Breakage Distribution Research Articles

Single particle impact breakage characteristics of clinkers related to mineral composition and grindability

Single particle impact breakage characteristics of clinkers from different plants were determined by the drop-weight technique. Impact breakage distribution parameter ( t 10) defined by Narayanan [Narayanan, S.S., Whiten, W.J., 1983. Breakage characteristics for ores for ball mill modelling. In: Proceeding of the Australasian Institute of Mining and Metallurgy, vol. 286, pp. 31–39.] was used to characterise the breakage distributions of different clinkers based on the specific comminution energy. The single particle impact breakage resistance parameter as represented by A · b [Man, Y.T., 2000. A model based scale-up procedure for wet, overflow ball mills. Ph.D. Thesis, Department of Mining, Minerals and Materials Engineering, The University of Queensland] based on the model parameters of the well known E cs− t 10 relationship given by Leung [Leung, K., 1987. An energy-based ore specific model for autogeneous and semi-autogeneous grinding. Ph.D. Thesis, The University of Queensland, JKMRC] was correlated with the standard bond index number, belite and alite mineral amounts.

The application of a simplified approach to modelling tumbling mills, stirred media mills and HPGR’s

Most modern population balance models for comminution invoke the concept of a specific breakage rate function and a breakage distribution function to describe breakage kinetics. One of the difficulties of this approach is that these functions are very difficult to measure directly. Consequently, it is usual to assume that these functions can be represented by simple equations with parameters that can easily be estimated from test data using back-calculation techniques. However, these estimates can be very sensitive to small measurement errors and are usually subject to very large variances. This paper presents a simplified approach to modelling comminution processes that invoke the concept of an energy-based cumulative breakage rate function to describe breakage kinetics. This function can be estimated directly from plant data and is well-suited to multi-component modelling of individual rock types and mineral species. Examples of the application of this simplified modelling approach are described for the treatment of platinum ores using ball mills, AG/SAG mills, HPGR’s and stirred media mills.

Kinetics of fine wet grinding of zeolite in a steel ball mill in comparison to dry grinding

Batch wet grinding of zeolite was studied with emphasis on a kinetic study in a laboratory size steel ball mill of 200 mm diameter. The breakage parameters were determined by using the single sized feed fractions of − 850 + 600 µm, − 600 + 425 µm and − 425 + 300 µm for the zeolite samples. The S i (specific rate of breakage) and B i , j (primary breakage distribution) values were obtained for those feed size fractions in order to predict the product size distributions by simulation for comparison to the experimental data. The specific rates of breakage values for wet grinding in the first-order breakage region were higher than the dry values reported previously by a factor 1.7 at the same experimental conditions, but the primary breakage distribution ( B i , j ) values were approximately the same. The simulations of the product size distributions of zeolite were in good agreement with the experimental data using a standard ball mill simulation program. The wet grinding of zeolite was subjected to slowing-down effect in the mill at 2 min of grinding, corresponding to an 80% passing size of about 400 µm. On the other hand, the slowing down effect in the dry grinding of zeolite was also seen at 4 min of grinding. In addition, effects of some operational parameters on dry and wet grinding of zeolite were determined by simulation using the breakage parameters obtained experimentally.

PTC/NTC Behavior of PVDF Composites Filled with GF and CF

Abstract Conductive polyvinylidene fluoride(PVDF) matrix composites filled with graphited fiber(GF) or carbon fiber(CF) were prepared by the melt-mixing method. The breakage and length distribution of the fibers in the polymer matrix were studied by scanning electron microscope(SEM) and optical microscope(OM) observations, respectively. The differences in the positive temperature coefficient(PTC) effects of the composites were mainly attributed to inter-fiber contact ability. The elimination of the negative temperature coefficient(NTC) effect for CF/PVDF composite was because of an increase in the viscosity of the polymer matrix. With the same filler content, CF could be more effective, to eliminate the NTC effect when compared with GF. Addition of 2% CF(mass fraction) in the PVDF composite with 7% GF(mass fraction) could effectively eliminate the NTC phenomenon of the composite.

