SUMMARY

A comparison is made of the results from a ball mill model simulation with those of the conventional Bond ball mill design method, for a material whose breakage characteristics and Work Index have been determined_ In order to perform the simulation, Zlormal’ values were chosen for make-up feed size distribution, mill residence time distribution, ball mix, classifier behaviour, etc_ At high flow rates through a mill (low reduction ratio), it is necessary to allow for the reduc- tion in breakage rates caused by over-filling of the mill, using an empirical mass transfer rela- tion: filling dfeed rate_ By suitable choice of the constant in this relation, it was found possible to duplicate the variation of mill capacity with feed size and product size (in wet closedcircuit grinding) predicted by the Bond method_ The simulation model is then used to predict the variation of performance with design variables not included in the Bond method.

Powder Technology, 34 (1983) 261 - 274 261

A Comparison of the Bond Method for Sizing Wet Tumbling Ball Mills with a Size-Mass Balance Simulation Model

L. G_ AUSTIN and K. BRAME

Department of dlineraZ Engineering. MineraZ Processing Section. PcnnsyZrionia State Lbiuersity, Cjniuersity hrk. PA I6802 (U_S_A_)

(Received March 11,1982)

INTRODUCTION

The Bond method [I, 23 for sizing tum- bling ball mills has been used successfully for many years. On the other hand, a more detaiied method of analysis of the grinding process has been developed in recent years [3 - 113, based on mathematical models of t.he process. This method of analysis has been variously called the phenomenological approach 131, the size-mass balance 14, 121 method or the population balance met.hod [5]_ The objective of this paper is to analyze the advantages and disadvantages of the two

methods, compare the predictions, and show how the advantageous features of each method can be combined.

The primary purpose of the Bond calcula- tion is to predict the mill size and mill power to give a desired capacity Q (t/h) from a wet overflow ball mill in normal closed circuit operated with a circulation ratio C = T,fQ of 2.5 (see Fig. 1). The input to t-he calculation is the make-up feed size characterized by the

Make-Up Feed Mill Producr

Fig_ 1_ A grinding circuit with recycle of the *aarse stream to the mill feed. The symbols represent mass flow rates of solid.

0032-5910/3310000-00001so3.00 0 Elsevier Sequoia/Printed in The Netherlands

262

80%-passing size xo, the desired circuit product size characterized by the SO%-passing size ;rg, and an empirical grindability number determined in a standard test: this number is called the Bond Work Index Wit,,. The primary purposes of the size-mass balance method are not, only to predict the mill size but to show the influence of the feed size distribution, the variation of the complete product size distribution with operating parameters, the influence of classification effi- ciency, etc_ The method essentially constructs a complete simulation model for any circuit, based on parameters measured in a small laboratory mill and scale-up laws developed by tests on small mills.

THE BOND METHOD

The standard method used in the U-S-A. for sizing wet ball mills is that of Bond Cl, 23. It requires a standard laboratory test on the material to determine the Bond Work Inda, using a special mill called the Bond test mill. The material is first reduced to prepare a feed of 100% < 6 U-S_ mesh and about 80% < 2000 Mm. 700 cm3 of this feed (tapped down according to a standard procedure to give a reproducible bulk density) is ground dry in a standard cylindrical test mill of 12 in X 12 in (305 mm X305 mm) with rounded comers, run at a fixed speed of 70 rpm (85% of the critical speed)_ The ball charge consists of a specified number of balls ranging from l/2 to l?& in in diameter (12.7 to 38 mm), with the total ball load weighing 20.125 kg. The pro- cedure involves grinding the charge for a short time, sieving at a desired screen size to remove the undersize, and replacing the weight of undersize with an equivalent weight of original feed. This’new mked feed is reground and the process continued, using estimations of a suitable grinding time (mill revolutions), until a constant mass ratio of oversize to undersize equal to 2.5 is achieved and the net grams of undersize produced per mill revolu- tion (denoted by GbP) becomes constant.

Screen analysis is performed on the under- size product. Then the Bond Work Index is calculated, in kilowatt hours per ton, from the formula

We,., = . (110)(44 5)/ j ( p,O-= )(Gbp)OS2 X - 1

(1)

where p1 is the opening of the classifying screen used in the test; xgT is the 80%- passing-size of the product in micrometers; and #oT the SO%-size of the original feed, which is near 2000 pm. The factor l-10 converts the original Bond Work Index in kilowatt ho&s per short ton to kilowatt hours per metric ton.

Bond correlated the results from the laboratory mill with those from a wet over- flow ball mill which was 8 ft (2-44 m) in diameter, operated in closed circuit, with a circulation ratio of 2.5. His tests showed that the specific grinding energy E was empirically related to the make-up feed and circuit product sizes by

E=WIy!&-g_ ( - - 1 (2)

where the value of 10 is actually- pm. E is the specific grinding energy (based on shaft power) to grind from a make-up feed of SO%-passing xo to a circuit product of 80%- passing size xg, in kilowatt hours per ton. Equation (1) converts the laboratory result (c;bP) to a WI*, which is appropriate in eqn_ (2) for the 8 ft i-d. mill circuit,. For another mill diameter, the scale-up relation is

WI = (WI-)( 2.44/D)‘-* D < 3.81 m

WZ = (W&,)(0.914) D 2 3.81 m > (3)

The empirical Bond equation for shaft mill power mP for this type of mill is given, in kilowatts, by the equation

mP = 7.33 J&(1 - 0.9375) x

P~LD*-~ (4)

where pb is the true density of the grinding medium (tons per cubic meter), 9, is the frac- tion of critical speed, J is the fractional volume of the mill filled by the ball bed (based on a formal bed porosity of 0.4) and L is the mill length (meters).

