Optimum Design for Fine and Ultrafine Grinding Mechanisms Using Grinding Media t

Ultrafine grinding in the submicron range has currently attracted attention in connection with the development of new ceramics and electronic materials, and quite a few investigators have reported experimental data of wet process milling using grinding media. As mills using grinding media, conventional ball mills, vibration mills, planetary mills, and stirred mills are typical machineries; special interests have been focused on ultrafine grinding using grinding balls smaller than 1 mm in diameter. Based on the experimental data presented recently in Japan including the classical data well known worldwide, a general form of the selection function applicable commonly to a wide range of particle sizes if possible for various kinds of grinding mills, and the optimum ball size to maximize the rate of grinding are first dealt with in this review. The comminution kinetics is referred to in order for the design of the ultrafine grinding mechanism to be emphasized, in which the size distribution of the finished product can be discussed in relation to the particle size and the time required. The size distribution governs the properties of the ground product so that the size distribution should be adjusted to meet the requirement by setting up a closed circuit grinding system. Various modes of the closed circuit system are considered together with the basic characteristics of the resulting size distribution as well as the basic design procedures. Furthermore, the improvement in the rate of grinding is considered from various points of view. One possibility involves composite balls of different sizes which have been declared in most text books on comminution to be ineffective. On the contrary, this review shows that a specific size distribution of the balls may lead to an remarkable improvement in the rate of milling. Finally, physicochemical consideration is also taken into account for the purpose of grinding rate promotion. Grinding aids can be used not only for improving the grinding rate but also for the modi-


Introduction
Ultrafine grinding in the submicron range has currently attracted attention in connection with the development of new ceramics and electronic materials, and quite a few investigators have reported experimental data of wet process milling using grinding media. As mills using grinding media, conventional ball mills, vibration mills, planetary mills, and stirred mills are typical machineries; special interests have been focused on ultrafine grinding using grinding balls smaller than 1 mm in diameter. Based on the experimental data presented recently in Japan including the classical data well known worldwide, a general form of the selection function applicable commonly to a wide range of particle sizes if possible for various kinds of grinding mills, and the optimum ball size to maximize the rate of grinding are first dealt with in this review.
The comminution kinetics is referred to in order for the design of the ultrafine grinding mechanism to be emphasized, in which the size distribution of the finished product can be discussed in relation to the particle size and the time required. The size distribution governs the properties of the ground product so that the size distribution should be adjusted to meet the requirement by setting up a closed circuit grinding system. Various modes of the closed circuit system are considered together with the basic characteristics of the resulting size distribution as well as the basic design procedures.
Furthermore, the improvement in the rate of grinding is considered from various points of view. One possibility involves composite balls of different sizes which have been declared in most text books on comminution to be ineffective. On the contrary, this review shows that a specific size distribution of the balls may lead to an remarkable improvement in the rate of milling. Finally, physicochemical consideration is also taken into account for the purpose of grinding rate promotion. Grinding aids can be used not only for improving the grinding rate but also for the modi-Tatsuo Tanaka* fication of the finished surface of the ground products.
1.1 Rate of ultrafine grinding related to ball size K. Tanaka et all.2l reported extensive experimental data of finely grinding BaTi0 3 to 1.9 micron by using a ball mill and a vibration mill, varying widely the ball diameter and the specific gravity, the ball filling degree, the rotating speed or vibrational conditions as well as the slurry concentration. Among them the effect of ball size on the rate of grinding is noteworthy. The rate of grinding calculated from the specific surface area increase per unit time measured with the BET method was expressed in the form of the following empirical equation.
On the other hand, Tashiro et al 3 l conducted a ball milling speed of 220 rpm with zirconia balls ranging in three stages from 1 to 15 mm in diameter D. They proposed the following empirical equation under some theoretical assumptions: wherein b is the size of particle and D is the ball diameter.
The formulae above are somewhat similar and it is of some interests to note that Eqs(1) through (3) were obtained by taking the rate of grinding as being the product of an increasing function and a decreasing function with increasing D (or r) although the resulting expressions are quite opposite. These works, however, are extremely important contributions because they derived a general form of the selection function applicable to grinding in the submicron range in which, if dry process had been performed, a grinding limit would have been probable.
From these studies it is suggested) that the ultrafine grinding predominantly proceeds by compression and shearing at the contact points between colliding balls, not by the impact of balls dropping onto the powder bed. Thus, the rate of grinding or the selection function may be proportional to the following items: (1) total number of contact points between balls, (2) collision frequency of a ball with another, (3) volume of particle nipped at a contact point for grinding, (4) pressure acting on the particles nipped by balls, (5) probability that the stress applied on the particles exceeds the inherent strength for possible fracture. Each item can be formulated as follows (1) the total number of contact points is inversely proportional to the cube of a ball diameter under a constant ball filling degree, varying with ]ID 3 . (2) the collision frequency can be given as a function of v/D, where v is the speed of a colliding and the mean free path varies with D. (3) the volume nipped by two balls is calculated to be proportional to Db 2 (4) the stress created by hitting balls can be expressed by the force divided by the effective contact area between a ball and the nipped particle bed. The surface area is found to be proportional to Db and to the force pD 3 a where p is the density of a ball and a is the acceleration following a collision. Thus, the stress is proportional to D 2 pa/b. (5) if the stress above is smaller than the strength required to crush the material then no grinding can be expected. For this purpose the following probability P should be multiplied: P= exp{-(bi7 5 rfpD 2 a)} =exp{-(Dm/D)n} (4) Dm =a function of (17 51 blpa) (5) a is unknown but the highest acceleration of a ball dropping onto a powder bed of the tightest packing was reported to be proportional to the hitting velocity, v. The exponent n in Eq (4) appears equal to 2 dimensionally but preferably can be adjusted from experimental data. Dm is the representative ball diameter for a given material to be nipped for possible fracture by overcoming the strength. The probability Pis plotted against D/Dm in Figure 1, where Dm is D corresponding Figure 1 to P = 0.368. Therefore, the rate of grinding can be expressed by the product of the items (1) through (5), divided by the total weight of the material in the mill, considering the frequency of the materials introduced into the comminution area. The total holdup W is proportional where fc is the fractional volume of the material inside the mill and equals ]U. Compared with Eq(1) to (3), the derived formula(6 ') looks like Eqs(1), (2) rather than Eq(3), as far as the effect of ball diameter on the rate R is concerned, the latter of which, however, involves b, being consistent with Eq(6 ').
Differentiating the above equation with respect to D and putting the derivative equal to zero gives Dopt= (1/n)-llnDm, wherein Dopt is the optimum diameter of the grinding balls and when n = 1, D opt = D m, and when n = 2, Dopt = 1.4 Dm. Therefore, if Eq(6 ') is to be confirmed by experimental data, D m becomes very important for determining the most efficient grinding condition for a given material.

