Numerical Analysis of Movement of Balls in a Vibration Mill in Relation with Its Grinding Rate t

Effects of grinding conditions and fractional ball filling on the behavior of balls and their collision characteristics in a vibration mill were investigated by image analysis with a video recorder as well as by a numerical calculation method. It was made clear that an increase in fractional ball filling causes an increase in frequency but a decrease in average intensity of ball collisions, and that it results in acceleration of the circulation speed of the balls in the reverse direction to that of circular vibration of the mill. A good correlation independent of vibration conditions and fractional ball filling was obtained between the rate constant of grinding determined by grinding experiments with glass beads as a feed material and the effective breaking collision frequency of balls calculated by the simula tion and defined as the frequency of ball collisions whose intensity is greater than the strength of feed particles. From these results, it was made clear that there is an optimum fractional ball filling, depending on the strength of the feed materials and vibration conditions of the mill, that maximizes the rate of grinding.


Introduction
There are various types of ball-media mills which lately attract attention as ultrafine grinding mills. In order to elucidate the grinding mechanism of these mills, knowledge of movement of balls is fundamental and essential. Since many years ago, there have been published a number of experimental and theoretical studies on the ball movement in vibration mills 1 ) which have about ten times higher grinding rate than tumbling ball mills, including the early work of Bachmann 2 l. As the recent analytical approaches, a group of balls were treated as an assemble by Kousaka et a!. 3), the one-dimensional vibration of a few balls was analyzed by Inoue 4 l, the effects of vibration frequency and amplitude on the collision velocity and frequency of a single ball in one-dimension was studied by Kuwahara et a!. 5 l and the motion of a single ball was analyzed in two dimensions by Suzuki et a!. 6 l On the other hand, some attempts have been made in experimental ways with stress sensors buried in the inner wall of mill pots 7 l or with Furo-cho, Chikusa-ku, Nagoya 464-01, japan t This report was originally printed in Kagaku Kogaku Ronbunshu, 17, 1026-1034 (1991) and 18, 78-86 (1992) in Japanese, before being translated into English with the permission of the editorial committee of the Society of Chemical Engineers, Japan.
KONA No.ll (1993) mechanical and electrical sensors fixed in the balls 8 ) to measure the intensity and frequency of their collisions. However, there have been seen few reports which analyzed the movement of individual balls or measured the collision intensity and frequency of small balls in a vibration mill.
In practical use, vibration mills are operated usually with the fractional ball filling of 0.8 to 0.99· 10. 11), which is quite larger than that of tumbling ball mills. Concerning the effects of fractinal ball filling, a number of reprots have been published with the discussions using a model of double actions 12 l, on the relation with the circular speed 13 l and on the effects of mill's shape 14 l. However, the effects of fractional ball filling on the grinding properties as well as the movement and collisions of individual balls have not been cleared yet.
In the present study, it was attempted to simulate the movement of balls in a vibration mill by means of a numerical calculation method taking accout of the interactions between the individual discrete elements 1 5. 1 6. 1 7. 18) and to compare the results with those obtained by experiments with a video tape recorder. In addition, the grinding performance was attempted to correlate with the intensity and frequency of ball collisions calculated from the simulation of ball movement. The focus was placed on the investigation of the effects of vibration amplitude, frequency and the fractional ball filling of a vibration mill on the grinding performance as well as the movement of balls.