The mill power, desired mill capacity Q and specific grinding energy E are related by

mP= QE (5)

265

where Q is in tons per hour. Combining eqns. (2), (3), (4) and (5) gives

03-5(L/o)PlJJ - 0.937J2) I

O-10, & - 2x

Q = 6.13 ) - D d 361 m

SW&es% go ( -- -2

)

(6)

D3-3(L/D)pb(J - 0.937J2) @, - 2;:;$c I.

Q = 8.00 >

D 2 3231 m

SWltest go -go (

- -

1 I

TABLE 1

Correcting factors for Bond sizing method [$(pr ) is desired % less than pr at open circuit I : < = &!I $a&$~

Fineness of grind XQ

(ml

Low reduction ratio

=G IxQ

Open circuit ‘a(p,)

(%I

70 1.00 >6 1.00 50 1.035 60 1.02 6 l-03 60 1.05 50 1.05 5 1.04 70 1.10 40 1.19 4 l-05 80 l-20 30 1.17 3 1.08 90 1.10 20 1.32 2.5 1.11 92 1.46 15 1.47 2.25 1.14 95 1.57 10 l-77 2.00 1-20 9s l-70

5 2.67 l-75 l-33 3 3.87 1.50 1.9

Over-sized feed: .$a = 1 + ($ -7) (2) ( 4ooo~~l_l~~13~~~~~~~~ - 1)

IO IOD 1000 IOOOD -I

f GO k-PASSING SIZE OF PRODUCT. Q. “m

Fig. 2. Prediction of Bond for mill of 3.8 m diameter (L/D= 1_6,J= 0_35;&= 70%;&= 7-9 t/m31 for

~w?st = 10 kWhIton_

The factor c is the product of a series of empirical correcting factors to WI for differ- ent conditions (see Table 1): (i) fineness of grind; (ii) low reduction ratio; (iii) over-sized feed; (iv) open circuit_ Figure 2 gives a typical result of the calculation_

ADVANTAGES AND DISADVANTAGES OF THE

BOND METHOD

The method has two major engineering advantages: (i) it is very simple; (ii) experi- ence has shown that it works for many circumstances, to a reasonable degree of accuracy, because it is based on plant measurements_

There are two logical problems involved in the Bond sizing method_ First, the specific grinding energy required to take a feed with a certain 8O%passing size to a product with a

264

certain SO%-passing size cannot be the same for a batch test, the standard Bond locked- cycle test, or a steady-state continuous mill with a real mill residence time distribution, yet Bond claimed that eqn. (2) was a ‘univer- sal law’_ The size-mass balance treatments of these three cases show that the shape of the product size distribution and the associated specific grinding energy is different for the three cases [ 133, and it is not possible to cor- relate the three different values exactly with- out using the size-mass balance method. However, these differences are in effect avoided in the Bond sizing method because the standard test result is empirically matched to actual plant data (on the 2.44 m i-d. mill), and it is thus not really assumed that WI is the same between the different types of test. Second, because the Bond method is purely empirical, it is not possible to assign physical meanings to the relation of capacity to x0 given in Fig. 2, or to the various correction factors_

There are other disadvantages of the Bond sizing method_ First, it is based on a mean empirical fit of data, and there will be a range of error for any specific mill and material and set of operating conditions_ It does not explicitly include several factors which are obviously important: (i) recycle ratio and classifier efficiency; (ii) mixture of ball sizes in the mill; (iii) variations of residence time distributions with mill geometzy and slurry density; (iv) the influence of lifter design; (v) the influence of slurry density and .slurry rheology on breakage rates, and chemical effects on rheology; (vi) variations caused by underfilling or overfilling of the mill as flow rate is changed, especially for grate or peripheral discharge mills.

Again, it is known that the specific grinding energy E is not independent of ball loading J, whereas the use of eqn. (2) explicitly assumes that E is not a function of J. Industrial practice and laboratory tests IlO] show that the specific grinding energy (to go from a specified feed to a specified product) is less for lower ball loadings than the ball loading for maximum mill capacity.

The method uses only the 80%-passing sizes of circuit feed and product to characterize the size distributions, whereas it is clear that mill capacity in general must depend on the shape of the feed size distribution and the product

size distribution. The prime example of this is the use of the reverse closed circuit shown in Fig. 3, which is advantageous when the make- up feed contains a significant quantity of material already fine enough to meet product specifications_ Conceptually, this circuit can be treated as if there were two identical class- ifiers, one classifying the make-up feed and the other classifying the mill product. The underflow from the fust classifier is the effec- tive make-up feed to the normal closed cir- cuit. To perform the Bond calculation on the normal closed circuit part of this circuit requires a knowledge of the classification ac- tion of the classifier on the makeup feed. The reverse circuit is more efficient than the normal circuit for a given make-up feed and product specifications (x0 and ~0) because the shape of the final product size distribu- tion is different, containing a smaller propor- tion of over-ground fine material_

f

CLDSSIFER RECElVlNG RETURN FROM MILL AND MAKE-UP FEED

ODUCT

Fig. 3. The reverse closed circuit treated as two identical classifiers.

