Breakage Kinetics Research Articles

Particle aggregation and breakage kinetics in cemented paste backfill

Particle aggregation and breakage kinetics in cemented paste backfill

Population Balance Modeling of Milling Processes: Are We Falsifying Breakage Kinetics and Distribution via Back-Calculation Methods?

Population balance models (PBMs) for milling processes are based on two fundamental concepts: specific breakage rate function and breakage distribution function, which vary with particle size as well as design–operation conditions. The solution of the inverse problem, i.e., the estimation of these two functions’ parameters, may cause falsified kinetics and breakage distribution mechanisms. This perspective article aims to expose and mitigate various aspects of potential falsification, thus enabling the development of a robust PBM. Through an in-depth analysis of historical approaches to the PBM inverse problem and experimental observations, as well as the author’s recent contributions to the inverse methodology within the context of back-calculation methods, six principles have been offered: (i) include the governing physical phenomena and reduce errors in model building; (ii) reduce the number of model parameters via size–operation-dependent functional forms, hybrid approaches for back-calculation, and combination with CFD–DEM and other mechanistic models; (iii) generate a dense particle size distribution data set obtained at various milling times and/or locations; (iv) ensure a grid-independent solution with a sufficient number of size classes; (v) use a global optimization-based back-calculation method for parameter estimation and provide standard errors of the estimates; and (vi) test the predictive capability of the PBM. This perspective article boosts awareness of various challenges involved in the solution of the inverse PBM problem as pertinent to milling processes and provides researchers with six principles to minimize falsified kinetics.

Do Mixtures of Beads with Different Sizes Improve Wet Stirred Media Milling of Drug Suspensions?

The impacts of bead sizes and bead mixtures on breakage kinetics, the number of milling cycles applied to prevent overheating, and power consumption during the nanomilling of drug (griseofulvin) suspensions were investigated from both an experimental and theoretical perspective. Narrowly sized zirconia beads with nominal sizes of 100, 200, and 400 µm and their half-and-half binary mixtures were used at 3000 and 4000 rpm with two bead loadings of 0.35 and 0.50. Particle size evolution was measured during the 3 h milling experiments using laser diffraction. An nth-order breakage model was fitted to the experimental median particle size evolution, and various microhydrodynamic parameters were calculated. In general, the beads and their mixtures with smaller median sizes achieved faster breakage. While the microhydrodynamic model explained the impacts of process parameters, it was limited in describing bead mixtures. For additional test runs performed, the kinetics model augmented with a decision tree model using process parameters outperformed that augmented with an elastic-net regression model using the microhydrodynamic parameters. The evaluation of the process merit scores suggests that the use of bead mixtures did not lead to notable process improvement; 100 µm beads generally outperformed bead mixtures and coarser beads in terms of fast breakage, low power consumption and heat generation, and low intermittent milling cycles.

Role of Flow Inertia in Aggregate Restructuring and Breakage at Finite Reynolds Numbers.

Forces acting on aggregates depend on their properties, such as size and structure. Breakage rate, stable size, and structure of fractal aggregates in multiphase flows are strongly related to the imposed hydrodynamic forces. While these forces are prevalently viscous for finite Reynolds number conditions, flow inertia cannot be ignored, thereby requiring one to fully resolve the Navier-Stokes equations. To highlight the effect of flow inertia on aggregate evolution, numerical investigation of aggregate evolution in simple shear flow at the finite Reynolds number is conducted. The evolution of aggregates exposed to shear flow is tracked over time. Particle coupling with the flow is resolved with an immersed boundary method, and flow dynamics are solved using a lattice Boltzmann method. Particle dynamics are tracked by a discrete element method, accounting for interactions between primary particles composing the aggregates. Over the range of aggregate-scale Reynolds numbers tested, the breakage rate appears to be governed by the combined effect of momentum diffusion and the ratio of particle interaction forces to the hydrodynamic forces. For higher shear stresses, even when no stable size exists, breakage is not instantaneous because of momentum diffusion kinetics. Simulations with particle interaction forces scaled with the viscous drag, to isolate the effect of finite Reynolds hydrodynamics on aggregate evolution, show that flow inertia at such moderate aggregate Reynolds numbers has no impact on the morphology of nonbreaking aggregates but significantly favors breakage probability. This is a first-of-its-kind study that establishes the role of flow inertia in aggregate evolution. The findings present a novel perspective into breakage kinetics for systems in low but finite Reynolds number conditions.

