On modelling of submicronic wet milling processes in bead mills

Powder Technology 143 – 144 (2004) 253 – 263 www.elsevier.com/locate/powtec

On modelling of submicronic wet milling processes in bead mills Christine Frances * Laboratoire de Ge´nie Chimique-UMR CNRS 5503, ENSIACET-5, rue Paulin Talabot-BP 1301, 31106 Toulouse Cedex 1, France Available online 20 June 2004

Abstract Methods used for modelling submicronic wet grinding processes are discussed. Simplified solutions of the population balance equation are first tested on experimental results obtained during batch runs performed with media mills on different materials. The use of statistical laws for modelling multimodal particle size distributions resulting from size reduction and aggregation processes is then presented. Finally, a method of macroscopic modelling to predict the particle size distribution of product from continuous ultrafine grinding is presented. It combines batch grinding kinetics and a solids flow model. D 2004 Elsevier B.V. All rights reserved. Keywords: Comminution; Ultrafine grinding; Stirred media mill; Particle size distribution; Grinding model

1. Introduction Comminution models are used as a basis to choose process conditions for milling, design mills or scale-up laboratory tests to industrial plants. The methodology to describe grinding operations is similar to the one developed for other unit chemical engineering operations, as for example chemical reaction. Thus, a grinding step is defined by the fragmentation kinetics and the mill is the reactor, in which this operation occurs, characterized by the hydrodynamic conditions inside the grinding chamber. The aim of grinding models is to predict the product quality as a function of the initial material properties and the operational conditions. The objective of this paper is to discuss how to extend the existing knowledge of modelling ultrafine grinding operations. This discussion is based on wet grinding tests with different materials, performed over the last 10 years at the Chemical Engineering Laboratory in Toulouse (UMR CNRS 5503).

2. Modelling of batch grinding 2.1. Population balance approach—fine grinding processes Batch grinding is generally described on the basis of the selection function (S), which represents the aptitude of * Tel.: +33-5-34-61-52-24; fax: +33-5-34-61-52-53. E-mail address: [email protected] (C. Frances). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.04.018

particles to break and the breakage distribution function (b), which describes the distribution of the resulting fragments. The general form of the overall mass balance equation for batch grinding is: Bmðx; tÞ ¼ SðxÞmðx; tÞ þ Bt

Z

l

SðyÞbðx; yÞmðy; tÞdy

ð1Þ

x

where m(x,t)dx is the mass fraction of particles in the size range x to x + dx at a time t. Cumulative distributions are sometimes more practical to use. The retained cumulative oversize distribution, corresponding to m(x,t), is: Z l Rðx; tÞ ¼ mðy; tÞdy ð2Þ x

Combining Eqs. (1) and (2), the following integrodifferential equation for batch grinding is written: BRðx; tÞ ¼ Bt

Z

l

SðyÞBðx; yÞ x

BRðy; tÞ dy By

ð3Þ

where B(x,y) is the cumulative breakage function. The general analytical solution of batch grinding equations (Eq. (1) or Eq. (3)) comprises a convergent series of iterated kernels. This solution is too complex and has found no practical application. Many workers have developed analytical solutions of that integro-differential equation, using simplified representations of the selection and breakage functions. Kapur [1], in particular, has shown that the

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grinding function function function

equation has a similarity solution if the selection is a power law and the breakage distribution is self-similar, i.e. if the cumulative breakage can be expressed as:   x Bðx; yÞ ¼ B ð4Þ y The interesting aspect of that solution is that the distributions, generated at different grinding times, coincide on a single curve when they are plotted as a function of the dimensionless particle size x/l1(t), where l1(t) is the first moment of the particle size distribution. Kapur has demonstrated that the first moment, the median particle size or any other quantile sizes are all proportional to one another and that any characteristic length could be used instead of l1. Fig. 1 shows an example of experimental size distributions having a self-similar character. It concerns gibbsite with a straight initial size distribution around 100 Am ground in a tumbling ball mill [2]. All the experimental size distributions reported in this paper were measured by means of a diffraction laser sizer (Malvern Mastersizer S) in appropriate dilution conditions. Here, the cumulative size distributions are plotted versus x/x63.2, using x63.2 as the characteristic size corresponding to 63.2% of particles smaller than this size. It can be easily shown that the classical Rosin – Rammler –Bennett (RRB) size distribution is a particular solution of Kapur similarity function. It is shown, in Fig. 1, that the size distributions of comminuted particles can be fitted with a good approximation by a RRB equation type such as:  2 ! x Rðx; tÞ ¼ exp  : ð5Þ x63:2 ðtÞ The self-preserving character of comminuted particles as shown in Fig. 1 implies that, in spite of the complex-