Finally, the empirical over-size correction factor applied to a feed with xc = 10 mm for example gives a very large reduction in mill capacity at low reduction ratios for materials with a high Work Index. The mill capacity according to the calculation procedure is . almost independent of x0 over a substantial range (see Fig. 11). This means that a mill - operated at a fixed flow rate would give large changes in product fineness x0 with minor fluctuations in feed rate or material grindabil- ity, which is clearly not in accord with plant practice or common sense_

Thus, the Bond sizing method does not incorporate a number of important second- order effects, and it cannot be used as a guide to the fme-tuning or optimization of a given system, from either.the operating point of view or the economic point of view. It can

only be valid as a gross method for mills oper- ating under normal closed circuit conditions.

THE SIZE-MASS BALANCE METHOD: FIRST-ORDER BREAKAGE

Esperimental batch grinding tests (see Fig. 4) show that the rate of breakage (under normal conditions) of material sized within a J?Z upper-to-lower screen interval f0110Ws a ‘first-order’ breakage law:

rate of breakage to smaller sizes = Szu(t)W

where w(t) is the mass fraction of the mill charge (hoid-up) W which is of the size interval examined, t is the time of grinding, and S is the specific rate of breakage of this size, with units of fraction per unit time. In order to construct a complete size-mass balance, it is convenient to split the total size range into ,/T screen intervals, numbered 1 for the top size, 2 for the second, etc, down to interval n for material les than, say, 400 mesh (38 pm).

Using this symbolism, a size interval which is breaking can be denoted by j, and a smaller size interval which is receiving the products of this breakage can be denoted by i, where R > i > j. It has been found experimentally that the mean set of primary breakage fiag- ments produced from breaking sizej does not change with the grinding time- These frag- ments mix into the charge, and can in turn be

o EXPERIMENTAL - CALCULATED

GRiNDING TIME @WdUTESI

Fig. 4. First order pIot for batch grinding of 16 x 20 mesh quartz in laboratory mill.

broken, but if they are measured before re- breakage occurs, the mass fraction arriving in size interval i from breakage of size interval j is symbolized by bi.i, where X$Z?+ 1 bi.i = l- This set of numbers varies from one material to another-

A balance of material being broken into and being broken out of size interval i is: rate of accumulation of size i material = sum rate of production from breakage of all larger sizes (j = 1 to i - 1) - rate of breakage of size i to smaller sizes. In symbols, this is

dw,(r)W = ri=l

dt 1

C bi_jSjWj(t)?V i=l i> 1 I

-SjWi(t)W

or

dw,(t) i- I

df = -Siwi(t) + C bi_jSiwi(t), TZ 2 i Z 1

i=, i>l ('7)

This is the basic set of equations for first- order batch grinding_ The set of numbers Si is a precise index of the weakness or ease of grindability of each size, and varies from one material to another_ The solution of the equa- tions with a known starting feed of w,(O), wZ(0), etc, for time t, gives ml(t), wz(t), etc., thus giving the size distribution produced.

This set of equations would apply to a continuous mill operating at steady state if the entering solid flowed through the mill as a ‘plug’ with all material staying in the mill for the same residence time T_ However, ball mills have a residence time distribut.ion (RTD) be- cause forward and backward mixing occurs along the mill axis. The model must therefore be extended to allow for the actual RTD, as first described by Reid [14]_ If a fraction Q(t) dt of the feed stays in the mill for a time t, and leaves between t and f + dt, this frac- tion will be broken as if it were batch g-round for time t. Its size distribution upon leaving will be the set of numbers wi(t), obtained by the solution of eqn_ (7) for each value of i.

Simultaneously, material that has been in the mill for different lengths of time will also be leaving; at steady state. the total product in size interval i, Fpi, which leaves the mill will be the weighted sum of all fractions of size i

product:

PiF= J w,(t)@(r)F dt 0

266

or

pi = j’ Wi(t)@(t) dt 0

(8)

This is the basic equation of steady-state, continuous, first-order grinding. The integral limit of infinity is, in practice, a time long enough to include all significant contributions to pi, and must be at least three mean residence times, 37. Note that there are two important additional assumptions implicit in eqn. (8). First, it is assumed that all particle sizes in the mill have the same distribution of residence time. Second, it is assumed that the slurry reaching the discharge flows out with no preferential return of larger sizes back into the mill; that is, there is no size classification due to discharge_

In order to incorporate this mill model into a closed circuit as shown in Fig. 1, it is neces- sary to describe the action of the classifier(s) in the circuit_ Under any given condition, it is assumed that the fraction of size i in the feed to the classifier which is recycled to the mill feed is si: the feed to the mill is then obtained from

Ffi = Qgi * F’iSi (9)

where fi is the fraction of size i in the mill feed and gi the fraction of size i in the make- up feed (see Fig_ 1 for symbolism). The set of si values &es the action of the classifier on all size intervals, and the values are called the classifier selectivity values. Luckie and Austin [ 153 showed how eqns. (7), (8) and (9) could be combined and computed sequentially start- ing at i = 1, to determine the size distributions around the circuit for a given value of mean residence time T, for specified Si, bi_i, gi, RTD and Si-

The computation gives the circulation ratio C, and since the mean residence time is related to the solid flow rate through the mill F and the c~erall mill hold-up W by T = W/F, the mill capacity follows from

Q = W/7(1 + C) (10)

In practice, the computation is normally performed to find the value of-r which will give a desired one-point match (e.g. 80% minus 100 mesh) in the circuit product.

