DEM–PBM modeling of impact dominated ribbon milling

Ribbon milling is a critical step in dry granulation using roll compaction as it determines the properties of granules, and subsequently the properties of final products. During ribbon milling, fragmentation of ribbons or flakes (i.e., compressed agglomerates from dry powders) are induced by either impact or abrasion. Understanding these fragmentation mechanisms is critical in optimizing ribbon milling processes. In the current study, the discrete element method (DEM) was used to model fragmentation at the microscopic level, providing a detailed insight into the underlying breakage mechanism. In DEM modeling, virtual ribbons were created by introducing an appropriate interfacial energy using the cohesive particle model based on the JKR theory. A set of three‐dimensional parallelepiped ribbons with solid fraction and surface energies ranging from and were created and then fractured during impacts with a plane at various impact velocities, to model impact dominated milling. The fragmentation rate, and the number and size of fragments (i.e., granules) resulting from the breakage of a ribbon during the impact were determined. The DEM simulations showed that the granules size distribution had a bimodal pattern and there was a strong correlation between the size of fines generated from fragmentation during impact and the size of the feed powder (i.e., the size of the primary particles in this study), which was consistent with the observation from physical experiments. Two quantities were calculated from the DEM simulations: the number of fragments p and the fraction of fines z for each breakage event which were then used as input parameters for population balance models (PBM) to develop a DEM–PBM modeling framework. Comparision with published experimental data shows that the developed DEM‐PBM model is a promising tool for analysing ribbon milling, but all breakage mechanisms involved need to to considered in order to achieve an accurate prediction. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3692–3705, 2017     

Coupling of CFD with PBM for a pilot-plant tubular loop polymerization reactor

A computational fluid dynamics model, coupled with population balance model (CFD–PBM), was developed to describe the liquid–solid two-phase flow in a pilot-plant tubular loop propylene polymerization reactor. The model combines the advantage of CFD to calculate the entire flow field and that of PBM to calculate the particle size distribution (PSD). Particle growth, aggregation and breakage were taken into account to describe the evolution of the PSD. The model was first validated by comparing simulation results with the classical calculated data. Furthermore, four cases studies, involving particle aggregation, particle breakage, particle growth or involving particle growth, breakage and aggregation, were designed to identify the model. The entire flow behavior and PSD in the tubular loop reactor, i.e. PSD, solid holdup and liquid phase velocity distribution, were also obtained numerically. The results showed that the model is effective in describing the entire flow behavior and in tracking the evolution of the PSD.     

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.     

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.     

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.     

Drop breakage in a single-pass through vortex-based cavitation device: Experiments and modeling

Liquid–liquid emulsions are used in many sectors such as personal care, home care, and food products. There is an increasing need for developing compact and modular devices for producing emulsions with desired droplet size distribution (DSD). In this work, we have experimentally and computationally investigated an application of vortex-based hydrodynamic cavitation (HC) device for producing emulsions. The focus is on understanding drop breakage occurring in a single-pass through the considered HC device. The experiments were performed for generating oil-in-water emulsion containing 1%–20% rapeseed oil. The effect of pressure drop across the HC device in the range of 50–250 kPa on drop breakage was examined. DSD of emulsions produced through a single pass was measured using the focussed beam reflectance measurements. Comprehensive computational fluid dynamics (CFD) model based on the Eulerian approach was developed to simulate multiphase cavitating flow. Using the simulated flow, population balance model (PBM) with appropriate breakage kernels was solved to simulate droplet breakage in a vortex-based HC device. The device showed an excellent drop breakage efficiency (nearly 1% which is much higher than other commercial devices such as rotor–stators or sonolators) and was able to reduce mean drop size from 66 to ~15 μm in a single pass. The CFD and PBM models were able to simulate DSD. The presented models and results will be useful for researchers and engineers interested in developing compact devices for producing emulsions of desired DSD.     

