Breakage Model Research Articles

Developing breakage models relating morphological data to the milling behaviour of flame synthesised titania particles

A detailed population balance model is used to relate the reactor conditions of flame synthesised titanium dioxide particles to their milling behaviour. Breakage models are developed that utilise morphological data captured by a detailed particle model to relate the structure of aggregate particles to their size-reduction behaviour in the post-synthesis milling process. Simulations of a laboratory-scale hot wall reactor are consistent with experimental data and milling curves predicted by the breakage models exhibit features consistent with experimental observations. The selected breakage model considers the overall fractal structure of the aggregate particles as well as the neck size between neighbouring primary particles. Application of the model to particles produced under different reactor residence times and temperatures demonstrates that the model can be used to relate reactor conditions to the milling performance of titanium dioxide particles.

A high-order moment-conserving method of classes (HMMC) based population balance model for mechanical flotation cells

In this paper, a high-order moment-conserving method of classes (HMMC) model is developed to predict bubble size distribution, number mean and Sauter mean diameters (SMD) in mechanical flotation cells. The population balance model (PBM) uses vessel averaged turbulence kinetic energy dissipation rate and gas hold-up as model inputs along with initial bubble size distribution. Experimental measurements are made in a 0.8m3 pilot scale XCELLTM mechanical flotation cell at different aeration rates and impeller tip speeds. All the experimental measurements using MIBC frother are performed at frother dosage value over critical coalescence concentration (CCC), therefore, coalescence process is not considered in the PBM. Nonlinear least squares function (lsqnonlin) and constrained nonlinear multivariable minimization function (fmincon) available in the MATLAB optimization toolbox are used. Three breakage models, namely, Coulaloglou and Tavlarides (1977), Chen et al. (1998), and Alopaeus et al. (2002), each requiring estimation of adjustable parameters from the experimental data are selected. Also, 95% confidence intervals for estimated breakage parameters are calculated using a bootstrap based re-sampling technique. The developed model is capable of providing acceptable predictions of bubble size distribution, number mean diameter, and SMD for a reasonably wide range of operating conditions and has the potential to be developed further for three phase air-water-solid systems.

Modeling of Chip Breakage in Machining of AISI 1045 Steel by Using an Improved Damage Mechanics Model

Modeling of chip breakage is challenging due to the complex loading conditions in the machining process. This paper is dedicated to the finite element modeling of chip breakage with damage mechanics approaches. In this study, the extended modified Bai–Wierzbicki model, which has been successfully implemented in modeling of chip formation and flow in a previous study, is supplemented with an effective damage concept to simulate chip breakage. The model takes the influence of temperature, strain rate, stress state, and the loading history of a chip breakage cycle into consideration. The emphasis is placed on the loading paths under fiercely varying stress states during external longitudinal turning processes, which has significant influence on the damage behavior of materials. An effective damage concept based on cut-off value is proposed, which differentiates the effect of accumulated plastic strains on damage in different phases of the chip breakage cycle in turning processes. The material model successfully predicts the chip breakage behavior in turning processes conducted with different process parameters and tools.

Copula-based approximation of particle breakage as link between DEM and PBM

Abstract In process engineering, the breakage behavior of particles is needed for the modeling and optimization of comminution processes. A popular tool to describe (dynamic) processes is population balance modeling (PBM), which captures the statistical distribution of particle properties and their evolution over time. It has been suggested previously to split up the description of breakage into a machine function (modeling of loading conditions) and a material function (modeling of particle response to mechanical stress). Based on this idea, we present a mathematical formulation of machine and material functions and a general approach to compute them. Both functions are modeled using multivariate probability distributions, where in particular so-called copulas are helpful. These can be fitted to data obtained by the discrete element method (DEM). In this paper, we describe the proposed copula-based breakage model, and we construct such a model for an artificial dataset that allows to evaluate the prediction quality.