RELATIONS FOR S AND B VALUES

A series of investigations [ lo,16 - 191 in laboratory mills has shown that the values of Si vary with mill conditions according to the following relations. The variation of Si values

with particle size can be described by the empirical expression

Sic= xi a

( I( 1 n>i>l (11)

X0 l* (xi/ilA 1

where xi is the upper size of interval i, and x0 is a standard dimension, here taken as 1 mm. The parameter CY is a characteristic of the material which does not vary with rotational speed, ball load, ball size, mill hold-up or slurry density over the normal recommended test ranges. The term U(xi/xo)= is the left-hand straight line portion in the log-log plot of Fig_ 5 The term l/l1 + (Xi/~)“] is 1 at smaller values of Xi, but is less than 1 at larger vahe~

of Xi and makes the curve bend over to the right-hand side. The parameters P and A de- scribe the size at which the bendover occurs and how steeply it falls to the right of the maximum in S: p is the size at which the term is 0.5.

The value of a also depends on the material and it is determined [16] for laboratory test conditions of Dr, d,, Jr, Gr and QcT, where dr is the ball diameter in the test mill and Ur is the formal interstitial filling of the ball charge with powder: it is converted to the desired pilot or full-scale mill conditions by the following equations. The effect of ball and powder fiiing is

Fig- 5_ Specific rates of breakage used in simulation: D=3m,J=0.35;U=1,~,=0_7;Bondballmix, 2 in make-up ball.

1 aa l+66J2_3 =~(--1.32w (12)

This applies for grinding at normal slurry densities with water, over the range J = 0.2 to 0.5 and U = 0.5 to 2.5. Because a specific rate of breakage is a fractional rate of breakage, the breakage per unit of mill volume is pro- portional to the specific rate times the amount of charge in the volume, which is proportional to aJU. Equation (12) shows that QJU goes through a flat maximum be- tween U = 0.5 to 1.1, but decreases for high values of U, Le. the mill is over-filled, leading to cushioning_

The effects of ball diameter and mill diam- eter are:

(i) a a (l/cZ”~) No= 1.0 (13) (ii) a=DNa (14)

N, is close to 0.5 for D < 12.5 ft (3.81 m), and by analogy with the Bond method, is 0.3 for D > 12.5 ft (see eqn. (6)).

(iii) p a d 2DN: (15)

where N2 is O-1 to O-2. This allows for the effect of ball size and mill diameter on breakage of large feed sizes. The values of the exponents N,-, and N+ may vary with the type of lifters, since the test mill will not tumble a given size of ball in exactly the same way as a larger mill with different lifter configuration_ Mill capacity can change as lifters wear down in operation_

The value of (I can be corrected for rota- tional speed by

1 a =(&-OS)

1+ exp[15_7(+, -0.94)] (16)

Again, it is expected that this equation may not apply precisely for different types of lifters.

Combining eqns. (12) - (16) gives

S,(d) = aT

where

C2 = (dT/d)No

N2 = 0.1 to O-2

(17a)

ATo’1 (17b)

c, = (D/DT)N, N1 = O-5 (17~)

C,= 1 + 6-6&Z-3-

1 + 6.652-3 esp[---1_32(U- Ur)] (17d)

c = 4, - 0.1 1 + exp[15_7(& - O-94)]

s @CT- o-1 1 + exp[ 15_7(& - O-94)]

(i7e)

Z&(d) is the specific rate of breakage for ball size d_

For a mixture of balls of mass fraction m, of size d,, m2 of size d,, ___, mb of size d,, etc., the overall values of breakage rates are given by

where sj is the overall. specific rate of break- age of size i for amisture of m different ball sizes. (Note that S, = 0, since material cannot break out of the sink interval.)

If the values of bi_i are cumulated from i = n, the cumulative values B&= Zi=,, bk_j) have the form shown in Fig_ 6_ This can be expressed as the sum of two straight lines of slope y and /3, with a fraction Q, of slope y and a fraction 1 - @ of slope /3s t Biej = aj;

i 1 x22 X,-,.3 + (1 - 61,) - Lri ( 1 Xi

n>i>j (19)

It is found for many materials that ai, y and @ are not dependent upon the breaking size j, so that the descriptive parameters are G’, y and &

Fig_ 6_ Cumulative primarv breakage function for quartz: 1 in balls.

266

from which biSj are readily calculated. It is assumed that the biSj values do not change with mill diameter or mill conditions, provid- ing that rotational speeds are reasonably close to the maximum power condition so that the balls are cascading properly, and that the slurry viscosity is low enough to avoid changes in 7 [19]. However, the values change somewhat for different ball diameters [ 191.