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     

Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling

Non-linear population balance models (PBMs), which have been recently introduced due to the limitations of the classical linear time-invariant (LTINV) model, account for multi-particle interactions and thus are capable of predicting many types of complex non-first order breakage kinetics during size reduction operations. No attempt has been made in the literature to estimate the non-linear model parameters by fitting the model to experimental data and to discriminate various models based on statistical analysis. In this study, a fully numerical back-calculation method was developed in the Matlab environment to determine the model parameters of the non-linear PBM. Not only does the back-calculation method identify the parameters of complicated non-linear PBMs, but also it gives the goodness of fit and certainty of the parameters. The performance of the back-calculation method was first assessed on computer-generated batch milling data with and without random error. The back-calculation method was then applied to experimental batch milling data exhibiting non-first order effects using both the LTINV model and two separate non-linear models. The back-calculation method was able to correctly determine the model parameters of relatively small sets of batch milling data with random errors. Applied to experimental batch milling data, the back-calculation method with a two-parameter non-linear model yielded parameters with reasonable certainty and accurately predicted the slowing-down phenomenon during dry batch milling. This study encourages experimenters to use advanced non-linear population balance models along with the back-calculation method toward estimating the breakage rate and distribution parameters from dense batch milling data sets.     

Agglomeration and breakage of nanoparticles in stirred media mills—a comparison of different methods and models

The increasing industrial demand for nanoparticles challenges the application of stirred media mills to grind in the sub-micron size range. It was shown recently [Mende et al., 2003. Mechanical production and stabilization of submicron particles in stirred media mills. Powder Technology 132, 64-73] that the grinding behavior of particles in the sub-micron size range in stirred media mills and the minimum achievable particle size is strongly influenced by the suspension stability and thus the agglomeration behavior of the suspension. Therefore, an appropriate modeling of the process must include a superposition of the two opposing processes in the mill i.e., breakage and agglomeration which can be done by means of population balance models. Modeling must now include the influence of colloidal surface forces and hydrodynamic forces on particle aggregation and breakup. The superposition of the population balance models for agglomeration and grinding with the appropriate kernels leads to a system of partial differential equations, which can be solved in various ways numerically. Here a modified h-p Galerkin algorithm which is implemented in the commercially available software package PARSIVAL developed by CiT (CiT GmbH, Rastede, Germany) and the moment methodology according to [Diemer and Olsen, 2002a. A moment methodology for coagulation and breakage problems: Part I—analytical solution of the steady-state population balance. Chemical Engineering Science 57 (12), 2193-2209; Diemer and Olsen, 2002b. A moment methodology for coagulation and breakage problems: Part II—moment models and distribution reconstruction. Chemical Engineering Science 57 (12), 2211-2288] are used and compared to explicit data on alumina. This includes a comparison of the derived particle size distributions, moments and its accuracy depending on the starting particle size distribution and the used agglomeration and breakage kernels. Finally, the computational effort of both methods in comparison to the prior mentioned parameters is evaluated in terms of practical application.     

A CFD‐PBM‐PMLM integrated model for the gas–solid flow fields in fluidized bed polymerization reactors

Although the use of computational fluid dynamics (CFD) model coupled with population balance (CFD‐PBM) is becoming a common approach for simulating gas–solid flows in polydisperse fluidized bed polymerization reactors, a number of issues still remain. One major issue is the absence of modeling the growth of a single polymeric particle. In this work a polymeric multilayer model (PMLM) was applied to describe the growth of a single particle under the intraparticle transfer limitations. The PMLM was solved together with a PBM (i.e. PBM‐PMLM) to predict the dynamic evolution of particle size distribution (PSD). In addition, a CFD model based on the Eulerian‐Eulerian two‐fluid model, coupled with PBM‐PMLM (CFD‐PBM‐PMLM), has been implemented to describe the gas–solid flow field in fluidized bed polymerization reactors. The CFD‐PBM‐PMLM model has been validated by comparing simulation results with some classical experimental data. Five cases including fluid dynamics coupled purely continuous PSD, pure particle growth, pure particle aggregation, pure particle breakage, and flow dynamics coupled with all the above factors were carried out to examine the model. The results showed that the CFD‐PBM‐PMLM model describes well the behavior of the gas–solid flow fields in polydisperse fluidized bed polymerization reactors. The results also showed that the intraparticle mass transfer limitation is an important factor in affecting the reactor flow fields. © 2011 American Institute of Chemical Engineers AIChE J, 58: 1717–1732, 2012     

Experimental and modelling study of gas dispersion in a double turbine stirred tank

Gas dispersion in a double turbine stirred tank is experimentally characterised by measuring local gas holdups and local bubble size distributions throughout the tank, for three liquid media: tap water, aqueous sulphate solution and aqueous sulphate solution with PEG. For all these media, bubble coalescence generally prevails over breakage. Where average bubble size decreases, this can be attributed to the difference in slip velocity between different sized bubbles. Most of the coalescence takes place in the turbine discharge stream.A compartment model that takes into account the combined effect of bubble coalescence and breakage is used to simulate gas dispersion. The model predicts spatial distribution of gas holdup and of average bubble size, with average bubble size at the turbines as an input. Reasonable agreement between experiment and simulation is achieved with optimisation of two parameters, one affecting mainly the slip velocity, the other related mainly to the bubble coalescence/breakage balance. Different sets of parameters are required for each of the three liquid systems under study, but are independent of stirring/aeration conditions. The model only fails to simulate the smaller average bubble diameters at the bottom of the tank.     