Effects of bead packing limit concentration on microhydrodynamics-based prediction of breakage kinetics in wet stirred media milling

We studied wet-milling kinetics of a drug in a stirred mill using a microhydrodynamic model with various radial distribution functions (RDFs). An nth-order kinetics model was fitted to the median size evolution data at several stirrer speeds and loadings of polystyrene/zirconia beads to identify the breakage rate constant k. Microhydrodynamic parameters were calculated using three RDFs: Carnahan–Starling, Lun, and Ma–Ahmadi. The first one, used in previous microhydrodynamic studies, does not account for the bead packing limit concentration. The Lun and the Ma–Ahmadi RDFs similarly predicted much higher frequency of less energetic–forceful bead–bead collisions than the Starling RDF. A subset selection algorithm determined the best multiple linear regression models (MLRMs) of k with the microhydrodynamic parameters (semi-theoretical) or the process–bead parameters (empirical) as predictors. The Lun RDF-based MLRM had the best fitting–predictions of the kinetics among the semi-theoretical MLRMs, which outperformed the empirical MLRMs.

Characterisation of spray dried microencapsules with amorphous lutein nanoparticles: Enhancement of processability, dissolution rate, and storage stability

Lutein has limited applicability in the food industry because of its poor water solubility and chemical instability. In this study, amorphous and crystalline lutein-loaded microencapsulated powders were prepared via wet media milling and spray drying techniques. The breakage kinetics, surface morphology, physicochemical characteristics, encapsulation efficiency, dissolution behaviour, and storage stability of the two types of microencapsules were determined. Compared with the crystalline formulation, amorphous lutein nanoparticles displayed better breakability (∼478.8 nm within 20 min) in the milling process and faster dissolution rates under both sink and supersaturation conditions (88.0 ± 1.7% and 47.0 ± 3.8%, respectively, within 2 min). Stability testing revealed that the amorphous formulation exhibited slower degradation rates, with decay constants k of 0.03 and 0.07 at 25 and 40 °C, respectively. Our study results suggest that microencapsules with amorphous lutein nanoparticles represent a commercially viable formulation for maintaining chemical stability and improving oral bioavailability.

Effect of operating conditions on the particle size distribution and specific energy input of fine grinding in a stirred mill: Modeling cumulative undersize distribution

Simulation of the fine or ultra-fine grinding process by Population Balance Model (PBM) has its restrictions because some coarse particles maybe disappear or the breakage rate exhibits complex non-first-order breakage kinetics during the grinding process. In this study, the influence of grinding media diameter, stirrer tip speed, media fulling rate, and powder filling rate on the uniformity of particle size distribution and specific energy input were studied in a stirred mill. The cumulative undersize distribution of ground products was simulated by the Rosin-Rammler-Benne (RRB) function, from which the parameter k and characteristic particle xe of the RRB function were obtained. The time-dependence of the parameter k was investigated. And, the Walker model for describing the reduction in characteristic particle xe with specific energy input was examined. Based on these results, combining the Walker model with the RRB function to simulate the cumulative undersize distribution of ground products. Thereby, a new cumulative undersize distribution of the prediction model was developed, which was successfully validated by the experimental data against predicted data.