ity of the gibbsite structure, the mechanism of size reduction remains probably the same during the size reduction of that material in the size range from 100 to about 10 Am. 2.2. Population balance approach—ultrafine wet grinding processes Batch grinding population balance is more often used in terms of discrete size classes as follows: i1 X dRi ðtÞ ¼ Si Ri ðtÞ þ ðSjþ1 Bi; jþ1  Sj Bi; j ÞRj ðtÞ dt j¼1

ð6Þ

with Ri ðtÞ ¼

i X

mj ðtÞ

ð7Þ

j¼1

and Bi; j ¼

X

bk; j

ð8Þ

kzi

where mj(t) is the mass fraction of particles in size class j, Si the probability of breakage of particles of class i, bk,j mass fraction of particles of class j, which break and end up in size class k. The analytical solution of that equation, assuming that the selection and the breakage functions were independent of time was given by Reid in 1965 [3]. The main problems in the population balance modelling concern the determination of the selection and breakage functions. Usual methods, used for coarse grinding, in particular those based on experiments done with monoclass size cuts, cannot be applied in the particle size range covered by fine ( < 50 Am) and ultrafine grinding (c 1 Am) due to the difficulty to prepare narrow sieve fractions in that range.

Fig. 1. Self similar size distribution—wet batch grinding of gibbsite in a tumbling ball mill.

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However, it is possible to avoid these problems, considering the following solution of the batch grinding equation given by Kapur [4]:  log

Ri ðtÞ Ri ð0Þ

 ¼ Si t þ

i1 X ðSjþ1 Bi; jþ1  Sj Bi; j Þ j¼1



 expðGj  Gi Þt  1 Rj ð0Þ  ðGj  Gi Þ Ri ð0Þ

ð9Þ

with Gi ¼ Si þ

i1 X Rj ð0Þ ðSjþ1 Bi; jþ1  Sj Bi; j Þ Ri ð0Þ j¼1

ð10Þ

The solution proposed by Kapur is best known under an approximate form obtained by stopping at order 2 the infinite series of the exponential terms of Eq. (9):   t2 Ri ðtÞ ¼ Ri ð0Þexp Gi t þ Hi ð11Þ 2 where  i1  X Rj ð0Þ ðSjþ1 Bi; jþ1  Sj Bi; j ÞðGj  Gi Þ Hi ¼ : Ri ð0Þ j¼1

ð12Þ

From experimental data on size distributions of ground samples, it is possible to derive the functions Ri = f(t) and to obtain Gi and Hi by fitting the experimental curves for each size class, assuming that these functions are independent of time. Compared to tumbling ball milling, stirred milling (Fig. 2) is a process more suitable for the production of submicronic dense suspensions. All the experimental results reported below have been obtained using a laboratory stirred media mill (volume 0.8 l) (Drais Perl Mill). An example of the mathematical method previously described is illustrated in Fig. 3 on the basis of experimental data obtained during a grinding run of gibbsite with an initial mean size around 100 Am [5]. Fig. 3a shows the

Fig. 2. Stirred media mill.

