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In the past, most of the Population Balance Model-based ball mill scale-up studies were carried out using a mono-size ball charge and a single size fraction of the particulate material as the starting feed. It was reported that the breakage distribution parameters were independent of the mill diameter, specific breakage rate parameters varied with mill diameter as Dγ, and the mill diameter exponent γ was independent of particle size. However, in the industrial grinding operations, we have significantly broad size distributions for both the grinding media and the particulate contents of the mill. Therefore, experimental data generated under these conditions were taken from the literature and analyzed. These data pertain to the dry and wet batch grinding of limestone, quartz, and ores of vanadium and pyrochlore. Our analysis of the data showed that in contrast to the findings of the earlier researchers, the breakage distribution parameters were not independent of the mill diameter, and the value of γ for the specific breakage rate parameters varied with particle size in the case of multicomponent ores. Because of these complications, a single parameter, the specific rate of production of material that is finer than a specified size, was used as the basis for scale-up. It was found that the value of the mill diameter scale-up factor varied quite significantly with the breakage properties of the material, the size distribution of the particulate contents of the mill, the size of the desired product, and the mode of grinding (dry/wet). Therefore, ball mill scale-up models presently available need to be modified in light of these new findings.

This paper presents the results of the experiments on the effect of particle size distribution and the volume ratio of components of the binary mixtures of calcite, hematite, and quartz on the grinding kinetics in ball mills. A total of forty-six grinding tests were carried out. These included forty-two dry grinding tests and four wet grinding tests. For each test, only one short-duration batch grinding experiment was carried out. The feed charge was prepared by pre-grinding the particulate material in the test ball mill. The results of these experiments have led to several important findings. Three of these findings are: (i) the size distribution of the ground product of the mixture of two minerals was practically the same as the mass-weighted sum of the size distributions of the ground product of these minerals when they were ground alone for the same duration, (ii) fine particles broke faster in the coarser environment, and (iii) while quartz fines were found to be most effective in increasing the specific breakage rate of coarse particles of quartz and hematite, in the case of calcite, hematite fines were found to be most effective. It shows that besides the hardness and density of the components, there are some other presently unknown properties that determine the fraction of the net energy input to the mill that is absorbed by the mixture.

A detailed analysis of the specific breakage rate and mill power data on dry ball milling of three individual minerals and their binary mixtures has been carried out. The energy split model of Kapur and Fuerstenau has been examined and shown to be incorrect. A new model has been proposed. The basic postulates are: (i) only a small fraction λ of the net energy input to the mill is absorbed by the particulate material (ii) value of λ is mainly determined by the hardness and specific gravity of the material(s) being ground, volume proportion of the components of the mixture, particle size, and mill operating conditions, and (iii) in the mixture, same size particles of different components receive the same amount of energy per unit volume. An important feature of the model is the use of volume-specific mill power. It has been shown that the effect of the operating variables such as the mill speed, particle load, and mill diameter on energy absorption is significantly different for different materials. These findings have been discussed in light of our current state of understanding of the mechanics of the ball milling operation. It has been concluded that the concept of energy absorption factor can be quite useful in the analysis of grinding systems, and mill scale-up design work.

In the dry ball milling operation, the specific breakage rate of particles has been found to vary significantly with the size distribution of the particulate material. However, due to the lack of suitable experimental data and a clear understanding of the role of inter-particle interactions in determining the specific breakage rate of particles, it has not been possible to develop a general mathematical model that can be used to describe the variation of the specific breakage rate of particles. These limitations were overcome in our investigation by using feed charges for batch grinding experiments that were prepared according to specially designed continuous and discontinuous particle size distributions. A detailed analysis of the experimental results led to several new findings about the nature of the non-linear grinding kinetics. In particular, it was found: (i) for a given size fraction of particles, while the effect of the coarser and immediately finer sizes on its specific breakage rate was negative, the effect of the other finer sizes was positive and it increased as the particle size decreased, (ii) the overall effect varied with the material type and particle size, being more pronounced in the case of the softer material and coarse particles, (iii) the breakage distribution parameters were also significantly affected by the particle size distribution, and (iv) a simple linear relationship in terms of the mass fractions of different sizes described the variation of the specific breakage rate parameters with particle size distribution quite satisfactorily.

