Fine grinding of aggregated powders

In many cases, including natural ores as well as synthetic powders, fine grinding involves the breakage of bound aggregates rather than solid particles. The characteristics of breakage in such systems have been investigated by experimental studies of grinding kinetics, in a model system of partially sintered alumina particles, ground in a laboratory centrifugal ball mill. The effects of aggregate strength (extent of sintering) and energy input (mill speed) on the breakage rates and breakage distributions have been evaluated. Breakage appears to occur primarily through splitting of the aggregated mass into two or three smaller aggregates accompanied by release of the primary particles, leading to strongly bimodal breakage distributions.     

Analysis of nonlinear batch grinding in stirred media mills using self-similarity solution

Generally, particle breakage rate is considered to be independent of the grinding environment, and hence, the system is referred to as a linear time-invariant grinding system with first-order grinding kinetics. However, time-dependent breakage rate exists and perhaps, is more critical for fine grinding of particles. The time-dependent breakage rate also introduces nonlinearity in the grinding phenomena. In the present work, a self-similarity based approach is described to model the evolution of fine particle size distributions in a batch stirred media milling with an emphasis on the nonlinear breakage rate function by considering the breakage rate to be a function of the grind time. The present approach yields analytical expressions for cumulative weight percent finer distributions for the continuous-size continuous-time population balance equation. The breakage parameters in the analytical solution can be estimated for a given system from any three measured size distributions that show self-similarity and these parameters can be used to predict distributions evolving at higher grind times. Several sets of published data of stirred media milling are employed to validate the model.     

A comparison of dry and wet grinding of a quartzite ground in a small batch ball mill

Classical grinding models involve the selection function (S), which gives the rates of breakage of particles of each screen size fraction, and the breakage function (B), which describes the instantaneous size distributions of fragments produced when the particles of each fraction are broken. In order to investigate the differences between dry and wet grinding as far as the selection and breakage functions are concerned, batch grinding experiments were performed on both dry and wet bases, on the same material, a quartzite, in a small ball mill under similar experimental conditions.On a dry basis, the rates of breakage were found to be time invariant and independent of the size environment in the mill. It is logical to postulate a similar behavior for the breakage function. On a wet basis (65% solids), an increase of the rates of breakage was observed as grinding proceeds. This behavior is essentially due to the variation of the size environment within the mill. This increase in breakage rates was, however, less and less important as the particle size decreased and was not observed for the smallest particles tested. These points were confirmed by considering the disappearance kinetics of samples of different screen size fractions of quartzite injected in the mill during the batch grinding of a limestone. Moreover, it is not impossible that the breakage function could also vary with grinding time, giving rise to finer instantaneous size distributions of fragments as the size environment in the mill becomes finer. As an overall result, wet grinding has appeared more selective than dry grinding for coarse material, while it did not produce more schlamms.     

Application of the population balance model to the grinding of mixtures of minerals

Batch experiments were carried out on the dry ball mill grinding of calcite, hematite and quartz, individually and as binary mixtures. In terms of the population balance grinding kinetics model, the breakage rate functions of the individual minerals were found to be time-independent but environmental-dependent. The breakage distribution functions of the individual minerals were the same, irrespective of whether the mineral was ground alone or as a component of a mixture. A modified form of the Charles energy—size reduction equation was used to resolve the net energy expended by the mill into that consumed by each of the components of the mixture. The breakage rate function, when normalized in terms of specific energy (energy consumed per unit mass of mineral), was found to be both time- and environment-independent for each of the minerals studied. These observations have potential for application in mill design, scale-up and control in the processing of complex ores.     

An explanation of abnormal breakage of large particle sizes in laboratory mills

Abnormal breakage in laboratory mills may be defined as departure from first-order kinetics, and occurs particularly for the larger particle sizes in the mill feed. The reason for this abnormal breakage is that all the particles within a size fraction do not have tire same strength, rather having a distribution of strengths which interacts with a distribution of applied loads on the part of the grinding media. The interrelation between these two distributions, of strengths and forces, is discussed statistically showing the conditions necessary for departure from first-order kinetics. Actual values of the distribution of strengths support the theory, and a final experiment on the grinding of the survival material from previous grinding tests validated the concepts presented.     

Population balance modelling of protein precipitates exhibiting turbulent flow through a circular pipe

The breakage of liquid-liquid, solid-liquid and solid-gas dispersions occurs in many industrial processes during the transport of particulate materials. In this work, breakage of whey protein precipitates passing through a capillary pipe is examined and an experimentally derived breakage frequency is applied to construct a suitable population balance model to characterize the breakage process. It has been shown that the breakage frequency of precipitate particles is highly dependent on their shear history and on the turbulent energy dissipation rate in the pipe. The population balance equation (PBE) uses a volume density based discrete method which is adapted from mass density based discretization. In addition to comparing the model with experimental data, predicted results at different velocities are presented. It was found that the population balance breakage model provides satisfactory results in terms of predicting particle size distributions for such processes.     