Figure 7 shows the agreement betweeen computed batch grinding results (using the solution of eqn. (7)) and the experimental data, for the characteristic breakage param- eters given in Table 2, which were determined in a small laboratory mill with 25.4 mm diam- eter steel balls.

RTD, MASS TRANSFER AND HOLD-UP

Measurements of residence time distribu- tion in mills [20] indicate that the residence time distribution in full-scale wet overflow ball mills can be approsimated by the simple equivalent system of one larger fully-mixed reactor in series with two equal smaller fully- mixed reactors_ However, when the measured values of mean residence time r are used to calculate mill hold-up W from r = W/F it is also apparent that many mills are operated in an over-filled condition. At high mass flow rates through the mill, i-e_, short residence times, the values of interstitial filling U exceeds 1, and the values of a and W to be

SIZE. PC=

Fig. 7_ Size distributions for quartz wet ground in batch laboratory mill (J = O-35, d = 25.4 mm, D = 195 mm): 40% solid by volume, 64% by weight.

TABLE 2

Characteristic breakage parameters for wet grinding of quartz: Bond Work Index = 19 kWh/metric ton

Weight of solid True specific gravity

Slurry density

= 1.16 kg = 2.65

= 64 wt.% soiid = 40 vol.% solid

Ball charge: d = 25.4 mm J = 0.35

Mill:

Breakage parameters:

D=195mm V = 5200 cm3 r.p.m. = 72 = 70% critical

D = 0.35 min-’ fY = 0.80 $1 ;y65 mm

y = 1.25 Q = 0.55

&~Iiaed

used in the simulation are functions of flow rate.

It is convenient to perform the computa- tions of the simulations using a constant a value corresponding to U = 1, aI say. The value of r thus obtained is a formal or false 7, rf say. It is clearly related to the real -r by

The real value of W is related to the formal hold-up W, by W = UWI_ From eqn (10) then, the circuit capacity is obtained from

Q = Wreall~rea~U + Cl = &W-JTf(l + C)

where (20)

Equation (12) gives a,,&, as a function of U, thus

K. = U exp(-1_32U)/exp(-1.32) (21)

Figure 8 shows K. as a function of U: K. is the over-filling factor, which gives a decrease in capacity as U becomes greater than l_

Marchand et aL 1211 have concluded that mill hold-up is related to flow rate through a mill (for constant feed slurry density) by the approximate empirical relation

U CE F”-5 (22)

269

TABLE 3

Slowing-do&x factors for quartz at 40% volume solids (65% weight solids): initial feed 16 x 20 mesh

80%passing size tvm)

Slowing-down factor X

FFOM U-05 TO 1.0

oi _ _.I._ .._ _.~ I _..___I _____I:.......~ 0 05 1.0 x.5 20 25

INTERSTITIAL FlLLlNG ”

Fig. 8. The over-filling factor K. as a function of ball interstitial filling by po-wder.

The value of the proportionality factor depends on slurry density, ball charge and size, etc_, as well as mill size_ It seems a reasonable approximation that relatively low flow rates to a mill can be considered to give U = 0.5 to 1.0, which from Fig_ 8 gives K. essentially equal to 1, while very high flow rates lead to over-filling of the mill and K,< 1.

Equation (22) can be written as

u= [(F/w,)T,J0-5 (2M

where T, = IV, /F,, with F1 being the flow rate which gives U = 1. Since F = (1 + C)Q, eqns. (20), (21) and (22a) give

71 1~~ = U exp[ 1.32( U - l)] (23)

If 71 is known for a given system, a value of rf gives the corresponding value of U from eqn_ (23), and hence K. and Q follow from eqns_ (21) and (20).

SLOWING-DOWN EFFECT

Examination of the long grinding times of Fig. 7 shows that the simulation predicts a finer grind than that obtained experimentally- This results from a slowing down of breakage rate as the charge becomes fine, due to the development of high effective viscosity in the slurry [ 22,231. -4ustin and Bagga [24] have shown that the results can be simulated using false first-order grinding times 8, giving a slowing-down factor K defined by

I = Sg(t)/Si(O) = de/dt (24)

> 100 1.0 90 0.97 80 0.91 70 0.53 60 0.74 50 O-61 40 0.5-i

t30 0.5-l

Table 3 gives the values of K as a function of the xP of the mill charge_ Mill capacity can be approximately corrected for this slowing- down effect by assuming that t.he slurry leav- ing the mill is characteristic of the slurry in the mill. This is valid for mill RTDs which are reasonably close to fully-mixed, as found for wet overflow ball mill& The normal capacity is reduced by multiplying by K_

COMPARISON OF BOND _4XD SIZE-MASS

BALANCE SIMULATIONS

It is not possible to perform an exact comparison because the conditions in the test mill and circuit used by Bond have never been published_ Therefore, a number of reasonable assumptions have to be made to construct- the equivalent simulation model. It was assumed that the mixture of balls in the mill corre- sponds to the equilibrium Bond [l] mixture with a top size of 2 in, at J = 0.35 The RTD used was the equivalent one-large/two-small model with relative sizes of 7, c: O-7, rl_ = O-15. The classifier selectivity values were assumed to fit the empirical relation (25):

