Grinding Rate of a Ball Mill Operated under Centrifugal Force t

Measurements of specific surface area of product were made on a new type of ball mill operated under centrifugal force. The results indicated that this type of mill was superior in grinding performance to the conventional ball mill operated under gravitational force and that the use of centrifugal force was effective in reducing the grinding time. An empirical equation was proposed for expressing the increase of specific surface area of product by the introduction of a non-dimensional parameter to evaluate the effect of centrifugal force. This equation was found to express well the grinding process under gravitational force as well as that under centrifugal force.


Introduction
A ball mill is one of the most commonly used devices for fine grinding on an industrial scale.Its grinding ability is based on gravitational force alone, and requires lengthy grinding time in a batchwise operation to produce fine particulate materials which have been more necessary to industrial use.It is expected that this operation time is reduced from the viewpoint of practical energy saving effort. 2> One of the effective methods of shortening this operation time is to employ an enhanced centrifugal force instead of a gravitational force.
It has been reported that this type of mill has much higher grinding capacity than conventional tumbling ball mills which are operated only under gravitational forces 1 • 3 • 5 • 9 >.Unfortunately, however, there are few papers describing the grinding rate quantitatively and in detail; in particular, there has been little knowledge of relation between a grinding rate and an added centrifugal force which is the most important to estimate the mill performance.
In the present work, a planetary-type ball mill was used to investigate experimentally the effects of an added centrifugal force, fractional ball filling, and fractional material filling on a grinding rate in the mill.

1 Experimental equipment
Fritsch planetary-type ball mill was used as an experimental equipment.The geometrical outline and dimensions are shown in revolution acted on the media as an acceleration force.The rotation speed was set to be 2.2 times as large as the revolution speed which could vary from 70 to 300 r.p.m.

2 Sample material
The sample material used in the present work was a feldspar powder with a specific gravity of 2.55 and a specific surface area of KONA No. 4 (1986) 26.5 m 2 .kg-1 • This powder was obtained by crushing a material produced in Niigata and sieving it to the particle size range from 7 4 to 149 pm.

3 Experimental method
Experiments on size reduction were carried out batch wise in a dry state.Change of specific surface area of the sample powder according to time was obtained by Surface area analyzer (Shimadzu Seisakusho Co., Ltd., Type SSlOO) to assess a grinding process.The experimental procedure employed was: -Given amounts of sample material and grinding media were put into the grinding bowls.-After grinding operation was conducted during a given period, the ground material of 3 g was sampled by the cone and quartering sampling method to be used for the specific surface area measurement.-Then, the material taken out for such a measurement was entirely returned into the grinding bowls.-The above procedure was repeated.In addition to this measurement, experiments on a conventional-type ball mill operated under a gravitational force-cr were made to compare these two results.

Packing volume of balls J = -
(1) Net volume of test material U= (2) Net volume of balls

Critical rotation speed of planetary-type ball mill
It is well known that there is a critical state for a conventional ball mill where the balls would rotate at the same speed as that of the bowl.By using a conventional method 4 ), the authors attempted to derive the critical condition of a planeatary-type ball mill.
We consider the rotating coordinate system * The experiment under gravitational force was entirely conducted at the rotation speed of 112 r.p.m.This corresponds to 69 percent of the critical rotation speed. ?

Direction of revolution
where m 8 is the mass of a ball and r is the distance between the revolution center and the ball center given by When the critical state is achieved as shown in Fig. 3 (C), the three forces can be balanced at e = 1T as follows: (7) Substituting Eqs. ( 3), ( 4), ( 5) and ( 6) into Eq.( 7), the critical condition can be obtained by using the revolution-to-rotation speed ratio: 56 The above form demonstrates that the critical condition of a planetary-type ball mill can be expressed by the revolution-to-rotation speed ratio.
The value of (()c obtained by substituting practical equipment dimensions for Eq. ( 9) was 2.9, and the present experimental condition ((() = 2.2) is found to correspond to 76 percent of the critical value.The experimental condition employed in the present work proves to have been adopted adequately, because it is recognized in general that a conventional ball mill runs effectively at 60 to 80 percent of its own critical rotation speed 7 ).

1 Increasing process of specific surface area
As the size reduction proceeded, the specific surface area S increased, as indicated in Fig. 4 where Nt of the horizontal axis denotes a summation of the rotation numbers of the grinding bowl.This figure shows clearly that the size reduction under a centrifugal force would permit a specific surface area to increase more rapidly and shorten the operation time, compared with that only under a gravitational force which is expressed as a plot of closed circle symbols.During each initial period of the size reductions, S appears to increase linearly with Nt, and this means the applicability of Rittinger's law7).On the other hand, after a certain period of operation was conducted, the grinding rate would decrease due to the production of fine fragments.On the basis of such an experimental result, the following grinding rate equation was introduced under the assumption of the general relation between particle size and grinding energy: Integration of Eq. ( 10) under the assumption that the initial surface area would be negligible in comparison with the value in the state after the grinding operation leads to The log-log plots of S against Nt obtained through Eq. ( 11) from the experimental result are illustrated in Fig. S and Fig. 6 for the cases under centrifugal and gravitational force, respectively.Each result is found to be consistent with the relation given by the Eq. ( 1 1) except the initial and the final period of size reduction.
10'r----.--------------.---------/ LoJ Fig. 6 Relationship between specific surface area of product and number of bowl rotations The slope of any straight line, n, in the both figures is close to the constant value of 0.8 (or 0.25 form), independently of the experimental conditions.This suggests probably that n in Eq. ( 11) or m in Eq. ( 1 0) tends to be dependent solely on the type of material used.As mentioned above, Eq. ( 11) expresses adequately the increasing processes on specific surface area.Thus, the effect of operating factors on grinding rate can be assessed by the coefficient of k used in Eq. (11 ).