Confirmation of the derived equation from various aspects
Current experimental data 5 -14 l including the classically well known grinding studies are listed in Table  1, in which the documentations are presented for ball mills, vibration mills, planetary mills and stirred mills. Most works here were carried out through a wet process and fortunately the selected data were obtained almost at the same hitting velocity, v =, 1 m/s, even with different types of mills of different sizes, so that we can cancel v temporarily in Eq(6 '). First, Eq(6 ') will be verified using the 4 kinds of mills data by varying the ball size to obtain the maximum value of R, the other variables remaining unchanged. Assuming D m first then R is calculated from Eq(6 ') as a function of D. Figure 2 shows the calculated curve as compared with K.Tanaka's 1 l wet ultrafine ballmilling data. A logarithmic plot reveals that a better coincidence can be brought to about n = 1 rather than n = 2, to be more parallel. D m is 2 mm. Likewise, vibration milling data by Tanaka confirmed Eq (6') where D m = 1 mm as shown in Figure 3 trend was also found with the data of Coghill's ball mill fine grinding as shown in Figure 4 where n = 1 is still better than n = 2. l2) As in Table 1 the materials used are quartz, calcite, coal, dolomite and some electronic materials like BaTi0 3 , of which the strengths are not always clear except for calcite and quartz. Thus, it is assumed here that the strength of materials as well as the size dependence except for quartz are the same as those of calcite. The strength of some materials was reported in detail by Yashima Table 1, an important correlation is found as shown in Figure 5, from which we have, regardless of the type of mills. 4 l where SI units are used for p (kg/m 3 ) and f/st (Pa) where q nearly equals 0.5, Eq(7) becomes Dopta:b 0 · 6 , which agrees with the conventional optimum ball size in relation to particle size.
Next, as an example for confirmation by varying b in Eq (6' (7), which enables one to calculate R as depicted in Figure 6. The dotted curve was drawn by using a complex probability function, which appears to be almost proportionally coincident with Eq(6'). The influence of the density of the balls on the rate of grinding should be almost linear in Eq(6'), whereas K.Tanaka 1 · 2 l varying p=(2.7-7.8)x103 kgfm3, supported this both for ball mills and vibration mills. Zhao et al 10 l also recognized this effect in the planetary mill experiment, and Mankosa 6 l pointed out that steel balls are clearly more efficient than glass balls with stirred mills at a lower speed of the impeller, v = 1.7 m/s. It is very interesting, however, to note the latest report of Kugimiya 8 l who stated that no effect of ball density had been recognized in the wet stirred milling with much higher rotating speed, e.g., v = 14 m/s. This means that the 4th term in Eq(6) can be replaced by a constant shearing stress arising from the rotating impellers, resulting in Ra:(b2JD3). In fact, Kugimiya 8 l and Hashi7l 's data for high speeds seem to confirm it in Figure 7, and more improvement would be expected with stirred mills for ultrafine grinding by varying the speed, because other types are more or less restricted in increasing the hitting speed freely due to their constructions and mechanisms.
As to the effect of ], the mill filling degree, in Eq(6'), Rose 13 l stated that the rate of milling is approximately linear with ] up to ] =50-60o/o for ball mills and]= 80-90o/o for vibration mills. 17 l K.Tanaka 1l indicated that at ] = 40 and 60o/o the rate of grinding was the same for ball mills. Furthermore, they also did experiments of vibration milling, varying the slurry concentration 2 l by holding a constant volume of water.
This corresponds to the relation between R and fc in order to check Eq(6 '), which is depicted in Figure  8. Therefore, the comfirmation of Eq(6') is reasonably satisfactory so far for the various terms concerned.