Modeling and measurement of granule attrition during pneumatic conveying in a laboratory scale system

A methodology combining theoretical and experimental techniques for characterizing and predicting the friability of granules in a laboratory scale pneumatic conveying systems is developed. Models of increasing mathematical complexity are used for analysis of experimental data. Firstly, a two-dimensional (2-D) computational fluid dynamics (CFD) model of the gas–solid flow within the Malvern Mastersizer laser diffraction equipment is developed to simulate impact of different inlet jet pressures on the flow properties and to calculate average velocity and average volume fraction of particles in the equipment. Secondly, a simple maximum-gradient population balance (MG-PB) mathematical model of breakage is developed. The model is solved using the Quadrature Method of Moments (QMOM) and used for evaluation of experimental data from the Malvern equipment. Different semi-empirical expressions for the breakage kernels and for the daughter distribution functions are tested. Multiple breakage distribution functions are needed to get satisfactory agreement with experimental data. Finally, a CFD-PB model combining CFD and QMOM methodologies is developed. The combined model employs different binary fragment distribution functions and a kernel with the breakage rate proportional to the characteristic particle size and to the square of the impact velocity between a particle and the equipment wall. Simulation results are compared with attrition experimental data indicating that the model is able to capture the qualitative trends and quantitatively predict the Sauter mean diameter d 32 at the outlet. However, the lower moments, in particular m 0 and m 1 are under predicted by the model. Based also on the MG-PB model results, it is our hypothesis that chipping, or breakage of particles in multiple fragments results in higher m 0 and m 1 than predicted. Further improvements of the model are proposed to incorporate multiple breakage effects. It is assumed that analogous physically based models combining properties of the gas–solid flow with the PB models can be employed to predict attrition and breakage in large-scale pneumatic conveying systems.

Effect of Grinding Media Shapes on Breakage Parameters

AbstractIn the present study, the effect of grinding media shape on breakage parameters was investigated. Balls and cylpebs were used as the grinding media. It was observed that the grinding of quartz obeyed first‐order breakage kinetics in the case of balls and cylpebs. Higher breakage rates were noted with cylpebs than with balls. Furthermore, it was found that the primary breakage distribution function is dependent on the feed size (i.e., non‐normalizable), but independent of the grinding media shape. The effect of grinding time on the product size distribution has also been investigated. Following four and ten minute grindings, cylpebs produced a relatively finer product compared to balls.

On the Consequences of Non‐First‐Order Breakage Kinetics in Comminution Processes: Absence of Self‐Similar Size Spectra

AbstractComminution or size reduction is a commonly used unit operation in a variety of industries including pharmaceuticals. In quantitative analysis of comminution at the process length scale, population balance modeling has been used with an assumption that the breakage rate of particles is first‐order and that the population balance model is linear. For certain forms of the specific breakage rate and breakage distribution functions, a similarity solution to the linear population balance exists, and self‐similarity plots have been used to show its validity in batch comminution processes. On the other hand, ample experimental batch milling data in the literature indicate the inadequacy of the linear theory. In this study, a theoretical investigation has been conducted within the framework of a recently proposed non‐linear theory. The non‐linear theory takes into account phenomenologically the non‐first‐order effects resulting from mechanical multi‐particle interactions. Specifically, the cushioning action of small particles on the coarser ones is considered. It is found that self‐similarity does not exist in the presence of the non‐first‐order breakage kinetics. The present study suggests the use of self‐similarity plots for an approximate assessment of the impact of non‐first‐order kinetics on the shape of the cumulative particle size distribution.