1 Si=~ ~(l-b)

1 -i- (xJdso)-h (25)

where X is related to the Sharpness Index by SI = expl-2_1972/h]_ Reasonable values of the by-pass fraction zi and Sharpness Index SI were taken as O-3 and 0.5, respectively_ The value of d,, was varied to give C = 2.5 for any desired product size xB_

B values for a m-kture of ball sizes were cal- culated from eqn_ (19) using y = l-10, @ = O-65 and fi = 5.8. The breakage parameters of

270

the quartz of Table 2 were scaled to a mill of U = Is this gives 7, = 4.94 min. The new

3.8 m diameter with an L/D of l-5, usingNz = values of Q are also shown in Fig. 9. They 0.2. The make-tip feed was assumed to fit a agree quite well with the Bond prediction. It Rosin-Rammler distribution with a charac- seems possible, therefore, that the Bond teristic Schuhmann slope of s = 0.75. Initial- method applies when the mill is over-filled at ly an SOY&passing size of the feed was chosen high flow rates. Figure 10 shows the agree-

to be 2 mm_ The value of K0 was taken ini- ment between the model simulation results tially as unity; U = 1 gives Wf = 14.4 metric and the Bond results for the same shape of tons. A comparison of the Bond calculation feed size distribution but xG of 1 mm or with the circuit simulation at low feed rates 500 pm_ Again, the agreement is excellent, suggested that the simulation was predicting showing that the agreement of Fig. 9 is not about 10% too low capacity values: since the fortuitous. mill and classifier conditions were arbitrary estimates, it would be fortuitous if the Bond ‘OOOL

t’ l-XT *I & 7

‘! : : 9.‘

and simulation results were identical. There- o SIMULATIDN

fore, the simulation values were multiplied by 4

1.1 (a increased by l-1) to give good agree- 1

ment for low flow rates. Figure 9 shows the

1 result. It is~clear that variation of the capacity 1

with fineness of grind for the simulation and 7 3

Bond methods agree quite closely in the 2 2

capacity range of 10 to 50 t/h. Especially, the Bond fineness-of-grind correction factor is the 1 natural consequence of the mill simulation_ It is also clear, however, that the simulation t ,oi 1 : i , : ,,.i method w+h K0 = 1 over-predicts mill

, XI,. ! 10 100 IODO

capacity for low degrees of size reduction_ PRODUCT GO X-PASSING SIZE. x0. “III

This is not surprising since a circuit capacity Fig. 10. Comparison of Bond and simulation results: of 200 t/h corresponds to a solids flow rate feed SO%-passing 1 mm and 0.5 mm (see Fig. 9).

through the mill of ‘700 t/h, which would clearly lead to over-filling of the mill. Figure 9 also shows the open circuit results

To allow for this factor, it was assumed that 175 t/h through the mill corresponds to

predicted by the model simulation. Because the mass flow rates through the mill are only l/3.5 of those in closed circuit, the over-

loo0 I , i I ,:*i ---7---z 1 filling factor does not come into effect until Q is greater than 175 t/h. Thus the increase of efficiency normally obtained by closing the circuit to prevent over-grinding of fines is offset at low reduction ratios (high flow rates) by the decrease of efficiency caused by over- filling of the mill. At low flow rates and fine grinds, the increase in output obtained by _ closing the circuit to C = 2.5 is approximately 2, which agrees with the result quoted by Taggart [26]_ It does not agree with the Bond

i sizing method (see Table 1) if the size at

l0l---!--LL_LliL_ : I : i LLLI/ which the Work Index was determined is 10 100 ,000 taken as 75 pm.

PRODUCT GO X-PASSING SIZE, a, pm Table 4 shows the model simulation results

Fig- 9_ Comparison of mill capacities by Bond and obtained by varying the classification effi-

simulation methods (D = 3.8 m, L/D = 1.5; J = 0.35; ciency as defined by by-pass and Sharpness

70% critical speed): feed 80%passing 2 mm, quartz Index. Although more efficient ciassification WI- = 19 kWh/metric ton. (low &, high S.I.) gives significantly improved

TABLE 4

Effect of classifier efficiency on closed circuit output at C = 2.5 (350% circulating load): base condition SI = 0.5, by-pass b = 0.3

Product xs,, S-I_ By-p& b Capacity factor

(pm) Q/Q-

420 to 38 0.6 0.2 l-10 0.6 0.3 1.04 0.6 o-4 0.97 O-5 0.2 l-05 OS o-3 1.00 0.5 0.4 0.94 o-4 o-2 l-00 o-4 0.3 0.95 0.4 0.4 0.89

output, the change is not sufficient to explain t.he discrepancy between model closed/open circuit ratios and Bond closed/open circuit ratios. However, if the screen size used in the Bond calculation is taken as 300 pm, and the percentage less than 300 pm taken from the simulation, the agreement between simulated and Bond capacity results is excellent, as shown in Fig. 9. The Bond calculation stops atxo= 75 pm for this case as correction factors are not given for finer grinds.

Figure 11 shows the model predictions vexsus Bond predictions for a feed with over- size particles, x F = 10 mm. Again, the two methods give substantially different results for relatively low reduction ratios.

It should be noted that the scaling laws fox mill diameter are

i 10 j . . . . . . .i , .: . . . . . 1 i

IO 100 ID00

PRODUCT GO X-PASSING SIZE. x0. ym

Fig_ 11. Comparison of Bond and simulation results for a large feed (see Fig. 9).