Simulation by discrete element method
The balls in the pots of vibration mill with circular oscillation are put into violent motion by receiving kinetic energy through the collisions against the inner wall of pot or other surrounding balls. Since the drag terms by fluid in the equations of motion for the larger balls in the air are negligible, their movement is determined almost by the contact force from the mill pot and other balls as well as by the gravity force. In fact, the individual balls repeat the collisions many times in a short time period and besides the collisions are not completely elastic because of local plastic deformation and damage in the vicinity of the contact points. It was attempted here to simulate the movement of individual balls by use of discrete element method developed by Cundall et al. 19  Assuming that each ball is a rigid sphere, the elastic and non-elastic characteristics at the ball collisions are expressed by modeling with elastic spring (stiffness k) and viscous dashpot (contact damping coefficient YJ) as demonstrated in Fig. 1, leading to the following equation of motion for the balls in the two-dimensional pot.
where u and¢ are translational and rotational displacements, the subscripts n and s denote the normal and tangential directions, and m, r 8 and I are mass, radius and inertia moment of a ball respectively.
The position and the velocity of individual balls at a time t are obtained by the numerical integration of the above equations. The time increment .:l t was set at 1.0 x 10· 4 s from the practical viewpoint taking account of the condition of .:l t < 2 ~as recommended by Cundall for the convergency and stability of the solution. The stiffness in the normal direction kn = 1.45 x 10 5 N/m was adopted by the fitting based on the comparison of absolute velocity of balls with those determined by video pictures. The stiffness in the tangential direction k s was assumed to be a quarter of kn as proposed by Kiyama et al. The contact damping coefficient YJ in both the normal and tangential directions were calculated from the following relation to maximize the damping of the abovementioned one-dimensional vibration equation.
As for the friction coefficient between balls, the value of 0.33 was used 20 l. The properties for collisions between the balls and the inner wall of mill pot were regarded the same as those between the balls.

Experimental
The vibration mill used for the experiments (model MB-1.6 made by Chuo-Kakouki Ltd.) is quipped with a pair of pots which oscillate in circle around a horizontal axis. The vibration amplitude and the frequency were controlled with unbalance weights and an inverter respectively and set at 3.6, 4.5 and 5.2 mm ing at an interval of 60 1/s. The video pictures were superimposed on a display connected with a computer and the position of all the balls in the cell were memorized manually with a mouse for successive 20 picture frames. The velocity of balls was obtained as the distance between the positions of each ball in two successive frames divided by the time interval of 1/60 s. In addition, the circular movement of balls in the cell was noticed and its speed was determined from the time required for a single ball to make one tum by the slow motion video pictures.
In order to evaluate the results obtained by the simulation, grinding experiments were carried out with glass beads screened between 42 mesh (350 .urn) and 28 mesh (590.um) beforehand. The fractional feed filling of ball bed voidage was kept constant at 0.2. The ground fraction at each batch grinding was obtained from the weight passing 350.um screen. In order to investigate the appropriateness of the simulation, the dispersed states of balls calculated were compared with those obtained by observation with a video camera under various conditions of the vibration amplitude and frequency of the mill, in the radial (Fig. 4) and the tangential directions (Fig. 5).

Comparison of the calculated with
These comparisons of ball number density show good agreement between the results obtained by simulation and experiments also under other vibrating conditions and fractional ball filling. As a general tendency, the ball number density near the inside wall is higher than that in the core. It is also clear from the figures that the ball number density is highest in the third quadrant and lowest in the first under all the vibrating conditions, though the balls are not well dispersed with the higher density near the pot bottom under the weaker vibrating conditions.  were modified corresponding to 1160 s. The modified ball velocity agreed well with those determined by the observation at every condition of vibration.
simulation and experiments agree well and a ffilmmum velocity is seen around J = 0.8.