Dry grinding kinetics of colemanite

The specific rates of breakage ( S i ) and primary breakage distribution ( B i,j ) values of colemanite were determined and compared to quartz ground under the same grinding conditions. It was found that colemanite follows a first-order breakage behavior for all feed sizes and is broken faster than quartz. Breakage function of colemanite is non-normalizable similar to quartz. [Display omitted]

Breakage behavior characterization of micron‐scale particles

AbstractA method for quantitative characterization of breakage of particulate material is presented. The approach developed uses a 3‐D population balance (PB) breakage model, which describes the changing population of particles in a “origin‐size‐toughness” domain, to extract breakage parameters from dynamic tracer and size evolution experimental data. The breakage behavior is captured in terms of the model parameters of breakage rate and breakage distribution function. It is shown that utilization of the dynamic tracer distribution information contributes significantly to the accuracy and robustness of the breakage parameter estimation. The method is validated by demonstrating good agreement of the model with measured dynamic particle size distributions and tracer distributions. It is also demonstrated that the 3‐D PB breakage model better captures the highly distributed nature of the dynamic experimental data than an equivalent 2‐D model. The usefulness of the technique for characterization of breakdown characteristics is demonstrated using commercial aluminium oxides; both for quantitatively differentiating the breakage behaviors and also for providing insights into the fundamentals of the breakage mechanisms. “Breakage maps” and “breakage activity” plots are introduced as useful formats for presenting the complex breakage information. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Particle size distributions a new approach

The consideration that the mass discontinuities of brittle materials statistically present fractal behavior makes available a new approach to the elucidation of crushing and grinding process. Through crushing and grinding experiments of characteristics brittle materials, the breakage distribution function (particle size distribution of products of a single breakage event) is defined as the sum of two distinct distribution (fragments and dust). The breakage distribution function follows a geometric dimensional scale and is depending on two new statistical characteristics constants of the material. Entering the number of breakage events, as the independent variable of the aforementioned function, a theoretical model of grinding process is formulated. The application of the theoretical model in any given size reduction process allows the definition of the process breakage probability distributions function and thus the theoretical model can be transformed to an actual simulation model of the process. The reliability of the new approach in describing the evolution of particle size distribution during the crushing or grinding was evaluated by its application in characteristics experimental and industrial installations.

Simulation of wet ball milling of iron ore at Carajas, Brazil

Laboratory-scale batch grinding tests were performed on 2 sieve size fractions of the almost pure hematite mined at the Carajas mine of CVRD Brasil. Good first-order kinetics of grinding were obtained, both for dry grinding and for grinding with water, using the Bond test levels of ball load and powder load. However, the primary breakage distribution functions were not dimensionally normalized, as smaller particles produced proportionally more fines on primary breakage, possibly due to internal porosity of the ore. The test results were scaled to the conditions of a pilot-scale mill in normal closed circuit, and a ball mill simulator was used to predict the capacity, circulating load and size distributions, employing the size selectivity values calculated from the balance around the hydrocyclone. The simulation results were in reasonable agreement with the observed values. The technique was then extended to predict the behavior of a full-scale mill at a capacity of 308 metric tons/h and a circulating load of nearly 600%. Agreement between simulated results and plant test results was reasonable except that the simulator predicted a product (overflow) size distribution with a higher sharpness index than observed. The reason for this discrepancy is not known.

Validation of a model for impact breakage incorporating particle size effect

Investigation of a breakage probability model published by Vogel and Peukert [Vogel, L. and Peukert, W., 2004. Determination of material properties relevant to grinding by practicable labscale milling tests. Int. J. Miner. Process., 74S, 329–338.] has led to a modification of their model to describe the degree of impact breakage, t 10. The modified model takes a form similar to the JKMRC prior art breakage model, but with particle size and breakage properties incorporated explicitly in the model. The present breakage model was validated using eight sets of detailed drop weight test data on various types of ore and quarry material over a wide range of impact specific energies and particle sizes. The validation revealed that the present breakage model fits all the data sets well with fewer model parameters. The results from the validation study suggest the present breakage model has a fundamentally better structure for describing the effect of particle size on the breakage distribution function.