Bond: Q 0: D=(L]D) D <

Simulation: Q = Wa a D3*x- (L/D)

D<

3.81 m

3-81 m

Since IV, = 0.5, both methods scale in the same way with respect to mill diameter and length and the conclusions from Fig_ 9 are un- changed, providing that the mass transfer/ over-filling relation scales in the same way_

DISCUSSION

Even though the Bond method has been of great use in sizing mills fox closed-circuit operation, the concepts behind the method

contain certain logical contradictions and in- consistencies. It was claimed by Bond that the Third Law of Comminution for batch grind- ing was a fundamental law related t.o fracture physics:

(26)

His argument can be reduced to this: eliarnination of the variation of 80%passing size of the product (xx) in a batch grinding test shows the E a lO&&relation to apply, and the Griffit.h crack theory of fracture [27] contains a l/(crack length)*” term, therefore eqn. (26) must be a fundamental law_ Such reasoning is clearly inadequate, but it is often repeated without question; for esample [ 283 : “The Bond theory states that work input is proportional to the new crack tip length produced in particle breakage___.“, which is an essentially meaningless statement. Even if eqn. (26) were generally true, it should logically be applied to the feed to the mill and the product from the mill not the circuit feed and circuit product_ Since it. was deduced for batch grinding, it should apply to a mill in plug flow, whereas RTD esperiments show mills to be closer to fully-mized. Again, if it were a fundamental law, why should it be necessary to use fineness-of-grind and low reduction ratio corrections to the law?

It must be concluded that the so-called Third Law of Comminution is not a funda- mental law; and therefore it does not have to hold under all circumstances as it is entirely empirical_ The results presented in Fig. 7 do indeed fit eqn. (26) (assuming E ” grinding time, ie_ constant mill power), but they are

272

TABLE 5

Effec: of Bond oversized feed correction factor

Test Work Index Feed Product Oversized Low reduction Capacity

(kwhit) s G X8 feed factor ratio factor &-& --

(t/h) (mm) (mm)

factor

14 10 1 1.92 1.0 0.216 216 10 2 2.83 1.04 0.124 247

19 10 1 2.96 1.0 0.216 104 10 2 4.90 1.04 0.124 105

24 10 1 4.33 1.0 0.216 56 10 2 7.64 1.04 0.124 53

also fitted by the simulation model. On the other hand, the closed circuit analysis suggests that it applies under closed-circuit conditions purely as a result of decreasing breakage rates due to over-filling of the mill and consequent cushioning. The region requiring the fineness- of-grind correction factors is actually the natural result of efficient breakage at proper filling level, while the lo/& relation should properly be considered as a corrected result. Similarly, Fig. 10 shows that the low reduc- tion ratio correction is also a natural result of the simulation.

Another logical problem occurs in the conversion of the law to open circuit. The correction factors involved (see Table 1) are a function of the percentage of product less than the sieve size used in the Work Index test, yet the Work Index is often virtually independent of the screen used in the test, over a considerable range of screen sizes. Thus, the prescribed Bond method of calcula- tion will give a different open circuit mill capacity for the same material depending on the arbitrary screen used in the test procedure for the Work Index The open circuit results predicted by the mill simulation model (see Fig. 9) agree with the Bond calculation only if the screen size is taken as 300 pm (even though the Work Index was actually deter- mined using 75 pm).

The simulation model shows that the reason that closing the circuit does not give such a big increase in output for coarse grinds as for fine grinds is that the high flow rates required to give the short residence times needed for coarse grinding in clcsed circuit leads to over-filling of the mill. This leads to cushioning and reduced breakage efficiency,

which counteracts the normal advantageous effect of closing the circuit.

It is clear that the Bond oversized feed correction factors become escessively large for high Work Inde... For example, Table 5 shows the predicted capacities for Work Indices of 14,19 and 24 kWh/metric ton. In the last case, the calculation gives a lower capacity for a coarser grind, demonstrating that the empirical Bond expression cannot be used for high Work Index.

Figure 12 shows the size distributions predicted by the model simulation for the oversized feed case, for grinding to 80% passing 50 mesh (300 pm). The notable characteristic is that the mill product still contains 9% of material above 10 mm, with a region from 1500 pm to 10 mm with only small amounts of material in each size interval. Figure 5 shows that the fastest break- age rates occur about 5000 pm, so material which is much larger than this will persist in the mill, while material of about this size will disappear most rapidly. In practice the size of

Fig. 12. Size distribution around closed circuit (C = 25) for over-sized feed 80% < 10 mm, product 80% < 300 pm.

make-up ball to the mill would be increased Bond method will over-predict capacity, un- to give higher breakage rates for the large less the slurry density is reduced and appro- sizes. priate additives used.

CONCLUSIONS ACKNOWLEDGEhIENTS

Although the Bond method for sizing mills has been very valuable, there are several logical inconsistencies involved in the dcriva- tion of the method_ A. detailed comparison with model simulations suggests that the so- called Bond’s Third Law is the fortuitous consequence of decrease of breakage rates caused by mill over-filling at high flow rates. The Bond result, including his fineness-of- grind correction factors, agrees with the model simulation for fine grinds and low flow rates where there is no over-filling. If it is further assumed that the Bond result under normal conditions gives the correct answer, the varia- tion of mill hold-up with flow rate can be deduced by comparison with model simula- tions, assuming hold-up is proportional to solid feed rate through the mill to the half- power, thus giving the proportionality constant for this relation.