AGGLOMERATION,BREAKAGE, POPULATION BALANCE,AND CRYSTALLIZATION KINETICS OF REACTIVE PRECIPITATION PROCESS

The agglomeration and breakage of particles play a significant role in determining final particle size distribution (PSD) and other qualities such as filtering characteristics and impurity content. In reactive precipitation processes, especially during the precipitation of fine particles, the agglomeration and breakage of particles normally cannot be neglected. In this study, the agglomeration and breakage of particles during the reactive precipitation process of procaine penicillin has been investigated experimentally through a continuous steady MSMPR crystallizer. Based on the population balance theory, a crystallization kinetics model including agglomeration and breakage is established, in which the breakage of particles is expressed by a two-body equal-volume birth function and a two-body power-law death function. The crystallization kinetics model is shown to be more suitable than size-dependent growth models as ASL and MJ2.     

AGGLOMERATION, BREAKAGE, POPULATION BALANCE, AND CRYSTALLIZATION KINETICS OF REACTIVE PRECIPITATION PROCESS

The agglomeration and breakage of particles play a significant role in determining final particle size distribution (PSD) and other qualities such as filtering characteristics and impurity content. In reactive precipitation processes, especially during the precipitation of fine particles, the agglomeration and breakage of particles normally cannot be neglected. In this study, the agglomeration and breakage of particles during the reactive precipitation process of procaine penicillin has been investigated experimentally through a continuous steady MSMPR crystallizer. Based on the population balance theory, a crystallization kinetics model including agglomeration and breakage is established, in which the breakage of particles is expressed by a two-body equal-volume birth function and a two-body power-law death function. The crystallization kinetics model is shown to be more suitable than size-dependent growth models as ASL and MJ2.     

Application of grinding kinetics analysis of inorganic powders by a stirred ball mill

The need for ultra fine particles has been increasing in the preparation field of raw powders such as fine ceramics and high functional products. A series of wet grinding experiments were carried out on inorganic powders such as calcite, pyrophyllite and talc by a stirred ball mill. The grinding rate constant K’ in the equation of grinding kinetics was examined based on the grinding kinetics analysis as the same type of function of a previous paper on a vertical type planetary ball mill. The experimental particle size distribution of the ground products was obtained in various grinding conditions. The grinding rate constants K and K’ were expressed by empirical equation involving experimental conditions by a stirred ball mill. The empirical equation on the grinding rate constant was expressed in terms of a function involving the ball diameter of grinding balls, the median diameter of feed material, and Bond’s work index of material, in the experimental conditions. The values of empirical constants C1 and C₂ were 21.13 and 0.0109 on K, while C₁ and C₂ were 120.99 and 0.0192 on K′, respectively. And the particle size distribution of ground products of each test material for a given grinding time was found to be expressing the selection function (the specific rate of breakage) which was obtained from the grinding kinetics analysis. In this study, the grinding rate change on calcite and pyrophyllite was similar at the same experimental operation condition. However, in the case of talc, it was observed that the grinding rate was not increased compared with other samples.     

氧化硼研磨的实验与理论研究

用行星式球磨机对13个不同尺寸区间氧化硼的研磨过程进行了实验与理论研究. 实验测量了13个不同尺寸区间氧化硼的粉碎速率常数及其分布系数,通过对测量结果的分析得到了不同尺寸氧化硼的粉碎速率函数及其分布系数函数,进而建立了粉碎过程质量分数的积分微分方程. 用四阶龙格-库塔法对氧化硼研磨过程的质量分数积分微分方程进行了数值计算,并与实验结果进行了比较. 计算与比较结果表明,氧化硼的研磨过程具有时变特征.     

The influence of grinding technique on the liberation of clinker minerals and cement properties

This paper describes a study of the relationship between the physical, chemical and mineralogical parameters of cement products obtained by different grinding mechanisms namely high pressure grinding rolls (HPGR) and ball milling, and their effects upon the properties of cements prepared from the ground clinker. Samples were prepared as narrow size fractions and also as distribution samples. Characterization parameters were ascertained by using XRF, laser sizing, Blaine and BET surface area and image analysis methods. HPGR grinding resulted in higher degrees of liberation of clinker phases arising from the intergranular breakage along the grain boundaries compared to ball mill grinding. As for service properties, water demand of HPGR products was higher than ball mill products resulting from high micro fissured structure. Despite high liberation of particularly alite mineral in HPGR grinding, the compressive strength of ball mill products was slightly higher than HPGR products for narrow size samples. Finally, particle size distribution effect on strength was more obvious for distribution samples; generally ball milling gave higher strength values.     