2 The relationship between grinding rate and fractional ball and powder fillings
As indicated in Fig. 7, k reached a maximum of J = 0.34 for the both cases under centrifugal and gravitational force.This value is substantially consistent with the optimum value obtained in a practical ball mill operated under a gravitational force.With an increase in U, which is not shown in the present paper, k was found to tend to decrease monotonously.This is probably because an increase in the materialto-ball mass ratio would result in reducing grinding energy applied to a material of a unit mass.
It may be considered that there is an optimum value of fractional powder filling at which the power input into the grinding system can be consumed most effectively.In this case, it will be convenient to use the product of k and the feed material weight, Wp to obtain an opti-

U[-]
Fig. 8 Effect of fractional powder filling on grinding rate 1.2 mum value of U; this product means the rate of increase in the whole surface area of all the particles.Figure 8 shows a typical relationship between kWp and U in the case an optimum fractional ball filling J, of 0.34.In this figure, kWp reached the maximum nearly at U= 0.4 when the operation was conducted under a centrifugal force, while the maximum value was not found under a gravitational force.If we obey Rose's concept 6 ) that a ball mill can be operated most effectively when the clearance among the grinding balls are filled with a powder, U defined by Eq. ( 2) will become 0.4tr, and this corresponds to the optimum value operated under centrifugal force.
Although the rotating direction of the centrifugal ball milling was different from that of the gravitational ball milling, the moving behavior of balls and bowl appeared to be similar in these two different operations.
Thus it is estimated that there might be no radical difference between their grinding mechanisms.However, relatively slow revolution under a centrifugal ball mill operation might cause dissimilar behavior to that mentioned above.This is because balls and a powder are accumulated at the bowl bottom due to innegligible gravitational influence.Therefore the effect of added centrifugal force on grinding rate is required to be examined.

3 The relationship between grinding rate and added centrifugal force
When grinding bowls are not so far from their revolution center, like the present experimental case, the centrifugal forces will not act uniformly on balls in the bowl.Thus there arises a problem on whichever characteristic quantity should be employed to estimate the influence of centrifugal force.The authors attempted to use, as a characteristic quantity, the following centrifugal effect Zm defined as the centrifugal force arising at the * If one takes P S• e S• and e 8 as the particle density, the porosity of powder bed, and the porosity of packed balls, he can get where V s, ap. is the apparent volume of the material and Vg is the gap volume among the balls.Thus, the optimum feed amount is given by the following form under the assumption of the optimum condition proposed by Rose: V s, ap.= Vg.

WF,opt, = VMPsleB(l-es)
Since U, on the other hand, is expressed from Eq. (2) as: It can be optimized by substituting Wp, opt.into Wp; that is, In general, the porosity is 0. For a conventional ball mill, Zm becomes unity, because gravitational force is the only force acting on the balls when centrifugal force is not applied.The authors investigated whether or not Zm might be suitable as a characteristic quantity.In Fig. 9, the specific surface area S, which is indicated in Fig. 4, is plotted against the product of Zm and Nt, that is, the quantity corresponding to the energy added by a centrifugal force.This figure shows that all the experimental values might be expressed by a single straight line.In the Rittinger region, or the initial grinding region, where grinding energy is proportional to the increase of specific surface area, all the measurements including the result of gravitational grinding could be expressed as a single straight line.In the Rittinger region, the relationship between the increment of specific surface area !:i.S and the increment of grinding energy !:i.E is written by The Rittinger number (!:i.S/ !:i.E) can be considered to be independent of Zm .Thus the result of Fig. 9 shows (14) This suggests that Zm could correspond to the grinding energy per one rotation of the bowl.
It is also found that the extrapolation of this relation would reach the individual value at Zm = 1.This implies that the use of Zm could connect the rate of gravitational grinding with that of centrifugal grinding.When Zm < 4, on the other hand, the poor effect of centrifugal force could produce small grinding rate.This suggests that efficient operation might be given in the region of larger centrifugal force.Substituting Eq. ( 15) for Eq. ( 11), the following equation of grinding rate is obtainable.The above equation can be rewritten by using differential form with respect to time.
Where K 1 , n1, K1, and m 1 depend on kinds or fractional fillings of materials.These values used in the present work were K 1 of 0.14 1 1 m 2 kg-1 ,n 1 of0.92,K 1 of6.9xi0-2mn,kg-n, and m 1 of 1.15 under the optimum condition (J = 0.34, U = 0.41) which was referred to in Section 3.2.
With regard to a ball mill under centrifugal force, the rotation speed of a bowl-NM varies proportionally with the revolution speed N 8 .Thus, NM is related to Zm as The above equation states that more added centrifugal force never fails to provide larger value of N M. Consequently, the duplicated effect caused by them tends to increase the grinding rate.For instance, if a feldspar is ground at Zm = 15 and under optimum fractional fillings of J and U, dS/dt will be 57 times as large as that under gravitational force (NM = 130 r.p.m.)

Conclusion
Size reduction by a ball mill under centrifugal force was investigated experimentally.The result shows that the addition of centrifugal force could reduce effectively operation time.The overall grinding rate equation was presented to describe the variation of specific surface area under both centrifugal and gravitational force, by the introduction of a typical descriptor to assess an extent of added centrifugal force.

Fig. 3 as 2 F1
Fig. 3 Description of forces acting on a ball

4 Fig. 4
Fig. 4 Relationship between specific surface area of product and number of bowl rotations in the beginning of grinding

Fig. 5
Fig. 5 Relationship between specific surface area of product and number of bowl rotations

F~>F 2
4 when spheres of equal size are packed at random.Therefore if one takes "S as "B• he can obtain Uopt. of about 0.4.