Multi-scale simulation of needle-shaped particle breakage under uniaxial compaction

The breakage of needle-shaped particles within a random packed bed subjected to uni-directional compaction has been simulated using the discrete element method (DEM). Elongated particles with a chosen aspect ratio have been created by linking individual spherical discrete elements by rigid bonds, characterized by a given ultimate bending strength. A randomly packed bed of these elongated particles has been formed and gradually compressed between two infinite parallel solid planes. The particle size distribution as function of the compaction ratio has been studied in dependence on the individual particle strength, the initial particle length, and their distribution. The simulations have shown that the fragmentation generally follows the sequential halving kinetics and that the formation of fines is most profound in systems with a distribution of particle strengths, both within and between individual particles.     

Agglomerate breakage and adhesion upon impact with complex-shaped particles

Understanding the breakage and adhesion of an agglomerate upon collision with a target particle is a primary step to fathom the adhesive mixing process. While the effect of several variables, such as collision velocity and particle interface energy, on collision behavior has been explored, the effects of target particle morphology have yet to be revealed. In this work, we generate three-dimensional particles with controllable shape and texture using Fourier harmonics and, using the discrete element method, we examine the collision of an agglomerate that impacts each target particle. Results show that the agglomerate breakage depends on the local curvature in the impact zone. We observe that the asperity and elongation factors of the target particle largely contribute to the extent of the deposition of fine particles and the size and number of generated fragments after impact, respectively. These results reveal the large potential error when approximating real particles as smooth spheres in fragmentation studies.     

Numerical simulation of particle breakage of angular particles using combined DEM and FEM

One of the effective parameters of the behavior of rockfill materials is particle breakage. As a result of particle breakage, both the stress–strain and deformability of materials change significantly. In this article, a novel approach for the two-dimensional numerical simulation of the phenomenon in rockfill (sharp-edge particles) has been developed using combined DEM and FEM. All particles are simulated by the discrete element method (DEM) as an assembly and after each step of DEM analysis, each particle is separately modeled by FEM to determine its possible breakage. If the particle fulfilled the proposed breakage criteria, the breakage path is assumed to be a straight line and is determined by a full finite element stress–strain analysis within that particle and two new particles are generated, replacing the original particle. These procedures are carried out on all particles in each time step of the DEM analysis. Novel approach for the numeric of breakage appears to produce reassuring physically consistent results that improve earlier made unnecessary simplistic assumptions about breakage. To evaluate the effect of particle breakage on rockfill's behavior, two test series with and without breakable particles have been simulated under a biaxial test with different confining pressures. Results indicate that particle breakage reduces the internal friction but increases the deformability of rockfill. Review of the v–p variation of the simulated samples shows that the specific volume has initially been reduced with the increase of mean pressures and then followed by an increase. Also, the increase of stress level reduces the growing length of the v–p path and it means that the dilation is reduced. Generally, any increase of confining stress decreases the internal friction angle of the assembly and the sample fail at higher values of axial stresses and promotes an increase in the deformability. The comparison between the simulations and the reported experimental data shows that the numerical simulation and experimental results are qualitatively in agreement. Overall the presented results show that the proposed model is capable with more accuracy to simulate the particle breakage in rockfill.     

Discrete Element Method Simulation of Effect of Aggregate Shape on Fragmentation of Particle Composite

How primary particles affect the fragmentation of particle composites using discrete element method simulation is described. Disk‐shaped composite particles consisting of large aggregate particles embedded in a matrix of smaller sand particles have been impacted against a solid surface at range of collision velocities. The results have been analyzed to determine whether the shapes of the aggregates in the mixture have an influence on the fragmentation of particle composites. It has been identified that the fracture depends on the aggregate shapes of the particle composites. Liberation degrees and particle size distributions have also been found to be controlled by the shape of the primary particles. Particle shapes have a remarkable influence on the fragmentation behavior of the specimen, especially at lower velocities.     

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.     

Breakage model development and application with CFD for predicting breakage of whey protein precipitate particles

Particle breakage due to fluid flow through various geometries can have a major influence on the performance of particle/fluid processes and on the product quality characteristics of particle/fluid products. In this study, whey protein precipitate dispersions were used as a case study to investigate the effect of flow intensity and exposure time on the breakage of these precipitate particles. Computational fluid dynamic (CFD) simulations were performed to evaluate the turbulent eddy dissipation rate (TED) and associated exposure time along various flow geometries. The focus of this work is on the predictive modelling of particle breakage in particle/fluid systems. A number of breakage models were developed to relate TED and exposure time to particle breakage. The suitability of these breakage models was evaluated for their ability to predict the experimentally determined breakage of the whey protein precipitate particles. A “power-law threshold” breakage model was found to provide a satisfactory capability for predicting the breakage of the whey protein precipitate particles. The whey protein precipitate dispersions were propelled through a number of different geometries such as bends, tees and elbows, and the model accurately predicted the mean particle size attained after flow through these geometries.     