Using the discrete element method to analyze the breakage rate in a centrifugal/vibration mill

The grinding characteristics of a centrifugal mill with varying G/D (gyration/mill diameter) ratios were investigated using the population balance model and the discrete element method (DEM). A series of grinding tests were conducted on illite samples using a centrifugal mill under various conditions, and the breakage parameters were calculated. Three-dimensional DEM simulations were also conducted. It was found that the specific rates of breakage estimated for various grinding conditions correlated well with the impact energy calculated from DEM simulations. This information was used to develop scale-up functions for the centrifugal mill in terms of G/D ratio, rotational speed, mill diameter, grinding media diameter, and ball loading.     

燃烧法和固相法所制备的红褐色陶瓷颜料ZnFe1.2Cr0.8O4的研磨效果研究

本文以燃烧法和固相法分别制备的红褐色陶瓷颜料ZnFe_1.2Cr_0.8O₄(燃烧法：C-ZnF;固相法：S-ZnF)为研究对象,采用正交试验的方法对颜料颗粒的研磨最优参数进行分析探讨。使用激光粒度仪表征研磨前后颜料颗粒的粒度及其分布,通过极差分析法来分析研磨的最优参数。结果表明,上述两种颜料颗粒的最优研磨条件均为：添加5wt%的分散剂WF211,研磨时间为100min,研磨转速为2000rpm。在相同的研磨条件下,对比固相法制备的颜料颗粒S-ZnF,燃烧法制备的颜料颗粒C-ZnF可以得到颗粒粒度细且分布窄的颜料颗粒产品。另外,采用一种粒数衡算模型(PBM)来模拟颜料颗粒的研磨过程破碎行为,计算颜料颗粒在研磨过程中的选择函数矩阵。通过模拟分析表明,颜料颗粒C-ZnF的研磨效果要优于颜料颗粒S-ZnF。     

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.     

Influence of process parameters on breakage kinetics and grinding limit at the nanoscale

In long‐term milling experiments, in a stirred media mill, a grinding limit where no further particle breakage occurs was identified. During mechanical stressing of the particles, defects are generated in the crystalline lattice, which allows real fracture of nanoparticles. Below a critical size, defects cannot be stored or generated in the crystallites and the overall limit of grinding is reached. This limit is strongly influenced by material properties and hardly affected by most of the process conditions. However, the breakage kinetics strongly depend on the process parameters and suspension conditions as long as the grinding limit is not reached. Based on these findings, two mechanisms of nanoparticle breakage are proposed. Proper choice of process parameters saves not only up to 90% of the energy input to reach the grinding limit but also leads to a higher product quality in terms of crystallinity and less milling bead wear. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A study on the specific rate of breakage of cement materials in a laboratory ball mill

The specific rate of breakage (S_i) in the widely accepted first-order expression of grinding rate is one of the important factors required to evaluate a grinding process, particularly for the initial grinding stage in various mill types.In this study, the effects of ball diameter and feed size on the specific rate of breakage were investigated on limestone, trass and clinker samples at batch grinding conditions based on a kinetic model. Eight different monosize fractions were prepared between 1.7 and 0.106 mm, using a √2 sieve series. The specific rates of breakage (S_i) were determined from the size distributions at different grinding times, and the specific rates of breakage were compared for three different ball diameters (41, 25.4 and 9.5 mm).The results indicated that the variation of the specific rate of breakage with feed size of cement materials could be expressed. For the specific rate of breakage of each material, empirical equations were developed to express it as a function of feed size and ball diameter.     

Implementation and analysis of numerical solution of the population balance equation in CFD packages

Simulation of polydisperse flows must include the effects of particle–particle interaction, as breakage and aggregation, coupling the population balance equation (PBE) with the multiphase modelling. In fact, the implementation of efficient and accurate new numerical techniques to solve the PBE is necessary. The direct quadrature method of moments, known as DQMOM, is a moment-based method that uses an optimal adaptive quadrature closure and came into view as a promising choice for this implementation. In the present work, DQMOM was implemented in two CFD packages: the commercial ANSYS CFX, through FORTRAN subroutines, and the open-source OpenFOAM, by directly coding the PBE solution. Transient zero-dimensional and steady one-dimensional simulations were performed in order to explore the PBE solution accuracy using several interpolation schemes. Simulation cases with dominant breakage, dominant aggregation and invariant solution (equivalent breakage and aggregation) were simulated and validated against an analytical solution. The solution of the population balance equation was then coupled to the two-fluid model, considering that all particles classes share the same velocity field. Momentum exchange terms were evaluated using the local instantaneous Sauter mean diameter of the size distribution function. The two-dimensional tests were performed in a backward facing step geometry where the vortex zones traps the particles and provides high rates of breakage and aggregation.