255

evolution of the retained cumulative fractions versus time for various size classes. From these data, the Gi values were derived and fitted by a linear function versus particle size as shown in Fig. 3b. In that example, Eq. (11) taken to order 1 (Hi = 0) was finally used to obtain a mathematical representation of the grinding kinetics. Using the fitting equation reported in Fig. 3b, giving the evolution of Gi versus the particle size xi, the calculated cumulative mass fractions agree well with the experimental ones. However, the discrepancy between simulated and experimental data increases drawing the local mass fraction versus particle size as illustrated in Fig. 3c. It is clear on that example that the model is not able to describe the multimodal character of the size distributions. A second example of such a mathematical treatment is presented in Fig. 4 concerning the dispersion of titanium dioxide in the stirred bead mill [5]. As reported previously, the Gi values are first calculated from the retained size distributions versus time for various size classes. In spite of the perfect fitting of Gi values as shown in Fig. 4a, the experimental size distributions of the product at the selected grinding times are not correctly simulated (see Fig. 4b). Here, the bad fitting of the experimental data results from an agglomeration phenomenon occurring during the dispersion process. Indeed, in its standard representation, the batch grinding equation (Eq. (1), Eq. (3) or Eq. (6)) does not take into account of the possible recombination of fine fragments to produce larger particles. In such situations, the fragmentation balance equation can be coupled with an agglomeration balance equation: dN ðw; tÞ 1 ¼ dt 2 

Z

wwmin

Kðw  w V; w VÞN ðww V; tÞN ðw V; tÞdw V w

Z min wmax

Kðw; w VÞN ðw; tÞN ðw V; tÞdw V

ð13Þ

wmin

where N(w,t) is the number of particles of mass w and K(w,wV) is the agglomeration kernel expressing the probability that two particles of mass w and wVagglomerate at a contact. The first term refers to the production of a particle of mass w resulting from the agglomeration of a particle of mass w – wV and a particle of mass wV. The second term expresses the disappearance of particles of mass w by the agglomeration of each of them with another particle of a different size. The solution of the system composed by both the fragmentation and the agglomeration equation gives the evolution of the size distribution of the product versus time. In practice, hypotheses are usually done to simplify the models and to allow a separate determination of the probability functions [6]. Studies are currently done at LGC Toulouse to obtain a general approach for modelling processes involving both size reduction and size enlargement.

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Fig. 3. Mathematical representation of grinding kinetics using the Kapur population balance solution [4]—wet batch grinding of gibbsite in a stirred bead mill. (a) Change in retained cumulative fractions with time for different size classes. (b) Evolution of the function G versus particle size. (c) Comparison between the experimental and the calculated size distribution obtained after 10 min of grinding.

2.3. Statistical size distributions An alternative way to obtain a mathematical representation of grinding kinetics is to use statistical laws. Classical statistical laws often used to fit size distributions are the Gaudin – Schuhmann law, the Rosin – Rammler law, and the normal or the log-normal law. When the distribu-

tions are multimodal, a single law cannot be used to model the kinetics and a combination of several laws sounds more appropriate. Of course, the higher the number of laws used for modelling, the better will be the fitting to the experimental data. However, the number of laws must keep a physical meaning and must be done on the basis of a quantitative or at least a qualitative characterization of

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agglomerates. For longer grinding times, a new granulometric population appears, corresponding to newly formed aggregates, the proportion of which increases with time. Such an evolution is quite typical for submicronic wet grinding of a material in absence of any dispersing agent. Indeed, during batch grinding, the number of fine particles increases, whereas the distance between particles decreases. Fines are subjected to interparticular interactions including attractive Van der Waals forces. In absence of electrostatic or steric repulsive forces, the agglomeration of fine particles rapidly occurs. The presence of these aggregates is clearly shown on the SEM photo of the sample taken after 45 min of grinding. On the basis of these observations, the experimental size distributions were decomposed into different sub-populations characterized by log-normal laws using the Origin Peak Fitting package (Integral Software, Paris France). The overall size distribution is then described by: fLN ðxÞ ¼

n X

wi fLNi ðxÞ

ð14Þ

i¼1

Fig. 4. Mathematical description of Kapur population balance equation— wet batch dispersion of titanium dioxide in a stirred bead mill. (a) Evolution of the function G versus particle size. (b) Comparison between the experimental and the calculated size distribution obtained after 6.3 and 15.3 min of grinding.