Impact of solvents during wet stirred media milling of cross-linked biopolymer suspensions

The aim of this study was to examine the impact of solvents during the wet stirred media milling of cross-linked biopolymers considering breakage kinetics, physical stability of the suspensions, and microhydrodynamics. To this end, a model cross-linked biopolymer, sodium starch glycolate (SSG), was milled in acetone and water wherein SSG was non-swellable and swellable, respectively. Suspensions containing SSG particles with and without two stabilizers, i.e., hydroxypropyl cellulose and sodium dodecyl sulfate, were prepared. The temporal evolution of the SSG particle size during the milling was tracked by laser diffraction. Swelling of the SSG particles in water was independently characterized using microscopy and laser diffraction, which revealed fast swelling within a few minutes. Results also suggest that SSG particles were broken faster in water than in acetone, and stabilizers could not prevent severe aggregation in acetone. Despite the greater viscous dampening in water, water was more favorable for faster production of stable suspensions than acetone. The superior performance of aqueous milling was primarily attributed to favorable nanoparticle stabilization in water and secondarily to swelling-induced softening and coarsening of SSG particles. Hence, also being environmentally benign and safe, it is preferred over organic solvent-based milling for the preparation of cross-linked biopolymer nanosuspensions.

Kinetic and Microhydrodynamic Modeling of Fenofibrate Nanosuspension Production in a Wet Stirred Media Mill.

This study examined the impact of stirrer speed and bead material loading on fenofibrate particle breakage during wet stirred media milling (WSMM) via three kinetic models and a microhydrodynamic model. Evolution of median particle size was tracked via laser diffraction during WSMM operating at 3000–4000 rpm with 35–50% (v/v) concentration of polystyrene or zirconia beads. Additional experiments were performed at the center points of the above conditions, as well as outside the range of these conditions, in order to test the predictive capability of the models. First-order, nth-order, and warped-time kinetic models were fitted to the data. Main effects plots helped to visualize the influence of the milling variables on the breakage kinetics and microhydrodynamic parameters. A subset selection algorithm was used along with a multiple linear regression model (MLRM) to delineate how the breakage rate constant k was affected by the microhydrodynamic parameters. As a comparison, a purely empirical correlation for k was also developed in terms of the process/bead parameters. The nth-order model was found to be the best model to describe the temporal evolution; nearly second-order kinetics (n ≅ 2) was observed. When the process was operated at a higher stirrer speed and/or higher loading with zirconia beads as opposed to polystyrene beads, the breakage occurred faster. A statistically significant (p-value ≤ 0.01) MLRM of three microhydrodynamic parameters explained the variation in the breakage rate constant best (R2 ≥ 0.99). Not only do the models and the nth-order kinetic–microhydrodynamic correlation enable deeper process understanding toward developing a WSMM process with reduced cycle time, but they also provide good predictive capability, while outperforming the purely empirical correlation.

Use of Bead Mixtures as a Novel Process Optimization Approach to Nanomilling of Drug Suspensions.

We aimed to evaluate the feasibility of cross-linked polystyrene (CPS)-yttrium-stabilized zirconia (YSZ) bead mixtures as a novel optimization approach for fast, effective production of drug nanosuspensions during wet stirred media milling (WSMM). Aqueous suspensions of 10% fenofibrate (FNB, drug), 7.5% HPC-L, and 0.05% SDS were wet-milled at 3000-4000rpm and 35%-50% volumetric loading of CPS:YSZ bead mixtures (CPS:YSZ 0:1-1:0 v:v). Laser diffraction, SEM, viscometry, DSC, and XRPD were used for characterization. An nth-order model described the breakage kinetics, while a microhydrodynamic model allowed us to gain insights into the impact of bead materials. CPS beads achieved the lowest specific power consumption, whereas YSZ beads led to the fastest breakage. Breakage followed second-order kinetics. Optimum conditions were identified as 3000rpm and 50% loading of 0.5:0.5v/v CPS:YSZ mixture from energy-cycle time-heat dissipation perspectives. The microhydrodynamic model suggests that YSZ beads experienced more energetic/forceful collisions with smaller contact area as compared with CPS beads owing to the higher density-elastic modulus of the former. We demonstrated the feasibility of CPS-YSZ bead mixtures and rationalized its optimal use in WSMM through their modulation of breakage kinetics, energy utilization, and heat dissipation.