3) Circulation speed
From observation of the movement of balls, it was noticed that the balls make circular movement as a whole in the mill pot and this was confirmed also by the simulation as shown in Fig. 8. This phenomenon of circulation of balls in the vibration mill, which would affects greatly the circulation of feed materials, has been long acknowledged qualitatively but its quantitative consideration has hardly been made. Fig. 9 shows the change of circulation speed of balls with the fractional ball filling under the standard vibrating condition off= 25 s· 1 and 2a = 5.2 mm. Both the simulation and the observation with VTR indicate that the balls tend to circulate in the reverse direction against the circular oscillation of the mill pot and that there is a point where the direction of circulation changes.
Furthermore, another new knowledge has been obtained on the circulation of balls in the mill in relation with the friction coefficient of balls P-· The results of circulation speed obtained by simulation with JJ-= 0.33 agreed well with those determined with VTR when the balls used repeatedly after washing and drying were applied for the tests. However, when entirely new balls were used for the experiments, the circulation speed was + 0.115 1r rad/s, while used balls showed -0.314 1r rad/s under the same condition. In order to clarify this phenomenon, the simulation was carried out with different frictional coefficients to calculate the circulation speed (Fig.  10). This figure shows clearly the tendency that the balls tend to circulate in the reverse direction against the mill's oscillation at each constant fractional ball filling, as the friction coefficient increases. It was confirmed that the friction coefficient of balls hardly affected the ball velocity, the intensity and frequency of ball collisions in spite of its considerable influence on the circulation movement of balls. From the above results of dispersion state, velocity and circulation speed of balls, this simulation method seems highly useful to estimate the movement of balls in a vibration mill at different vibrating conditions and fractional ball filling.

Relations with the grinding rate 1) Effects of vibrating conditions
This numerical calculation method has a great advantage to provide with information of frequency and intensity of ball collisions, the measurement of which has hardly been possible conventionally. An example of distribution of ball collision intensity obtained by the simulation is shown in Fig. 11, where the intensity ranges mostly from 1 to 100 N at the fractional ball filling of 0.9 under the standard vibration condition. The total number of collisions per unit time Z is given by the integration of Z F with the collision intensity and amounted to 11,800 to 15,000 per second at the ball filling of 0.8 within the experimental range, which correspond to 220 to 280 collisions per ball in a second. As seen from Fig. 12, the total collision frequency does not change so much as vibration amplitude and frequency themselves in the experimental range of amplitude up to 5.2 mm.
On the other hand, the average collision intensity changes greatly depending on the vibration amplitude and frequency in the experimental range, though it tends to saturate above the amplitude of 7 to 8 mm (Fig. 13). Fig. 14 shows the relationship between the total collision frequency and vibration velocity, which are well correlated independent of the vibrating conditions. As seen in Fig. 15, however, the average collision intensity is not so well correlated with only the vibration intensity defined as K =a w 2 /g.
Then, the effects of vibration amplitude and frequency on the total frequency and average intensity of ball collisions were examined keeping the vibration intensity constant as shown in Figs. 16 and 17. It is seen from these figures that under the normal operational range of amplitude and frequency both the total collision frequency and the average intensity increase with the increasing amplitude but that both of them decrease with the increasing vibration    Effect of vibration amplitude on total collision frequency and average intensity. K-5.64 [-] collision intensity at of feed on the movement of balls. Fig. 18 shows the relationships of the average intensity and total frequency of ball collisions with the fractional ball filling ] at different vibration frequencies obtained by the calculation. As seen from the figure, the total collision frequency increases but the average collision intensity decreases with the increasing ] . Therefore it is probable that there should be an optimum fractional ball filling and vibrating conditions depending on the size and strength of feed particles. The grinding experiments have been carried out with glass beads as feed under different vibrating conditions to study the effect of fractional ball filling on the grinding rate. Fig. 19 shows the plot of residues in the product R over 350~tm, which is the minimum particle size in the feed, against the grinding time t on a semi-log paper. Assuming the following firstorder equation of disappearance the rate constant of grinding k 1 is determined from the gradient of lines in Fig. 19 as follows. logR = -log e · k 1 • t The plot of this k 1 against the fractional ball filling ] is shown for two vibrating conditions in Fig. 20, which indicates maximum values of grinding rate around]= 0.9. On the other hand, the grinding rate has been conventionally evaluated using selection functions. The selection function for vibration ball mills S (x), which corresponds to the rate constant of grinding with feeds of narrow size distributions, has been discussed in the following equation from the statistical standpoint 22 · 23 · 24 l S (x) = c · P · Pr · Z sp n (6) where P n is the probability of nipping of a feed particle between balls, Pr the probability of a nipped particle being broken, Z ball collision frequency and csp a constant. It is assumed that Pr is determined by the relationship between the strength of feed particles and ball collision intensity. Consequently, the relationship between the rate constant of grinding obtained from the grinding experiments k 1 and the ball collision frequency Z obtained by the simulation KONA No.ll (1993) should differ depending on the vibrating conditions and the fractional ball filling.
In order to relate these factors, the following frequency of ball collisions effective to break a particle (hereafter called effective breaking collision frequency) was defined taking account of the distribution of ball collision intensity and breakage strength of a feed particle F cr as follows :