Kinetics of comminution and the selection function
The rate of grinding based on the specific surface increase per unit time has been dealt with in comparison with fine and ultrafine grinding mechanisms using balls as grinding media. On the other side, the particle size distribution of the finished product is also important as it influences the properties of the product. To discuss the size distribution in the lapse of time the kinetics of comminution is significant. Defining the selection function, S(x, t) and the breakage function, B(y, x), a dynamic mass-size KONA No.l3 (1995) ? where D(x, t) is the cumulative undersize, t the time, x the particle size, and -y is the size of a single particle to be broken. xm is the maximum particle size present. Assuming B( -y, x) = (xl-y)m, and S(x, t) = Kxn, wherein K is the grinding rate constant and m, n are the exponents, respectively, 18 l then Eq(8) can be analytically integrated into the following forms, R(x, t) being the cumulative oversize: When m =n, (mostly n = 1)

R(x, t) = R(x, O)exp(-Kxnt)
When mJn, then where Jl and P are determined from m/n, see Figure  9. Eq(9), (9 ') have been accepted for many years as the Rosin-Rammler size distribution as a function of time and if the selection function S(x, t) is replaced and is assumed to be proportional toR in Eq(6'), i.e. the particle size x is replaced by b in Eq(6 ') in the range where R is proportional to b as in Figure 6, then the rate constant K can be regarded as a definite function of the various parameters involved. This might make it possible to regulate the size distribution of a product by adjusting the operating variables of the mills. Namely, where Ra:b and x( =b) are constant, Ra:S(x, t) a:K. Then, we have from Eq(6 ')

S(a:Kfca: (v 2 ]p!D)exp{ -(D miD)} (10)
This suggests a possible contribution of the operating variables of a mill to the rate constant K which con- trois the size distribution of a product. Figure 10 represents the product of the experimental selection function of a ball mill by the fractional holdup fc together with] and U, proposed by Shoji et al 2 0l. Eq (6 ') indicates that Sfc varies with ], but is independent of U, whereas Figure 10 indicates a similar tendency.