Impact of media material and process parameters on breakage kinetics–energy consumption during wet media milling of drugs

While crosslinked polystyrene (CPS) beads and yttrium-stabilized zirconia (YSZ) beads have been commonly used as media for wet milling of poorly soluble drugs and their dissolution enhancement, no first-principle rationale exists for selecting the bead material. The aim of this study is to investigate the impact of stirrer speed (2000–4000 rpm) and CPS bead loading (~20–60%) on the breakage kinetics–energy consumption during milling of griseofulvin, a model poorly soluble drug, and compare the performance of CPS vs. YSZ at the highest bead loading. Laser diffraction, SEM, rheological analysis, and XRPD were used for characterization. The milling was most effective at the highest bead loading–stirrer speed. A microhydrodynamic model suggests that an increase in stirrer speed led to faster breakage due to more frequent and forceful CPS bead–bead collisions. Despite causing slight decrease in maximum contact pressure, an increase in CPS bead loading caused a dramatic increase in average frequency of drug particle compressions, which dominated the faster breakage observed. While YSZ generally required higher specific energy consumption than CPS, it achieved the same product fineness faster than CPS. The microhydrodynamic model rationalized the favorable use of YSZ over CPS beads within the experimental domain studied.

Selection function in breakage processes: PBM and Monte Carlo modeling

Breakage, i.e., formation of smaller fragments from larger initial particles, is an important phenomenon – wanted or unwanted – in many particulate processes. In this work, the volume and time dependent selection function for pure breakage process is modeled based on the population balance modeling approach. It is found that the selection function is directly correlated with the stressing frequency, probability of successful events, and some integral properties of the number density function. A discrete population balance equation is applied to compute the total number of particles and particle size distribution numerically. Moreover, an event-driven constant number Monte Carlo simulation algorithm is presented, and the simulation results are used as an alternative to experimental results. The volume dependency of the selection function is incorporated successfully in the Monte Carlo simulation while selecting particles for stressing event. Some important properties of any particulate process, such as the total number of particles and the size distribution of particles are validated successfully using the Monte Carlo results. This offers new insights into the estimation and interpretation of breakage kinetics.

A CFD-PBM-EMMS integrated model applicable for heterogeneous gas-solid flow

Accurate prediction of the structure-dependent interphase drag coefficient in a computational cell is of importance for the simulation of heterogeneous gas-solid flow. However, the cluster diameter in the cell was usually correlated with macroscopic hydrodynamic parameters or simply approximated as a constant value, though there is a cluster size distribution (CSD) at the sub-grid scale. Based on the energy minimization multiscale (EMMS) model and the population balance model (PBM), a CFD-PBM-EMMS integrated model was proposed to account for the effect of CSD on the interphase drag within the computational grid in this article, in which the PBM was used to describe the spatio-temporal evolution of CSD by introducing an EMMS-based cluster growth rate model. The CFD-PBM-EMMS model was validated by the experiments in a pilot-scale circulating fluidized bed riser of Geldart A and/or B particles. Comparing with the two-fluid model using the EMMS drag or homogeneous drag law, the CFD-PBM-EMMS integrated model shows the best agreement with the experiments under various operating conditions. Much effort is being devoted to incorporating the coalescence and breakage kinetics of clusters to the PBM further to improve the simulation accuracy of the CFD-PBM-EMMS integrated model, especially for dense gas-solid flow.