the particle morphology. The procedure, used to model grinding kinetics, is illustrated in the following for wet grinding of calcite in a stirred bead mill. More details concerning the device and experimental conditions are given elsewhere [7]. The evolution of the size distributions and the morphology of the ground product is presented in Fig. 5. Initially, the powder is constituted of compact agglomerates of calcite particles and is characterized by a multimodal size distribution. The main granulometric population corresponds to agglomerates having a median size around 30 Am. Elementary crystals or fragments (mean size around 4 Am) and fines (around 0.35 Am) are also revealed with a minor importance by the size distribution analysis. As grinding proceeds, the percentage of initial agglomerates decreases a lot, whereas the amount of finer particles increases. After 3 min of grinding, three types of fragments were identified, using Scanning Electronic Microscopy: some large particles with a cubic characteristic shape, numerous fragments, having an irregular shape and very small fine particles. Some agglomerates, made of fine particles, can also be observed. They are identifiable on the SEM pictures due to their brightness, since they present a great number of facets comparatively to fragments. After 10 min, the size distribution is the narrowest with a peak centred at 0.3 or 0.4 Am. Two kinds of particles can be distinguished: very fine fragments and

P with ni¼1 wi ¼ 1, where wi is the mass fraction of the subpopulation i and: " # ðlog x  log xgi Þ2 dF 1 pffiffiffiffiffiffi exp  ¼ fLNi ðxÞ ¼ d log x log rgi 2p 2ðlog rgi Þ2 ð15Þ where fLN is the frequence distribution, F the cumulated mass fraction of particles of size less than x, xg the geometric mean size and rg the geometric standard deviation. An example of such a treatment is given in Fig. 6a, corresponding to a sample taken after 1 min of grinding. At that time, three types of particles were observed: coarse particles, intermediate population constituted by fragments and fines. The morphological analysis is used to estimate the number of sub-populations, which is a parameter that must be introduced into the Origin Peak Fitting software. The experimental size distribution is thus decomposed into three log-normal distributions. The curve in Fig. 6a is the calculated size distribution using Eq. (14) showing a good agreement between experimental and calculated size distributions. The evolution of the characteristic parameters of the log-normal law, xgi and rgi and the proportion of each sub-population, wi, can help understanding the fragmentation mechanism and reveal at which time agglomeration occurs. Concerning the previous example, the evolution of the mean geometric size and the sub-populations fractions versus grinding time are respectively reported in Fig. 6b and c. During the first 2 min, the size and the fraction of the initial population (coarser particles) de-

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Fig. 5. Evolution of the properties of calcite ground in a stirred bead mill—wet batch grinding of calcite in a stirred bead mill. (a) Evolution of the size distribution of calcite versus grinding time. (b) SEM photos of ground calcite; (a) t = 0 min, (b) t = 3 min, (c) t = 10 min, (d) t = 45 min.

crease rapidly and we can estimate that these particles have totally disappeared after that period of time. The proportion of the fragments (called intermediate particles) increases first during the first minutes of grinding because of the fragmentation of the coarser particles but decreases at the time when no more large particles are left. At the same time, the agglomeration of fine particles, giving a new intermediate population begins to occur as soon as the number of finer particles is high enough. As fragmentation proceeds, the fragment size reaches the aggregates one and the intermediate population is then composed both of small fragments and aggregates. After 10 min, competition between fragmen-

tation and agglomeration leads to a bimodal size distribution, resulting from fines and aggregates. Thus, the log-normal law may be used to fit experimental size distributions of a product obtained during an ultrafine grinding process. However, the log-normal distribution is skewed to the right and the mode and the spread of the distribution are dependent on each other. For processes involving both fragmentation and agglomeration processes, which is an usual situation in submicronic grinding processes, the population mode and spread may vary independently and the skew of the distribution can shift from right to left or vice versa. In such situations, the beta or log-beta law can be used

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259

Fig. 6. Modelling of calcite grinding using log-normal statistical laws. (a) Decomposition of the size distribution of calcite after 1 min grinding into three subpopulations. (b) Evolution of the mean sizes of the sub-populations versus grinding time. (c) Evolution of the sub-population fractions versus grinding time.

instead of more classical statistical laws [8]. They are written as: fb ðX Þ ¼

C½mða þ 1Þ þ 2 X am ð1  X Þm Cðam þ 1ÞCðm þ 1Þ

with 0 < X < 1 and fb(0) = fb(1) = 0.