Operating Regimes and Hydrodynamics of a Rotating‐Disc Contactor

The operating regimes for a pilot‐scale rotating‐disc contactor (RDC) were investigated by a computational fluid dynamics/population balance model (CFD‐PBM) simulation. The model successfully predicted the critical rotor speed, which divided the entire operating range into two regions. In the low rotor speed region, the input energy was insufficient to break droplets, resulting in an almost constant droplet diameter. Therefore, the increasing revolution slightly affected the interfacial area, while the axial mixing became severe. In contrast, the interfacial area increased significantly in the high rotor speed region because of the increased breakage rate. Moreover, the axial mixing extent increased slightly because the dispersed‐phase accumulation enhanced the advection effect. The results indicate that the CFD‐PBM approach can be applied to engineering practice for extractors.     

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.     

Continuous grinding in a small wet ball mill. Part IV. A study of the influence of grinding media load and density

TXX influence of changes in grinding media load and density on the grinding behaviour of trace quantities of quartz within an environment of calcite in a small continuous wet ball mill have been studied using (a) ball loads ranging from 45% to 100% of the standard load of 1-in. balls, (b) a standard load of 1-in. pebbles and (c) standard loads of equicylinders with specific gravities ranging from 2.82 to 9.49. A mill overload condition (125% ball load) has also been studied using a feed comprising 5% by weight of full-size-range quartz and 95% standard calcite.The results show that variations in breakage behaviour were best followed by a comparison of the adjusted breakage rate constants (k′) for individual sizes corresponding to a standard hold-up-weight. The changes in k′ with ball load could be explained in terms of changes in number of impacts per unit time and in the environmental size distribution. A reduction in the density of the grinding media caused a disproportionate decrease in k′ for the coarse sizes compared with the fine sizes, and breakage rates tended to zero for all sizes as the density of the medium approached that of the pulp.Whilst under some of the test conditions the flow of tracer solids behaved anomalously, under most conditions there was an approximately linear relationship between average residence time and total weight of media charged to the mill.Media density per se had no effect on breakage function, but there was a distinct change due to change in media shape.     

Forthcoming meetings

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.     

Determination of breakage rates of oil droplets in a continuous oscillatory baffled tube

Following on our previous studies, the population balance model that was built on the earlier work from Jareš and Procházka [Break-up of droplets in Karr reciprocating plate extraction column. Chemical Engineering Science 42, 283-292] was modified to include the viscoelastic effect on droplet size distribution and to evaluate the breakage rates of oil-in-water dispersions in a continuous oscillatory baffled tube. In this work, experiments were performed showing that the breakage of droplets is the dominant mechanism in the system, and the physical properties of different oils had no significant influence on droplet size distributions. Under those conditions the model can be used to focus only on breakage rate constants, keeping the number of fitted parameters in the modelling process to a minimum. The droplet breakage results from this work suggest that the oscillation amplitude has more influence on the breakage rates than the oscillation frequency. This work is a major extension and includes droplet data from our previous studies so that the breakage rates can be compared; and the consistency of the rate constants is examined.     

Using the attainable region analysis to determine the effect of process parameters on breakage in a ball mill

Ball milling is one of the most common unit operations used for size reduction across a range of industries. However, it is also a notoriously inefficient process, often contributing substantially to operational costs. In this work, we investigate the influence of rotation rate, grinding media fill level and grinding media size on the optimal production of a product of intermediate size. We find that changing the grinding media size at otherwise identical conditions produces different breakage products as well as nonmonotonic trends with varying rotation rate and grinding media fill levels. In addition, we show how to use the attainable region analysis to explore the parameter space in a reduced time without having to perform tests at every parameter combination. Finally, we examine how the complex interplay between rotation rate, grinding media fill level and grinding media size can control the mechanism of breakage occurring inside a ball mill. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

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.