A modified method for modelling, optimization and control of an anti-solvent crystallization process

Crystallization is a complex process of heat and mass transfer, its product quality is affected by many factors, thus it is a challenge to control this process. Modelling and optimization of the manipulated variables is a feasible way to improve the product quality of crystallization. In this paper, a modified numerical scheme is developed to solve the process model of anti-solvent batch crystallization. The proposed numerical scheme can handle growth, nucleation, agglomeration and breakage kinetics by means of the Method of Characteristics (MOC), which provides an accurate and efficient result of the crystal size distribution (CSD). Then Non-linear Model Predictive Control (NMPC) and non-linear Moving Horizon Estimation (MHE) are presented to provide a temperature profile for optimizing the CSD. The result of case studies indicates that the calculated results of the MOC modified process model were in agreement with the experimental results, which indicated that the MOC method could deal with the specific crystallization process efficiently and accurately. The NMPC-MHE optimized temperature control strategy was tested using the verification experiment of anti-solvent crystallization of β-Artemether, the validated result indicated that the method developed had high feasibility and accuracy. The computational time for the MHE and NMPC formulations with 90 control and sampling steps is well within the sampling interval, which allows for real time process control.

Measurement and simulation of comminution rate in granular materials subjected to creep tests

The subcritical growth of intra-particle cracks and their consequent rupture are often cited among the processes responsible for creep in granular solids. However, classical testing procedures do not involve measurements of grain size distribution, thus preventing a connection of creep strains to delayed breakage. Here, a rate-dependent Breakage Mechanics framework inspired by the physics of delayed crack growth is used to interpret creep measurements through continuum variables reflecting the release of elastic energy during comminution. Creep experiments were conducted on water-saturated materials with different particle size and shape. Each specimen was subjected to stages of oedometric compression and creep at stress levels sufficient to induce grain fracture. During each test, the evolution of the grain size distribution was monitored to quantify the progression of comminution. The experiments were finally interpreted and simulated with a breakage kinetics model characterized by a power law expression similar to those used in subcritical crack growth theories. The results show that the proposed model is able to replicate the phenomenology of delayed breakage. The power law coefficient linking the breakage growth rate to the elastic stored energy was found independent of the size of the grains and characterized by values within the range reported for the stress corrosion index of quartz. By contrast, the calibration of the breakage model revealed that the timescale of comminution was size-dependent, thus signaling an increase of elapsed creep time in materials with larger grains. This finding suggests that samples consisting of larger grains may require longer time until all particles contributing to creep strain attain failure, thus stressing the connection between the processes occurring at the grain scale and those measurable at the continuum scale.

Factors affecting nickel upgrade during selective grinding of low-grade limonitic laterites

ABSTRACT In this study wet grinding batch experiments were performed in a laboratory ball mill using laterites from Evia island, Greece, in order to determine the optimal grinding conditions for potential nickel upgrade. Selective grinding which exploits the different grindability between soft and hard minerals was considered. Kinetic models were determined based on population balance considerations for different feed and ball sizes. The efficiency of selective grinding was evaluated by taking into account breakage kinetics as well as the grade and recovery of nickel in the products. The results showed that, in general, nickel tends to concentrate in the finer fraction of −0.075 mm. In this fraction nickel recovery is increased significantly with increasing grinding times, while only a slight decrease in its grade is noticed. The results of breakage rate analysis indicate that nickel bearing phases are ground with a higher rate when shorter grinding times are used.

Predicting effects of operating condition variations on breakage rates in stirred media mills

In order to model predictively the breakage kinetics in stirred media mills, it is required to determine the effect of operating conditions on the breakage rate. In this study specific breakage rates are measured for grinding experiments in wet-operated stirred media mills for various operating parameters. The experimental specific breakage rates are compared against two different models using yeast cells and limestone as test materials. A simple stressing energy model is able to predict the effect of changes in grinding bead size, grinding bead material, stirrer speed and mass concentration on the specific breakage rate. A more detailed model, which takes stressing energy distributions achieved by mill simulations and material breakage energy distributions from material tests as input, shows promising results. However, the latter is less robust and needs precise input data. Yeast cells prove to be a good test material for the purpose of model validation because of its well described breakage behaviour.