Towards optimising ball-milling capacity: Effect of lifter design

The effect of liner/lifter profile on kinetics of batch grinding and the milling capacity in general was assessed using mono-size quartz material of 30 × 40 mesh (−600 + 425 μm) as feed. The liner profiles tested were, (i) bevel with 60° lifter face angle to represent the new liners, (ii) bevel with 45° lifter face angle to represent the worn liners and (iii) worn bevel modified with cone-lifters. The tests were conducted under identical conditions to allow a comparative analysis of the results. In all cases, the breakage followed the first order hypothesis. The experimental size distribution data was well predicted by the S and B model, thus allowing for estimation of breakage and selection parameters (i.e., γ and a T ) for the three liner situations tested. The optimised values of the specific rate of breakage, S i for the three liner profiles tested were 0.381, 0.287 and 0.365 min −1, respectively, which clearly indicates the benefit of cone-lifters. The breakage distribution function ( B ij ) values did not vary significantly with liner profile, which echoes the findings by other researchers.

Effects of grinding media shapes on load behaviour and mill power in a dry ball mill

Little attention has been given to the study of grinding media shapes and how they affect load behaviour and power draw in grinding mills. To a large extent the focus has been on how grinding media shapes affect milling kinetics in terms of specific rate of breakage and breakage distribution parameterisation. This study investigated the effects of three media shapes (cylpebs, spherical and worn balls) on load behaviour and mill power draw at various mill speeds and load filling. An inductive proximity probe was used to determine the load orientation of the grinding media charge while a load beam enabled measurement of power draw. The variations in toe and shoulder positions among media shapes were observed. This was also reflected in the power draw. The power increases to a maximum with increasing mill speed for all media shapes. The maximum power draw was reached at different mill speeds for the three studied media shapes.

Grinding of ceramic raw materials by a standard Bond mill: quartz, kaolin and K-feldspar

The kinetics of batch dry grinding of ceramic raw materials, which are quartz, kaolin and K-feldspar, from the feeds of sieve sizes −3·350+2·360, −2·360+1·400, −1·400+0·850, −0·850+0·600, −0·500+0·425 and −0·300+0·212 mm have been determined using a Bond mill with a mixture of five ball sizes. The mill used was a size of 30·5 cm diameter, 30·5 cm length, providing a total mill volume of 22 272 cm3 with a total mass of 22 648 g steel ball mixtures of 38·10, 31·75, 25·40, 19·05 and 12·70 mm diameters, occupying 22% of mill volume and with a speed of rotation of 70 rpm. The specific rates of breakage (Si) and primary breakage distribution (Bi,j) values, called grinding breakage parameters, were determined for those feed size fractions to predict the product size distributions by simulation for comparison to the experimentally obtained data. As the feed sizes increase, the Si values also increase, that is, faster breakage values from higher to lower values were in the order of quartz, K-feldspar and kaolin by comparison with their α values. The Bi,j values obtained for these minerals were different in terms of characteristic fineness factor, γ. Thus, kaolin produced more fines, while K-feldspar and quartz produced less fines in the mill. In other words, the simulations of product size distributions for these minerals were in good agreement with the experimental data using a ball mill simulation program, called PSUSIM. However, the slowing down effect in the mill started earlier than expected for the minerals studied, unlike the laboratory size ball mill used extensively in our studies. Nevertheless, there exist linear relationships between the simulated time (θ) and experimental time (t) for these minerals.

Milling of Organic Solids in a Jet Mill. Part 1: Determination of the Selection Function and Related Mechanical Material Properties

AbstractThe particle size distribution of pharmaceutically active materials and other fine chemicals determines the performance of the final product to a large extent. Often milling of these particles is necessary. It is not possible to determine the milling conditions solely on the basis of the particle size distribution of the starting material, because the (mechanical) properties of the material also determine the desired milling conditions. It is often not possible to optimize milling conditions experimentally because the amount of material available is frequently highly limited. A theoretical approach towards predicting the best milling conditions is needed. The purpose of this study was to develop a method to predict the desired milling conditions given a specific (organic) solid material. The selection function and the breakage distribution function are usually the starting points in modeling the milling process. The selection function is the parameter that includes the material and mill properties. Dimensional analysis made it possible to correlate the selection function with material properties. A set of theories available in the literature enable prediction of the material properties. For different compounds (lactose, paracetamol, a steroidal compound, and two heterocyclic compounds) the selection functions were calculated. The calculations predict differences: lactose reduces slowly in size, while one of the heterocyclic compounds shows the most intense fracture.