Hydrate agglomeration modeling and pipeline hydrate slurry flow behavior simulation

Dynamic modeling and numerical simulation of hydrate slurry flow behavior are of great importance to offshore hydrate management.For this purpose, a dynamic model of hydrate agglomeration was proposed in this paper.Based on population balance equation, the frame of the dynamic model was established first, which took both hydrate agglomeration and hydrate breakage into consideration.Then, the calculating methods of four key parameters involved in the dynamic model were given according to hydrate agglomeration dynamics.The four key parameters are collision frequency, agglomeration efficiency, breakage frequency and the size distribution of sub particles resulting from particle breakage.After the whole dynamic model was built, it was combined with several traditional solid–liquid flow models and then together solved by the CFD software FLUENT 14.5.Finally, using this method, the influences of flow rate and hydrate volume fraction on hydrate particle size distribution, hydrate volume concentration distribution and pipeline pressure drop were simulated and analyzed.     

A proposal for discrete modeling of mechanical effects during drying,combining pore networks with DEM

A discrete modeling approach is introduced to investigate the influence of liquid phase distributions on damage and deformation of particle aggregates during convective drying. The approach is illustrated on a simple 3D aggregate structure, in which monosized spherical particles are arranged in a cubic packing and bonded together at their contacts; the mechanical behavior of this aggregate is simulated by discrete element method (DEM). Liquid phase distributions in the void space are obtained from drying simulations for a pore network. In a one‐way coupling approach, capillary forces are computed over time from the filling state of pores and applied as loads on each particle in DEM. A nonlinear bond model is used to compute interparticular forces. Simulations are conducted for various drying conditions and for aggregates with different mechanical properties. Microcracks induced by bond breakage are observed in stiff material, whereas soft material tends to shrink reversibly without damage. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Quantifying growth and breakage of agglomerates in fluid-particle flow using discrete particle method

The cohesive solids in liquid flows are featured by the dynamic growth and breakage of agglomerates, and the difficulties in the development, design and optimization of these systems are related to this significant feature.In this paper, discrete particle method is used to simulate a solid–liquid flow system including millions of cohesive particles, the growth rate and breakage rate of agglomerates are then systematically investigated. It was found that the most probable size of the agglomerates is determined by the balance of growth and breakage of the agglomerates the cross point of the lines of growth rate and breakage rate as a function of the particle numbers in an agglomerate, marks the most stable agglomerate size. The finding here provides a feasible way to quantify the dynamic behaviors of growth and breakage of agglomerates, and therefore offers the possibility of quantifying the effects of agglomerates on the hydrodynamics of fluid flows with cohesive particles.     

Emergence of falsified kinetics as a consequence of multi-particle interactions in dense-phase comminution processes

Particle breakage during dense-phase comminution processes is significantly affected by mechanical multi-particle interactions, which are neglected in traditional discrete linear population model (DL-PBM). A discrete non-linear PBM (DNL-PBM) has been recently proposed to account for multi-particle interactions; however, the inverse problem, i.e., the estimation of the model parameters, has not been addressed. In this paper, a method for the estimation of DNL-PBM parameters is presented with a purpose of determining the consequences of neglecting multi-particle interactions in the traditional DL-PBM. The model parameters were obtained from a constrained, non-linear, least-squares minimization of the residuals between comminution data and discrete PBM prediction. Comminution data exhibiting multi-particle interactions were obtained from a DNL-PBM simulation followed by addition of 0%, 10%, and 20% random error. A comprehensive statistical analysis of the goodness of fit and certainty of the parameters was performed to discriminate the models. Using the estimated parameters, predictive capability of both models was further assessed by comparing their prediction with additional computer-generated data obtained with a different feed particle size distribution. The parameter estimation method was shown to be highly accurate and robust. DNL-PBM can predict the influence of different feed conditions better than DL-PBM when multi-particle interactions are significant. This study has demonstrated that neglecting multi-particle interactions in dense-phase comminution processes via the use of DL-PBM can lead to falsified kinetics with erroneous breakage functions.     

Numerical simulation of particle breakage in dry impact pulverizer

A novel method to simultaneously simulate particle motion and its breakage in a dry impact pulverizer was developed. The motion of particles in the pulverizer was calculated using a discrete phase model (DPM)‐computational fluid dynamics (CFD) coupling model. When the particle impacts against a vessel wall, impact stress acting on the particle is calculated from Hertz's theory as a function of the impact velocity. At the same time, the particle strength as a function of the particle size is calculated from Griffith's theory. If the impact stress is larger than the particle strength, the particle is broken and replaced with smaller fragments. The size distribution of the fragments is obtained from a breakage function proposed. The motion of the fragments is calculated again by using the DPM‐CFD coupling model. By repeating the above calculations over the whole particles, the grinding phenomenon can be simulated. The calculated results showed good agreement with the experimental one, and validity of the proposed method was confirmed. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3601–3611, 2013     