Figs. 21(a) and (b)
show the change of effective breaking collision frequency with the fractional ball filling at different vibrating conditions. Although the total frequency of ball collisions at the stronger vibration (Fig. 21 (a)) hardly differs from that at the weaker one (Fig. 21 (b)), the frequency of ball collisions with the intensity larger than some decades of N radically decreases at the weaker vibration. In addition, the effective collision frequency shows its maximum around J = 0. 9 in the similar way as the grinding rate does. The relation between the rate constant of grinding k 1 determined by the experiments and the effective breaking collision frequency Zcr obtained by the simulation for Fer= 60 N from literature 25 l is shown in Fig. 22, which gives the following linear equation independent of the vibrating conditions and the fractional ball filling of the mill (8) .!!!.
.!!!. where ckz is a constant related with the probability of particle nipping depending on the ball diameter and size of feed particles. From these results, it is concluded that the grinding rate could be estimated from the ball collision intensity obtained by the simulation of ball movement as well as the breakage strength of feed particles to some extent. It would be necessary, however, to take account of the effects of feed materials on the movement of balls as the fractional feed filling increases to influence it.

Conclusions
The results obtained by the numerical calculation of movement of balls in a vibration mill using the discrete element method agreed well with those determined by the observation with a video tape recorder in terms of dispersion state, velocity and circulation speed of balls in the pot and it was found that this numerical calculation method is useful for describing the behavior of balls in the vibration mill.
Additionally, it has become possible by this numerical simulation to calculate the intensity and frequency of ball collisions, which have been difficult to measure directly and the following points have been made clear: (1) This numerical simulation describes the movement of balls well with the same parameters at different fractinal ball filling as well as vibrating conditions. (2) The frequency of ball collisions is determined definitely by the vibration velocity proportional to the product of both vibration amplitude and frequency, although the average intensity of ball collisions is not determined definitely by neither vibration velocity nor vibration intensity.
(3) As the fractional ball filling increases, the frequency of ball collisions increases but their average intensity decreases. (4) As the fractional ball filling increases, the balls tend to circulate in the reverse direction against the circular vibration of mill pot.
(5) The circulation of balls is influenced by the friction coefficient of balls 11-· With the larger Jt, the balls tend to circulate in the reverse direction against the circular vibration of the mill pot. (6) At a costant vibration intensity, both intensity and frequency of ball collisions increase with the increasing vibration amplitude but decrease with the increasing vibration frequency under usual grinding conditions. This results obtained by the numerical calculation explains well the experimental results under the similar grinding conditions. (7) A linear relation was found between the rate constant of grinding and the effective breaking collision frequency of balls defined as the ball collision frequency with intensity larger than the particle breakage strength, independent of the vibrating conditions and the fractional ball filling. (8) The optimum fractinal ball filling to maximize the grinding rate was regarded as that to maximize the effective breaking collision frequency depending on both the feed particle strength and grinding conditions and was calculated to be around 0.9, which agreed well with the experimental results.
Although the numerical simulation here has been carried out for the 2-dimensional pot to realize the observation of the ball movement directly for the comparison of simulation with experimental data, the results would be possibly extended to the 3-dimensional cases to a great extent. This matter would be future subjects to be investigated together with the effects of other conditions such as the feed and surrounding fluid.