Size distribution controlled by various modes of closed circuit system
Size distribution can be regulated by adopting the closed circuit grinding system composed of a mill and a classifier as schematically shown in Figure 11. The theory has been dealt with elsewhere 21 l regarding the case where both a clean cut classifier and a nonideal classifier are involved, and, a nomography was given for the former as in Figure 12, to readily obtain the necessary parameters for designing the circuit. The basic equations are Xc *n = F/KW (13) where RP is the cumulative oversize of the product, C1 is the circulating load ratio (=mill return/product), xc the cutoff size of the classifier, and xc * is the characteristic parting size when C1 tends to infinity. been considered in industry as in Figures 13 and  14, 22 l the characteristics of them are that the feed materials are first introduced into a classifier before entering a mill. The latter uses two classifiers. As a result of the analysis of these circuits, the nomograph in Figure 12 can be used as it stands, except that C1 = 1 + '' C1'' for Figure 13 and C1 1 (1 + C1 2 ) = 1 + "C1" for Figure 14, where "C1" is the value of C1 derived from Figure 12 to fulfill the same design requirement with the use of a standard scheme as shown in Figure 11. The suffixes 1, 2 denote the number of the classifiers indicated, resp. Generally, the product of Cl added by 1 means an advantage in the grinding capacity. On the other hand, two other schemes of closed circuit grinding systems are introduced in Figures 15 and 16, both of which are featured by using two mills and one classifier. The analysis of those circuits provides another nomograph to find out the necessary parameters for the design of those schemes (Type 1 and Type 2, respectively) as shown in Figure 17. Figure 18 illustrates the calculated size distribution of the product with each scheme as well as the circulating load. It should be emphasized here that Cl being infinity appears  impossible in a real closed circuit but this is practically realized in a mill of an air swept system, for example. This corresponds to the mechanism where only fine particles producing smaller than xc should instantaneously be discharged from the mill, the theory of which was reported by Ouchiyama et al. 23 l 26 5. Possibility for increasing the rate of grinding Now let's come back to the rate of grinding affected by the size of the grinding media. In view of the most experimental data in the past we feel that the size of balls was uniform and very few studies dealt with composite balls. Some books on comminution 15,16) stated that the exponent q should be nearly 3 in the Gaudin-Schuhmann expression, P(D)a:Dq, where P(D) is the cumulative undersize weight distribution with respect to ball size D. Based on this equation, the number frequency f(D) becomes (14) The average ball diameter D is therefore given by where k=DsfD 1 , the ra~ of the smallest size, D 5 , to the largest one, D 1 • D can be used in items (2) through (5) Substituting Eq(15) through (18) into Eq(6 ') we have the rate of grinding at constant b and fc as well as P=l for the case of a composite ball size.

Ra:(W 'ID/(qll-kG)[3{ (q-3)/(l-kG-3)} ·
{ (l-kG-2 J!(q-2J F +(l-kG-1 JI(q-lJ]!4 (19) Setting the ratio of W '/D 1 to be unity, R is computed as a function of q and k and given in Figure 19. It is at once noted that as q tends to be small R is likely to increase, so that the larger size distribution of the balls appears to be preferable.
Only three runs of experiments listed in table 1,  Figure 20, the necessary parameters are obtained as q = 0.5 and k = 0.06, which yield R = 3.2 from Eq (19). The experimental data demonstrated that the grinding rate was 3 times that obtained with single sized balls, indicating an outstanding increase and a satisfactory agreement with the calculation. In Figure 19 we see that R never change too much with q greater than 3. This does not contradict the experience which has been stated in textbooks which indicate composite balls were not effective. It can be concluded that for fine and ultrafine grinding considerable improvements may still be expected by selecting appropriate grinding mechanisms as well as operating variables as described above. Moreover, an emphasis should be put upon the physicochemical devises that make it possible to increase the rate of grinding. The strength of a material 0' 5 t is given by (20) where Yis the Young's modules, A the surface energy and c is the crack length. Wet process and addition of some surfactants serve as reducing A by adsorption effect on the solid surface, leading to a decrease in the strength, O'st· These additives are called grinding aids which have been noted for use to modify a fresh surface exposed to grinding, taking advantage of the reactivity. This assists fine particles to be well dispersed, resulting in better grinding efficiency. It has been reported that about a 20 time improvement was secured in the rate of grinding. Cryogenic grinding is also an application of physical chemistry, which reduces the toughness m to assist crushing.

Conclusion
Contribution of various parameters regarding grinding mechanisms using balls was reviewed theoretically and compared with current experimental data of fine and ultrafine grinding in the submicron range. An important correlation was obtained between the optimum ball size and density, and the size and the strength of a material regardless of the kind of machinery. The comminution kinetics was discussed not only in relation to the rate constant as a function of these parameters for design procedure, but also for the adjustment of particle size of the product by varying the closed circuit grinding schemes. A new proposal for the size distribution of balls has been made for securing the improvement of the rate of grinding, although it appears generally pessimistic in text books. Other various trials are to be considered from different aspects of comminution techniques.