Energy and population balances in comminution process modelling based on the informational entropy

The results of theoretical and experimental studies of a comminution process are presented. There are two random functions: the selection function and the breakage function in the stochastic model based on a population balance. This model enables prediction of particle size distributions of comminution products after determination of both random functions. Maximum entropy method is used in the entropy model for determining the breakage function. Two cases are analysed, based on continuous and discrete particle size distribution functions of the fed material. Apart from mass balance, the energy balance of comminution process is also used. Searched form of breakage function is determined with the application of methodology of calculus of variations. The results of experimental identification of both models are presented. The parameters that occur in the discrete form of the selection and breakage functions were the identification objects. The results of experimental investigations of quartz sand single comminution in a laboratory jet mill provided an identification base. The experimentally identified results of the entropy model confirmed the adequacy of the theoretical analysis and demonstrated the possibility of adequate prediction of particle size distributions resulting from single comminution.     

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.     

Breakage probability of irregularly shaped particles

The investigation of breakage probability by compression of single particles was carried out. The spherical glass particles and irregularly shaped particles of NaCl, sugar, basalt and marble were subjected to a breakage test. The breakage test includes the compression up to breakage of 100 particles to obtain the distribution of the breakage probability depending on the breakage force or compression work. The breakage test was conducted for five particle size fractions from each individual material, at two stressing rates. Thus obtained 50 breakage force distributions and corresponding 50 breakage work distributions were fitted with log-normal distribution function.Usually, the breakage probability distribution can be found by means of stress or energy approach. The first one uses the stress to calculate the breakage probability distribution. The second approach uses the mass-related work done to break the particle. We prefer to use the breakage force and energy as essential variables. The correlation between the force and energy at their breakage points is obtained by integrating the characteristic force–displacement curve, i.e. the constitutive function of elastic–plastic mechanical behavior of the particle. The irregularly shaped particle is approximated by comparatively “large” hemispherical asperities. In terms of elastic–plastic deformation of the contacting asperities with the plate, a transition from elastic to inelastic deformation behavior was considered. Thus, one may apply the model of soft contact behavior of comparatively stiff hemispheres. Based on this model a relationship between the breakage force distributions and corresponding energy distributions was analyzed. Every tested material exhibits a linear relationship between average breakage energy and average breakage force calculated for every size fraction.For future consideration both force and energy distributions were normalized by division by average force or energy, consequently. The relationship between the fit parameters of normalized energy distribution and corresponding fit parameters of normalized force distribution was established. The mean value and standard deviation of normalized force distribution can be found from mean value and standard deviation of normalized energy distribution by means of system of two linear equations. The coefficients of those linear equations remain the same for all of the above tested materials; particle size fractions and stressing rates. As a result the simple transformation algorithm of distributions is developed. According to this algorithm the force distribution can be transformed into energy distribution and vice versa.     

竖式移动床层中散料颗粒破碎的离散元分析

建立了竖式移动床层中颗粒破碎的炉料下降运动模型,采用离散单元法对干法熄焦炉和烧结余热回收竖罐两种竖冷装置内不同形状、尺寸及强度颗粒料的下落运动和破碎过程进行了数值计算。结果表明,颗粒在下落过程中所受压力先逐渐增大,进入出料区后又逐渐减小,颗粒破碎情况与所受压力密切相关;焦炭在干熄炉内下落到斜道区时,由于炉体直径扩张,料层所受平均压力减小,破碎速率有所减慢。由于烧结矿强度相对较小,刚进入烧结竖罐就发生破碎现象,而炉体直径扩张对破碎影响不大;受固定炉墙的影响,颗粒在靠近炉墙位置处更容易破碎。分析不同形状颗粒的破碎过程发现,正方形颗粒从某一顶点沿对角线逐渐破碎,长条形颗粒从一侧向另一侧逐渐破碎,而缺角的不规则颗粒从形状缺失一侧开始向内破碎。     

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.     

Interfacial phenomena and droplet size of particle stabilized emulsions in oscillatory shear

Production of particle stabilized oil in water emulsions has been investigated both theoretically and experimentally under oscillatory shear conditions using different stabilizing particles (SPs). The investigation included analysis of the interaction between particles interfacial stability and droplets breakage and coalescence. For hydrophobic SPs, droplets maintained their sizes as determined by torque balance (TB) without significant breakage or coalescence. For the more hydrophilic SPs, larger droplets formed that broke by eddies in the inertial subrange. At higher fluid shear stresses, loss of the SPs occurred during droplet formation leading to near bare droplet surface and coalescence to much larger sizes with subsequent fragmentation by capillary instabilities. The final droplet size in both cases was very different from TB model predictions. A modeling approach is proposed that combined both TB and droplet breakage and coalescence mechanisms. Comparison between the experimental results and the models predictions showed satisfactory agreement. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2902–2911, 2016