Single crystal fragmentation: Visualizing breakage model performance for pharmaceutical processes

The development of accurate crystal breakage kinetics is paramount to extending the capability of optimization and modelling methods within the pharmaceutical drug-product sector. However, there exists little experimental data on the performance on typical breakage models when applied to small-scale pharmaceutical crystals. Therefore, experimental investigations of crystal impacts on a steel target were carried out across a range of crystal sizes and impact velocities in order to determine critical breakage parameters for a given model. Direct observation of each impact was captured via a high-speed Shadowgraphy imaging technique, capable of 1μm crystal size resolution and up to a 125,000 frames-per-second capture rate. Crystals of Darunavir Ethanolate between a nominal 100→500μm size range were subject to impact velocities of 1→10m/s normal to the target surface. Furthermore, upon collision crystals were classified into one of three breakage modes: intact (Mode I), chipped (Mode II), or split/disintegrated (Mode III). The resulting data was then combined with a post-processing methodology utilizing a bi-variate histogram function and Kernel Density Estimation (KDE). This in turn enabled the extraction of experimentally observed crystal failure velocities, breakage modes, and selection functions detailing how crystal breakage varies with both crystal size and impact velocity. This data was then used to evaluate the performance of current literature in describing crystal breakage within pharmaceutical applications. Furthermore, the methodology implemented in this work can be applied to a range of particulate systems, yielding a flexible tool for pharmaceutical modelling capabilities.

Particle Breakage Kinetics and Mechanisms in Attrition-Enhanced Deracemization

In this study, we report on experiments designed to deconvolute the particle breakage kinetics and mechanism from the parallel phenomena (growth-dissolution, agglomeration) in attrition-enhanced deracemization processes. Through such experiments, we derived the specific breakage rates and cumulative breakage distribution functions for three grinding methods typically used in deracemization experiments: (a) bead grinding, (b) ultrasound grinding, and (c) the combination of bead and ultrasound grinding. Subsequently, we tested these methods on their ability to induce deracemization. We show that in the conventional bead grinding process, breakage occurs mostly by fracture. This results in slow deracemization rates due to the delayed formation of submicron particles that are essential to the process. Conversely, ultrasound grinding very efficiently breaks particles by abrasion. This leads to fast generation of an abundance of submicron fragments resulting in fast deracemization. However, using ultrasound, la...

Comparative analysis on flocculation performance in unbaffled square stirred tanks with different height-to-width ratios: Experimental and CFD investigations

The effect of liquid-level height (H) on floc growth during flocculation has been investigated in unbaffled square stirred tanks with a fixed bottom width (D). Firstly, flocculation tests were performed by using an in situ recognition system for floc morphology to evaluate flocculation performance within each tank at three typical shear rates of Gave=10, 30 and 70s−1. Then, turbulent flow fields generated under all flocculation-test conditions were predicted by computational fluid dynamics (CFD) simulations, followed by a detailed discussion based on experimental and numerical data. It was found that stirred tanks with different H values (characterized by H-to-D ratio) caused distinct turbulent flow fields at the same shear rates examined, reflected by different turbulent intensities in the impeller zone and different non-uniform nature of turbulence in the overall flocculating system. Accordingly, the evolution of floc size and structure was greatly affected due to various aggregation and breakage kinetics. Moreover, the effect of H-to-D ratio on floc growth appeared to be related to the predominant growth mechanism(s), i.e., for the shear rates of Gave=10 and 30s−1, little or no significant breakage occurred and more significant non-uniformity of turbulence generated in the stirred tank with a higher H-to-D ratio led to increasingly pronounced behavior of restructuring, forming aggregates with more compact structure caused by primary-particle rearrangement, whereas at Gave=70s−1, breakage dominated over aggregation and resultant aggregates became smaller and more compact with increasing H-to-D ratios, possibly due to the irreversibility of floc breakage. Therefore, an appropriate liquid-level height of tank should be designed to enhance floc formation and subsequent particle removal efficiency. The combined means of flocculation tests and CFD simulations may provide useful insights for optimizing the design and operation of stirred-tank reactors for flocculation.