We thank the sponsoring firms of the Coal Cooperative Program of The Pennsylvania State University for financial support of this study.

REFEREXCES

I

2

F. C. Bond, British Chcm. Eng.. 6 (1965) 378 - 391.

3

4

.5

C. A. Roalands and 1). Xl. Kjos, Nincral Rocess- ing PZanl Design. SME. Denver. Colorado. 2nd edn.. 1980, Chapter 12. R. P. Gardner and L. G. Austin. 140~. 1st Euro-

pean Symposium on Sizr Rcdcrction. Verlai! Chemic. Dusseldorf (1962) 232 - 268. L. G. Austin. Powder Tcchnol.. 5 (1971/72) 1 - 1’7.

The simulation model can then be used to esamine the effect of variation of milling con- ditions, e.g. the effects of classifier efficiency, ball size, etc., whereas the empirical Bond method applies only .for mean conditions over limited ranges. It can also be used under cir- cumstances where the Bond method clearly does not give a sensible answer.

9

10

Analysis of the milling process using size- mass balance simulation models instead of the Third Law of Comminution has the great advantage of forcing a consideration of the physical reasons behind mill and circuit capac- ity variations from the norm. When these reasons are understood, it is possible to make adjustments to mill conditions based on educated guesses as to behavior, thus speeding up the process of fine tuning of a circuit by the operators. For example, closed-circuit operation with a classifier with a high level of solid by-pass will have an optimum circu!ation ratio for relatively coarse grinds. Attempts to improve efficiency by increasing the circulat- ing load will lead to further over-filling of the mill and reduced efficiency. Again, the slowing-down effect seen for very fine grind- ing must be taken into account, otherwise the

11

12

13

1-f 15

16

17

1s

19

J. A. Iierbst, G. A. Grands and D. W. Fucrstenau. I’roc. 10th Infernational _Vincral Processing Congress. Alden Press, London (1973) 23 - 15. L. G. Austin. P. T. Luckie and D. Wightman. Inl. J_ of Jlinrral Processing. 2 (1975) 12i - 150. A. J. Lynch. Zlincral Crushing and Grinding Cir-

cuits. Etsevier. Sew Tork. 1977. D. Iiodouin. J. ~lc~Iulh?n and Xl. D. Everall. I’rc- prints Eur. Symp. Particlc Technology. z’ol. -4. Amsterdam (1960) 686 - i02. D. F. Kelsall. C. J. Reslarick. I’. S. B. Stewart and K. Weller. Rot. Australas. Inst. .\lin. _Vefali. CG,Zf, Xcwcastlc (19i2) 337 - 347. K. Shoji. I,. G. Austin. F. Smaila. K. Brame and P. T. Luckie. Powder Technol.. 31 (1982) 121 - 126. P. T. Luckie and L. G. hus~in. Coal Grinding Technology: _-I Jfanuai for hGC6X.S Engineers. U.S. Dept. of Energy. FE-2175-25. available from National Technical Information Service. Spring- field \‘a. (19i9) 92 pp_ . . L. G. Austin. Ind. Eng_ Chem. Proc. Des. Develop.. 12 (1973) 121 - 139_ L_ G. Austin and P_ T. Luckic. Trans. AIJIE. 252 (1972) 259 - 266. Ii. J. Reid_ Chcm. Enp. Sri.. 20 (1965) 953 - 963. _ P. T. Luckie and L. G. Austin. .Viner_ Sci. Eng_. 4 (1972) 2-I- 51. L. G. Austin and V. K. Bhatia. Powder Technol..

5 (1972) 261 - 266_ K. Shoji. S. Lohrasb and L. G. Austin. Powder TechnoL. 25 (1979) 109 - 114 L. G. Austin and P. T_ Luckie, Powder Technol.. 5 (19i2) 215 - 222. L. G. Austin, R. R. Klimpel and P. T. Luckie, The Process Engineering of Size Reduction: Ball hfilling. to be published by Society of Mining Engineers, AIBIE.

20 K_ R. Weller. Proc. 3rd IF-AC Symp. Automation

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21 J. C. Marchand, D. Hodouin and M. D. Evereli, 29 (1981) 277 - 281_ Proc_ 3rd IFAC Symp.. Automation in hfining. 26 A. F_ Taggart, Eahdbooh of Mineral Dressing, Mineral and Metal Processing, Pergamon Press Ores and Industrici: hfinerals. Wiley, New York, (1980) 295 - 302. 1945. p_ 100_ .

22 R. R. Klimpel, Powder TechnoL (in press). 27 A. A_ Griffith. P&Z_ Trans. Roy. Sot. (London). 23 R. R. Klimpel and L. G. Austin, Powder Technor.. 22ZA (1920) 163.

32 (1982) 239 - 253. 28 P. E. Snack and F_ B. Raymer. hfining Congress 24 L. G. Austin and P_ Bagga, Powder TechnoL. 28 Journal. 68 (1982) 28.