C is the gamma function. a and m are the two characteristic parameters of the law and X is a reduced size defined by:

ð16Þ X ¼

x  xmin for a beta law xmax  xmin

ð17Þ

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or:





x log xmin   for a log  beta law X ¼ xmax log xmin

ð18Þ

x is the actual particle size (Am), belonging to the finite size range [xmin, xmax]. The mode xm and the variance V of the distribution are derived as follows: Xm ¼

V ¼

a aþ1

ð19Þ ðam þ 1Þðm þ 1Þ

½ða þ 1Þm þ 2 2 ½ða þ 1Þm þ 3

:

ð20Þ

Eq. (16) implies that, if a>1, the distribution is skewed to the right, and if a < 1, it is skewed to the left. If a = 1, the distribution is symmetrical. Multimodal distributions may be represented by the combination of several beta or log-beta equations [9]: fb ðX Þ ¼

n X

wi fbi ðX Þ:

ð21Þ

constituted by a pile of pseudo-hexagonal plates. It was found that the main mechanism is, firstly, the rupture of the grains joints, leading to the dissociation of crystallites, and secondly, the chipping and the breakage of these crystallites. This initial population disappears, giving birth to two sub-populations. The coarser fraction, the median size of which varies versus grinding time from 100 to 10 Am, is constituted by fragments more elongated. They are less compact and have more concavities and more facets than the raw material. The finer fraction, which becomes above all predominant after a certain period of grinding in the stirred bead mill, consists of very small debris of a median size less than 1 Am. These fragments are simple convex polyhedrons with a few number of facets. Based on that morphological characterization, the size distributions of the ground product were divided into three sub-populations: initial particles, coarse fragments and finer ones, each of them described by a log-beta distribution. Fig. 7 presents a comparison of simulated and experimental size distributions putting in evidence a correct fitting of the experimental points at the times considered.

3. Modelling continuous grinding

i¼1

3.1. Mass transport –residence time distribution Such a treatment has been applied to data concerning the grinding of gibbsite in a stirred bead mill [10]. The quantitative characterization of the morphology of the ground gibbsite was conducted using a set of mathematical tools in order to define the path of fragmentation occurring in the media mill (for more details see Ref. [11]). The initial population of gibbsite comprises particles presenting a global spherical shape. They are compact agglomerates formed by an association of large crystallites having a platelet shape. The crystallites are themselves

During continuous grinding processes, the relation between the size distribution of the ground product and the one of the initial material depends on both the fragmentation kinetics and on the time particles spend in the grinding chamber. Particles coming through the mill stay in the chamber for different periods depending on their transport. Consequently, particles leaving the mill at a given time stayed in the mill for different periods and thus were differently stressed.

Fig. 7. Modelling of size distributions of gibbsite by log-beta laws: simulated (lines) and experimental (symbols).

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Two models have been reported describing the residence time distribution inside a stirred bead mill. 3.1.1. Axial dispersion model The first model is based on the assumption that mixing superposes upon convective axial mass transport between the inlet and the outlet of the mill. The dispersion model is given by the dimensionless differential equation as follows: BC BC 1 B2 C þ Diff ¼ BtR Bz vz l Bz2

ð22Þ

C is the ratio of the solid concentration to the inlet solid concentration, z the distance from the inlet of the mill divided by the mill length l, Diff the axial dispersion coefficient, vz the axial speed and tR the time divided by the ideal filling time, which is the ratio of the free volume of the chamber filled with suspension on the suspension flow rate. From the work of Schwedes and Bunge [12], it is known that the dimensionless Peclet number (Pe = vzl/Diff) for stirred bead mills varies from 0.6 to 5 depending on operational conditions. These values are small and indicate that global mass transport in stirred media mills is close to the one of a perfectly mixed reactor. 3.1.2. Cells model Heitzmann [13] put in evidence the existence of mixed areas centred on the agitation discs inside of the grinding chamber. He proposed a model based on a series of perfectly mixed cells with recycling in order to take into account of the back mixing of solids from one cell to the previous one. Fig. 8 illustrates the adaptation of that model to a Drais Perl Mill geometry. V2 is the cell volume perfectly mixed by an agitator disk. This volume is equivalent to the volume delimited by two agitator discs. The volumes of the first and the last cell (volume V1) have been increased to take into account the inlet and outlet zones. The mill transfer function is calculated, solving the system composed by mass equations on each cell. The residence time distribution is then calculated from the inverse Laplace function of the mill transfer function. For

Fig. 9. Residence time distribution in a laboratory stirred bead mill.