For evaluation of the performance of a continuous flow reactor, a description of the residence time distribution (RTD) of the reactants is required in the form of a suitable mathematical expression. For closed reactors, the analytic expression obtained from the well known axial dispersion model is too complicated for practical use. The \two-parameter empirical model that involves one large and two small perfectly mixed sub-reactors along with a small plug flow sub-reactor provides an accurate description of RTD in the form of a relatively much simpler mathematical expression. However, due to the lack of specification of the sequence in which the sub-reactors are arranged, this model cannot be used for modeling of systems characterized by nonlinear reaction kinetics. Therefore, in this paper we propose a new one-parameter empirical model that meets both the requirements. The model involves partitioning the process unit into a number of unequal-volume perfect mixers arranged in a definite order. The first perfect mixer at the feed end is the largest, and the volumes of the successive smaller perfect mixers form a geometric progression, the inverse of the common ratio being the sole parameter of the model. Several sets of available experimental RTD data on two closed reactors, rod mills and ball mills, were analyzed using the proposed model. In most cases the model was found to fit the experimental data on a par with the dispersion and two-parameter models. Moreover, the proposed order of arrangement of different perfect mixers could be validated by demonstrating that the simulated trend of variation in the particle size distribution along the mill matches closely with that observed in practice.

For a widely used combination of the functional forms for describing the size dependence of particle breakage rate and breakage distribution functions in the ball mill grinding operation, Kapur [13] has derived an energy–size relationship from a similarity solution to the fundamental size–mass balance equation of batch grinding kinetics. In spite of its significant relevance from the practical application point of view, this relationship has not been properly validated so far. Therefore, a detailed study was undertaken for its validation using some simulated batch grinding size distribution data as well as actual experimental data. The relationship has now been comprehensively validated. It is concluded that as compared to various known empirical energy–size relationships the one presented by Kapur can provide a more rational and sound basis for estimation of energy requirements for industrial ball milling operations.

Using the back-calculation approach, simultaneous estimation of rate and breakage distribution parameters has been carried out for a complex pyritic ore from the batch ball mill size distribution data. The ore showed several deviations from the normal grinding behavior: (i) the slope of the Gaudin-Schumann plots in the fine size range varied with the feed size and the grinding time, (ii) the first-order plots showed marked curvature in the initial period of grinding, (iii) the rate parameters did not conform to the relationship S i = Ax α i , even in the fine size range, and (iv) the breakage distribution parameters exhibited a high degree of non-normalizability. The main problem encountered was the choice of a suitable functional form for the breakage distribution function. This paper illustrates the limitations of the two existing functional forms and several other functional forms used in this work, and emphasizes the need for having a more general and practical functional form for the breakage distribution function.

Batch and continuous wet grinding experiments were carried out in a 40 × 40 cm grate-discharge ball bill. Well-distributed −4 and −20 mesh feeds of two complex pyritic ores were used. The vol. % solids in the pulp ranged from 30 to 40, the particle load from 5 to 8 kg, and the solids feed rate from 70 to 110 kg/h. Analysis of the experimental data showed that over the range of operating conditions investigated, a linearized model could be used to predict the size distributions of both the mill product as well as the mill hold-up of the solids, very accurately. The breakage rate parameters, Si, were found to be dependent on the mode of operation, independent of the percent solids in the pulp, and approximately inversely proportional to the mill hold-up weight of the solids. In the case of the continuous mode of operation, with decrease in particle size the rate parameter values became increasingly smaller by comparison with the corresponding expected batch operation values.

Simple graphical procedures for determination of empirical correlations for the variations in selection parameters of the discretized batch grinding equation with the particulate mass and the size of the grinding media balls are described. The influence of the particulate mass is given by a single correlation function which is valid for all particle size intervals, as well as both starved and saturated mill loadings. The search for the correlation for the effect of ball size can be considerably simplified by plotting the data in a particular fashion which results in a self-preserving curve, independent of the particle size.

For dry ball mill grinding operation, the effect of ball and mill diameters on grinding rate parameters of the size-discretized population balance model has been investigated for quartz, limestone, a soft cement clinker and a hard cement clinker. Experiments were performed in three mills of 29.2, 40.6 and 61.0 cm diameter. The diameter of the balls used ranged from 1.27 to 3.81 cm. The particle size range covered was 8 10 to 100 150 mesh. The rate parameter values were determined very accurately using a special technique. It has been shown that the particle size exponent α in equation Si = Axiα is independent of ball and mill diameters. Based on this fact, a new correlation has been developed to describe the effect of ball and mill diameters on the rate parameters. The various constants in this correlation are strongly material dependent.

A solution to the integro-differential equation of batch grinding has been presented which maps the trajectory of particle size spectra of continuous size and time domains. The solution, which is the most general analytical closed form expression available at present, employs Kapur's similarity solution to the grinding equation along with standard transformation techniques. The resulting size spectra, a modified form of the generalized gamma distribution, is not self-preserving and the energy—size reduction relationship does not conform to Walker's power law. These aspects have been illustrated by simulation studies.