller balls than predicted by the Bond ex- pression, leading to a much flatter ball size distribution. It also appeared that wear rates were faster in the equilibrium charge of smaller balls. For example, the lmear wear rate of 75 mm balls was 11 pm/h for the larger ball mix and 17 pm/h for the smaller, corresponding to wear rates of 1.6 and 2 6 g per ball per hour. This is perhaps due to the greater number of ball-ball collisions per unit tune for the smaller ball mrx The total num- ber of balls per unit mass of charge, NT, IS given by

1 4-A dmax'-d-dmml-A

~Pb"&= 1-_a d _{4-A-d 4-A} (15)

max mln

For A = 2, the ratio of NT for d,,, = 75 mm to NT for d,,, = 100 mm, assummg d,, = 12 mm, is 1.7: the wear ratio of 2.6 to 1.5 is = 17 also.

The method of followmg ball wear with time is illustrated by the followmg data. Two

hundred and twenty balls (1 toil) were tagged (l-5/100):24 = 0.6 X 10m3_ Smce this data by drilling a 6 3 mm dia. hole into each bail fellows the Bond wear law, the kg/kWh of and flllmg the hole with low melting point Table 3 can be compared with the value allow Using eqn. (lo), integration of predicted for quartzite from Table 1, that IS, --d(47rr3p,/3)/dt = dp,,4rr ’ + p gives 0 062 with O-C20 kg/kWh.

Vermeulen et al [12] have also reported marked bail wear tests on 60 mm dra. balls in a 2.4 m X 2.4 m (8 ft X 8 ft) rubber-lined ball mill running a; 85% of cntical speed with a ball charge of 22 ton (metric), grinding about 65C t per day of an ore. Samples of 10 of the marked balls were weighed at various time intervals corresponding to known tonnages milled. Equation (16) for A = 0 or 1 can be put as 1-A _{= (1 - 4)H} 1-4 A# I.0 r0 (16) h-l

5 =

where r. IS the initial size of the ball at tune t = 0, which was 50 mm. For a continuous overflow discharge mill of 4 m id. by 4 8 m long, wet grmdmg a hard gold ore at over 1GOO t per day (see Table 3), the varration of the radius of balls as a function of grinding time is shown in Pig 2. It is clear that this result is consistent with A = 0. the loss of OS of ball radius represents a loss of half of the ball weight_ and gives a wear rate K of about 19 pm/h This value in eqn. (13) gives a predicted mmlmum fractional wear rate of the total charge of mr = 1.44 X 10m3 fraction per hour, assummg that ~rmm,kma,J4 < 1.0. However_ the actual fractional wear rate was

TABLE 3

Ball wear Jata on 4 m dia by 4 8 m long wet over- rlow ml!1 grIndIng gold ore

Tyue or ball steel, specific gra\1ty 5’i5 Brmell hardness

Diameter of make-up ball. mm 100

Dally addltlon, t 1.5

Dal!y throughput. t 1100 - 1200 Steel per ton throughput, kg/t 1 3 Steel loss, hg/kWh 0.062

TIME IN MILL days

Fig. 2 Change of ball dimension with time in *mill, r,, = 50 mm (see Table 3) II3 r z _ = 1-E A = 1.0 r0 r0

IAL =

=

3ln(mlmd

where m is the mass of balls at time t, with the effective grinding time being propor- tional to the amount of ore milled Analysis of therr data shows that A = 0 fits the results in a similar manner to that of Fig. 2, see Fig 3, but with time about 15 times as long, giving K = 0.7 pm/h_

Vermeulen et al claimed that the fit

of data to A = 1.0 was as good as that of A = 0, based on a comparison of correla- tion coeffxients, and therefore rt was not pos-

077

TONS MILLED =lO-=

Fig_ 3. Change in ball diameter with tons milled (650 t/d), r0 = 30 mm, data of Vermeulen et al [12]

sible to dlstingulsh between A = 0 and A = 1.0. However, they appeared +a have-per- formed the balysis cm (r + error) in one case and ln[r3 + error] in the other, which is clear- ly not statistically correct. Equation (16) can be put as

r/r0 = 1 - z = fl(t) A=0

r/r0 = exp(-Kct) = f2(t) A=1

The sums of squares of [(r/ro)(expt) - (r/r,) modeilZ are 0.00053 for A = 0 and more than double this at 0.00113 for A = 1, clearly indicating the better fit :>i the model vvlth A = 0 More precise statistical analysis would require rephcated test data.

The value of K for A = 0 gives a predicted mmimum fractlonal wear rate of the toti charge of 0.9 X 10m4 fraction per hour, where- as the actual mean value was about 1.5 X 1W4 fraction per hour. Measurements of the balls and coarse metal fragments discharged from the mill with badly worn discharge g-rates gave a value of about U 4 X lo-” fraction per hour, thus giving a comparison of minimum weaT rate of 1.1 X 10m4 fraction ped hour for the direct expeiimensal data to 0.9 X 10e4 predlcted &o,n the K vzlue. The mean

measured size of discharge material was sven as an equivalent diameter of 28 mm.

As an example of the third method qf de- terminmg ball wear, typlcal data from a pilot- scale mill might be as follows. A charge of 380 kg of 60.8 mm (2 in) balls a;ld 120 kg of 25.4 mm (1 in) balls was tumbled for 48 h. The loss in weight of 50 8 mm bJls was 10 kg and that of 25.4 mm was 7.9 kg. The relation between weight loss and change in ball radius is

= 1 - fra.ctlonal weight loss

This gives r/r0 = 0.991 for the 50 8 mm balls

and 0.978 for the 25 4 mm. Equation (16) eves the relation between r/r0 for two sizes as

1 -

(rllrolY-*

0

(17)

ln(r,,/r,)/ln(r,,/r,) = 1-O A=1

The data g-we the following values of the left- hand side of this equation- 1.2 for A = 0, 2 5 for A = 1, 5.0 for A = 2. Thus the results

are consistent unth A = 0 and the hnear wear rate is 11.2 Pm/h.

DISCUSSION AND CONCLUSIONS

The situation is clearly confused The data given by Davis [4] give A = 1, whereas Loren- zetti [ll] reports A = 0; the Vermeulen et al [12] data and one of O-Z data sets for a large 1nil.l also give A = 0. The other data set gives A = 2_ Discussions with plant superintendents m the South tican gold fields (for example, Ref. 13) suggest that industrial experience is that equihbrium ball charges do not contam as big a diction of balls near to the make-up ball size as predicted by the Bond wear law, A = 0 It IS possible that an inconsistency exists between the taggmg method of deter- mining ball wear and the method of dumpmg or samphng the ball charge. The wear predicted from the tagged ball test for the Vermeulen et al. data was in rough accord urlth the required make-up If allowance was made for discharge of 28 mm equivalent diameter material. However, for the data set of Fig. 2, the predicted minimum wear rate was more than twice that actually measured_ To resolve this problem, it would be an advantage if industnal experience on dumped charges were openly reported along with the grindmg con dltlons leadmg to the ball size distnbrltlon. The tlsatment of the unsteady state change from a starting ball size distnbutlon to the equlllbnum ball size distribution lvlll be treated m a later publication

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