more details on the mathematical framework used for calculating the mill transfer function, see for example Ref. [14]. The residence time distribution of a continuous phase is easy to determine experimentally using soluble salts (NaCl, KCl) [15] or dyes [13] as tracer and measuring the tracer concentration at the outlet of the mill. The determination of the residence time distribution of the solid phase is more difficult. The use of a short-lived radioactive tracer seems to be an appropriate method but it is not so simple to put in practice. Therefore, as a tracer, small glass beads (100 – 125 Am in size) were put in the place of a gibbsite fraction. Grinding experiments were then conducted in a Drais Perl Mill using glass beads as media (1.18 – 1.4 mm in size), thus avoiding the breakage of the tracer. The overall size distributions taken at different times were then decomposed in order to extract the size distribution of the tracer. Fig. 9 shows an example of experimental results compared to theoretical predictions, where the stirred bead mill is considered to be in plug flow, as a perfectly mixed reactor, or modelled by a series of cells. It can be observed that the experimental residence time distribution is very close to that of a perfectly mixed reactor. Such findings must probably be related to the small volume of the grinding chamber used in that study. For larger stirred media mills, the residence time distribution would probably differ more than shown in our case. The axial dispersion or cells models would then be more suitable to describe the solids flow inside the mill. 3.2. Prediction of product properties ground in a continuous stirred bead mill The formalism used to model continuous milling processes is similar to that used to model continuous reactors. Thus, if the residence time distribution E(t) of particles inside the mill is known, the transfer function of the model is expressed by: Z l pi ¼ mi ðtÞEðtÞdt ð23Þ 0

Fig. 8. Simulation of mass transport in a stirred media mill by a series of stirred cells with recycling.

pi is the mass fraction of particles of size class i leaving the mill running in a continuous way. E(t)dt expresses the mass

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Fig. 10. Comparison of calculated and experimental size distribution of continuous grinding of gibbsite in a stirred media mill: simulated (line) and experimental values (symbols).

fraction of particles having stayed in the mill for a period t and appearing at the outlet between t and t + dt. One considers that this fraction has undergone the same treatment as if it stayed in a batch process for the same period. Hence, the fraction of the ground particles in each size class is determined by integration of Eq. (23). An example is given is Fig. 10 concerning the grinding of gibbsite using the Drais Perl Mill. The simulated size distribution was determined, using Eq. (23), in which the fragmentation kinetics was fitted by a combination of log-beta laws as reported in Section 2.3 and the mass transport described by a series of four cells with recycling. In that model, the fraction of recycled flow H is the only adjustable parameter varying as a function of operational conditions (feed rate, agitation speed, filling beads). It may be calculated comparing the experimental size distribution obtained during a continuous run with the one calculated using Eq. (23). Always considering the stirred media mill with a small volume, it was observed that the sensitiveness of the parameter H is low and a mean value (around 5) could be used for modelling continuous grinding process independent of the experimental conditions. The low sensitiveness of the parameter H is due to the fact that the hydrodynamics behaviour inside the grinding chamber having a small volume is quasi similar to that of a perfectly mixed reactor whatever the operational conditions.

4. Conclusions Many papers deal with modelling of coarse or fine grinding processes. The prediction of the product size distribution is usually done using a population balance approach. Concerning fine and submicronic wet grinding processes, problems arise due to practical difficulties in the determination of the selection and breakage functions and to agglomeration phenomena occurring during size reduction.

Simplified solutions of the population balance equation as the similarity solution are very interesting to use for modelling when they suitably predict the experimental results. If the size distribution of the ground product presents a notable bi- or multimodal character, the use of statistical laws, such as the log-normal or the log-beta laws can be used to fit the fragmentation kinetics. A morphological analysis of the material can then help to give a physical sense to the resulting kinetics equation and to define the fragmentation path occurring during the size reduction process. A general chemical engineering approach for modelling continuous processes, combining batch grinding kinetics and a mass transport model, can be used to predict the particle size distribution of the product from continuous fine or submicronic grinding. The accuracy of the model mainly depends on the precision of the batch kinetics equation. So, further research work will attempt to develop a more generalized population balance representation of submicronic grinding processes involving agglomeration phenomena. Nomenclature a characteristic parameter of the log-beta law b(x,y) breakage distribution function bi,j breakage function related to class i and j B(x,y) cumulative breakage function Bi,j cumulative breakage function related to class i and j C ratio of the solid concentration on the maximal one Diff axial dispersion coefficient (m2 s 1) E(t) residence time distribution (s 1) fb beta function fLN frequence distribution of log-normal law F cumulated mass fraction of particles of size less than x Gi Kapur function Eq. (11) (s 1) H reaction of recycled flow Hi Kapur function Eq. (11) (s 2) K agglomeration kernel