Nano-milling of pigment agglomerates using a wet stirred media mill: Elucidation of the kinetics and breakage mechanisms

The present study concerns the production of pigment nanoparticles in a wet-batch stirred media mill with polymeric media. The breakage kinetics and mechanisms were investigated using size-discrete population balance models (PBMs). The temporal variation of the particle size distribution was measured via dynamic light scattering. Considering the G–H model, a time-invariant PBM, and a time-variant PBM, the specific breakage rate parameters and breakage distribution parameters were identified. It is found that the breakage rate is not first-order and that a delay time exists for the breakage of nanoparticles. The time-variant PBM captures all these features and suggests a transition from deagglomeration of agglomerates to the breakage of primary particles. The analysis of the breakage distribution parameters suggests splitting as the dominant mechanism as opposed to attrition or massive fracture.

Formulation of a non-linear framework for population balance modeling of batch grinding: Beyond first-order kinetics

Population balance models (PBMs) for batch grinding are based on the concepts of specific breakage rate and breakage distribution. In the traditional PBMs, the breakage rate is assumed first-order, thus neglecting the effects of the temporally evolving material properties and multi-particle interactions. As an attempt to explain some of the above effects, a time-dependent specific breakage rate was introduced in the literature. The time-variant PBMs are inadequate to explain the multi-particle interactions explicitly and thoroughly. In this paper, we formulate a non-linear population balance framework to explain the non-first-order breakage rates that originate from multi-particle interactions. Based on this framework, four size-discrete non-linear models with varying complexity have been derived. A simple non-linear model with non-uniform kinetics assumption, Model B, was used to simulate the slowing-down phenomenon commonly observed in dry grinding processes. Not only does the model explain the effects of the fines accumulation on the specific breakage rate of the coarse, but also it is capable of predicting the significant influence of the initial population density. Identification of the proposed models, i.e., the solution of the inverse problem is also discussed.

Population balance modeling of non-linear effects in milling processes

Specific breakage rate (selection) and breakage distribution functions are used to describe the birth and death terms in population balance models (PBMs) for milling processes. Traditional PBMs for milling processes are inherently linear because the breakage rate is assumed first-order. The specific breakage rate is independent of the population density while it depends on particle size and possibly on time. Even though the linear theory has been applied with some success to the modeling, optimization, and design of various mills in the last 50 years, many researchers have indicated its restrictions and subjected it to serious criticism. In this paper, we first categorize the experimentally observed deviations from the linear theory and suggest the multi-particle interactions as the origin of these deviations. To account for the peculiar non-linear effects, a phenomenological theory has been proposed via multiplicative decomposition of the specific breakage into a size-dependent apparent breakage rate and a population density dependent functional. The proposed theory recovers the first-order breakage kinetics in the limit, yet it is sufficiently general to explain all experimentally observed deviations. Numerical simulations of a batch milling process have demonstrated the potential of the non-linear theory.

Study of Fiber Breakage and Length Distribution During Compounding Glass-NBR-Phenolic Composites

: One of the main criteria in preparation of short fiber composites in fiber length retention and distribution during compounding. This is because the mechanical properties of short fiber composites depend strongly on the fiber length and distribution. In this study, 50 mm glass fibers were used to make a glass-nitrile rubber (NBR)-Phenolic composite on a two-roll mill. Fiber breakage during the manufacturing process was investigated and a fiber length distribution was obtained at various mixing times. A reduction in fiber length was observed as mixing proceeded and an average fiber length was determined for different mixing times. It was concluded that for a particular matrix processing, conditions must be optimized in order to achieve the maximum reinforcing effect of the fiber.