Modeling Agglomeration Behavior in High Temperature Gas–Solid Fluidized Beds via Monte Carlo Method

Particles and bubbles are important discrete elements in gas–solid fluidized beds. The interactions between them affect the evolution of agglomerates. Based on the analyses of microscopic physical processes, mathematical models of particle coalescence and agglomerate breakage caused by bubble expansion were developed. Coupling with the constant-number Monte Carlo method, the evolution of particle size distribution was simulated. Modeling results matched well with cold-mode experiment results, which demonstrated that the approach is a promising way to predict agglomeration behavior in the fluidized beds. Also, the coalescence efficiency β0 and breakage efficiency βb extracted from Monte Carlo results can provide rate constants for the kernels in population balance modeling. Further investigations of the effects of process parameters show that decreasing bed temperature or increasing superficial gas velocity reduces the coalescence efficiency and increases the breakage efficiency, which are beneficial to pr...

A lognormal distribution of the lengths of terminal twigs on self-similar branches of elm trees.

Lognormal distributions and self-similarity are characteristics associated with a wide range of biological systems. The sequential breakage model has established a link between lognormal distributions and self-similarity and has been used to explain species abundance distributions. To date, however, there has been no similar evidence in studies of multicellular organismal forms. We tested the hypotheses that the distribution of the lengths of terminal stems of Japanese elm trees (Ulmus davidiana), the end products of a self-similar branching process, approaches a lognormal distribution. We measured the length of the stem segments of three elm branches and obtained the following results: (i) each occurrence of branching caused variations or errors in the lengths of the child stems relative to their parent stems; (ii) the branches showed statistical self-similarity; the observed error distributions were similar at all scales within each branch and (iii) the multiplicative effect of these errors generated variations of the lengths of terminal twigs that were well approximated by a lognormal distribution, although some statistically significant deviations from strict lognormality were observed for one branch. Our results provide the first empirical evidence that statistical self-similarity of an organismal form generates a lognormal distribution of organ sizes.

Grain breakage under uniaxial compression, through 3D DEM modelling

A breakable grain model is presented, using the concept of particles assembly. Grains of polyhedral shapes are generated, formed by joining together tetrahedral subgrains using cohesive bonds. Single grain crushing simulations are performed for multiple values of the intra-granular cohesion to study the effect on the grain’s strength. The same effect of intra-granular cohesion is studied under oedometric compression on samples of around 800 grains, which allows the evaluation of grain breakage model on the macroscopic behaviour. Grain size distribution curves and grain breakage ratios are monitored throughout the simulations.

A combined DEM-SBFEM for particle breakage modelling of rock-fill materials

Both experimental and numerical studies show that particle breakage has significant influence on the macro mechanical response of granular soils. This study presents a novel computational method that combines two numerical techniques viz., the discrete element method (DEM) and the scaled boundary finite element method (SBFEM), with an aim to model the particle breakage phenomenon in granular soils. Individual grains in the soil are modelled by a single star-convex polygon having an arbitrary number of sides. The DEM, which is well-known for its ability in addressing problems involving the response of granular and discontinuous media, is used to determine the motion of- and interaction between the particles. At the end of each time step, the SBFEM is used to determine the stress state in the grains. As the SBFEM can be flexibly formulated on polygons with arbitrary number of sides, only a single polygon is required to replicate the morphology of each grain. This reduces the computational resources necessary for a stress analysis of each grain. The particle breakage condition is determined if the stress state in a polygon satisfies a mechanically driven criterion. Breakage is executed by splitting a polygon into two. The resulting new polygons are directly modelled by the SBFEM without any change to the formulation. The feasibility of the developed technique is demonstrated by a numerical benchmark.

Numerical investigations on breakage behaviour of granular materials under triaxial stresses

The effect of particle breakage and intermediate principal stress ratio on the behaviour of crushable granular assemblies under true triaxial stress conditions is studied using the discrete element method. Numerical results show that the increase of intermediate principal stress ratio <TEX>$b(b=({\sigma}_2-{\sigma}_3)/({\sigma}_1-{\sigma}_3))$</TEX> results in the increase of dilatancy at low confining pressures but the decrease of dilatancy at high confining pressures, which stems from the distinct increasing compaction caused by breakage with b. The influence of b on the evolution of the peak apparent friction angle is also weakened by particle breakage. For low relative breakage, the relationship between the peak apparent friction angle and b is close to the Lade-Duncan failure model, whereas it conforms to the Matsuoka-Nakai failure model for high relative breakage. In addition, the increasing tendency of relative breakage, calculated based on a fractal particle size distribution with the fractal dimension being 2.5, declines with the increasing confining pressure and axial strain, which implies the existence of an ultimate graduation. Finally, the relationship between particle breakage and plastic work is found to conform to a unique hyperbolic correlation regardless of the test conditions.