C. Frances / Powder Technology 143 – 144 (2004) 253–263

l m mi(t) m(x,t) N(w,t) pi Q R(x,t) Ri(t) S(x) Si t tR vz V w, w V wi x xgi xm X z l1(t) rgi C

mill length (m) parameter of log-beta law mass fraction of particles of size class i mass fraction of particles of size x at time t number of particles of mass w at time t fraction of continuous ground product of size i flow rate (l/h) cumulative retained distribution at time t cumulative retained distribution related to size class i selection function (s 1) selection function related to size class i (s 1) time (min) actual time divided by the ideal filling time axial velocity (m/s) variance (m2) particle mass (kg) weight fraction of the sub-population i particle size (m) geometric mean size (m) mode (m) reduced size distance from the inlet of the mill divided by the mill length first moment of the particle size distribution (m) standard geometric deviation (m) Gamma function

References [1] P.C. Kapur, Self preserving size spectra of communited particles, Chem. Eng. Sci. 27 (1972) 425 – 431.

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[2] C. Frances, C. Laguerie, Fine wet grinding of an alumina hydrate in a ball mill, Powder Technol. 99 (1998) 147 – 153. [3] K.J. Reid, A solution to the batch grinding equation, Chem. Eng. Sci. 20 (1965) 953 – 963. [4] P.C. Kapur, Kinetics of batch grinding—Part A: reduction of the grinding equation, Trans. Soc. Min. Eng. AIME 247 (1990) 299 – 303; Part B: an approximate solution to the grinding equation, Trans. Soc. Min. Eng. AIME 247 (1990) 309 – 313. [5] H. BelFadhel, Approche expe´rimentale et mode´lisation du broyage fin et ultra-fin de solides en voie humide dans un broyeur a` billes agite´, PhD thesis, INPT, Toulouse (1998). [6] F. Stenger, W. Peukert, Nanomilling—the influence of suspension stability, Comm. World Congress on Particle Technology 4. CDRom 085 825 7947. [7] F. Garcia, N. LeBolay, C. Frances, Changes of surface and volume properties of calcite during a batch wet grinding process, Chem. Eng. J. 85/2 – 3 (2002) 177 – 187. [8] M. Peleg, M.D. Normand, J.R. Rosenau, A distribution function for particle populations having a finite size range and a mode independent of the spread, Powder Technol. 46 (1986) 209 – 214. [9] L.M. Poppewell, O.H. Campanella, M. Peleg, Simulation of bimodal size descriptions in aggregation processes, Chem. Eng. Prog., (1989 August ) 56 – 62. [10] H. BelFadhel, C. Frances, Wet batch grinding of alumina hydrate in a stirred bead mill, Powder Technol. 119 (2001) 257 – 268. [11] C. Frances, N. LeBolay, K. Belaroui, M.N. Pons, Particle morphology of ground gibbsite in different grinding environments, Int. J. Miner. Process. 61 2001, pp. 41 – 56. [12] J. Schwedes, F. Bunge, Operation of agitated bead mill, Hung. J. Ind. Chem. Veszprem 21 (1993) 129 – 147. [13] D. Heitzmann, Caracte´risation des ope´rations de dispersion-broyage. Cas d’un broyeur a` billes continu pour la dispersion des pigments, PhD thesis, INPL, Nancy (1992). [14] H. BelFadhel, C. Frances, A. Mamourian, Investigations on ultra-fine grinding of titanium dioxide in a stirred media mill, Powder Technol. 105 (1999) 362 – 373. [15] D.F. Kelsall, K.J. Reid, P.S.B. Stewart, A study of grinding processes by dynamic modelling, Inst. Eng. Australia, Elect. Eng. Trans. EES, (1969) 173 – 186.