Ball wear and ball size distributions in tumbling ball mills_Austin

Powder

Ball

Technology,

41 (1965)

279

279 - 286

Wear and Ball Size Distributions

in Tumbling

Ball Mills

L. G. AUSTIN The Pennsylunnrn State University,

University

Park, PA (US

A_)

R. R. KLIMPEL The Dow

Chemical

(Received

Octaber

Company, 17.1983;

Midhnd.

MI (USA.)

,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 differenteal 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 illustratedExperimental 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 addition of make-up balls to give a correct eqmhbrium 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 emprncal 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

0 Elsevier Sequoia/Printed

in The Netherlands

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 (I) Lmerskg/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 microsurfaces 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 compared 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

Dolomite Shale L.S. for cement Llmestone Cement cbnker Magnesite Heavy sulfides Copper ore Hematite Magnetite Gravel Trap rock Gramte Taconite Quartzlte Alumina

2.7 2.62 2.7 25 3 15 30 3 56 2 95 4 17 3.7 2 68 2 80 2 72 3.37 27 3.9

gra-ity

*i 0.016 O-021 0 024 0 032 0 071 0 078 0 128 0.147 0 165 0.222 0.283 0 364 0 388 O-624 0.775 0 891

possible to place error limits on their use_ In addition, the abrasion test does not give information on the ball wear laws, so the abrasion mdex cannot be used to predict the eqmlibrmm distnbution of ball sizes unless the wear law is known. Another type of test involves the measurement of weight loss of a ball charge m a laboratory or pilot-scale mill, under conditions comparable with those expected in the full-scale mill. Under some circumstances, data from this type of test can be used to obtam 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 diameters in a miu. The treatment by Bond [lo] makes two major assumptions. (i) that the wear rate of a ball is proportional to its SLEface area, and (ii) that bail makeup consists only of a single large size of ball The forrnulation gwen below extends this treatment to allow for other cases of wear laws and ball additions. In ball wear, there is no problem m drstinguishing between balls and the wear powder, so the wear products can he considered

281

simply as mass lost from the ball charge. Consider unit mass of balls in the mill, containing 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 where f(r) is mass per unit time. The number rate of balls weanng out is thus

T

= nr[l

--h(r)]

cWr) -~

f(r)

dr

(5)

47i+p,

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

Clearly, also rate of conversion element

= NTf@)

--n(r)]

cwr) dr

=

d.Mr)

dr

f(r)

K= (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 equation u-nplicitly defines the relation between ball addition rate nr and the wear rate. Experimental measurements of N(r), NT and nT for a known addition of balls n(r) enables f(r) to be calculated.

fraction

in the make-

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 = m,K4xp,

‘n-lax

NT

in the

(8)

7

where, since gn,np, [1-nn(r)3r2

1 (7)

of mass to powder

Also, mass and number up are related by

=NT

nTpb4m

4 --m3pP, 3

= ~&&,~&~(r)

dr

giving

+ &)3&

Equating eqns. (7) and (8) mves an alternative derivation of eqn. (6). The relation between cumulative mass fraction M(r) in the charge and number fraction N(r) is

w(r) nT[l

4 -(r 3

= n,[l - n(r)] pb4mr2 dr

dM(r)

= f(r) dt

=N

The differencein mass rates of balls entering and leaving the element by wear is the conversion of mass to powder, which at steady state equals the addition of mass of make-up balls Thus rate of conversion of mass to powder in the element

[l-n(r)]?

f(r) = mTJ(l/r3)dm(r),

1

7 am(r)

_f

(9)

‘mln

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) =

(pb4m2)(Kr”)

(10)

where fI can be positive or negative. Since ratI2 1s -d(47+&/3)/dt, Kpb4irr2+* = (-4npb/3)dr3/dt = -4mpbr2dr/dt and the wear

We=

292

dzstcnce

per

unit

0 the wear distance tion (6a) becomes = (m,K/K)

AI(r)

(-dr/dt) is KT* (for A = per unit time = K). Equa-

time

[l -

i

dr

n(r)]r3-a

(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 I;

(94

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

+nk,

m--l

=

+nk

_

r,