Experimental Investigation and CFD Modeling of Hydrodynamic Parameters in a Pulsed Packed Column

ABSTRACTIn this work, a combination of computational fluid dynamics (CFD) and droplet population-balance model (DPBM) in the framework of Fluent was applied to simulate the drop-size distributions and flow fields in a pilot-plant liquid–liquid extraction pulsed packed column. The three-dimensional unsteady-state liquid–liquid flow was modeled using the Eulerian two-fluid equations in conjunction with the realizable k – ε turbulence model. The classes method (CM) was chosen for solving population-balance equations. Two models for breakage and coalescence, the models of Luo and Garthe, were used in the CFD code. The model was validated by comparing the simulated drop-size distributions and holdup with experimental measurements. After the validation of the model, the effects of the operating conditions (feed rates and pulsation) on the dispersed phase holdup and drop-size distributions were studied. The results of linked CFD-DPBM model and experiments revealed that the dispersed phase holdup was increased when the organic and aqueous flow rates increased and when the intensity of pulse was increased, the holdup increased. Increasing the dispersed and continuous feed rates caused the Sauter mean diameter of the drops decreased and when the intensity of pulse was increased, because of high droplets break up rate, the Sauter mean diameter decreased. Results of linked CFD-DPBM model show that the CFD-DPBM tool is able to predict hydrodynamic parameters in a pulsed packed column.

3D printed agglomerates for granule breakage tests

In the research into agglomeration, a long term barrier is the lack of a universally accepted method to evaluate the breakage propensity of agglomerates. Computer simulation is often used but is limited by the lack of identical, controlled agglomerates to test and validate simple models, let alone replicate the complex structure of real industrial agglomerates.This paper presents work on the characterisation of strength of model test agglomerates prepared by a 3D printing production method enabling fully reproducible structures. Agglomerates were designed using Solidworks 2014 software and printed by an Objet500 Connex 3D printer. Materials with different mechanical properties were used to print the particles and the inter particle bonds, allowing a series of combinations of bond strength, particle strength and agglomerate structure to be tested. Compression and impact tests were performed to investigate the breakage behaviour of the printed agglomerates in terms of agglomerate orientations, bond properties and strain rates. This method will allow more rigorous testing of agglomerate breakage models.

Turbulence modeling in the wide energy spectrum: Explicit formulas for Reynolds number dependent energy spectrum parameters

The energy spectrum function is an important turbulence quantity in the breakage and coalescence models. An energy spectrum model valid in the entire spectrum of turbulence has been proposed by Pope (2000). However, the use of this energy spectrum model requires re-fitting of model parameters when the turbulent flow; characterized by Reynolds numbers, is changed. This re-fitting results in the need for an optimization routine which may be time consuming to implement and also increases the computational cost of solving the breakage and coalescence models. For this reason, in the present study, novel approximate explicit formulas are proposed that allow the calculation of model parameters in the Pope model energy spectrum without the need for optimization techniques.

AN INVESTIGATION ON SINGLE-PARTICLE IMPACT BREAKAGE FUNCTIONS OF A GOLD ORE BY DROP-WEIGHT TECHNIQUE

Impact breakage functions for a gold ore from Bergama region (Turkey) was estimated based on the single-particle impact breakage distributions of particles obtained from drop weight tests. For this purpose, a modified manual version of a JK Tech drop-weight test device was used. Effect of specific comminution energy (Ecs) on impact breakage distributions were investigated. Relationship between specific comminution energy and impact breakage product fineness (t10) parameter was establised. Ecs-t10 model given in the literature was successfully fitted to the breakage test results. Ecs-t10 impact breakage model for the gold ore was proposed to be used in breakage function estimation. t-family curve approach was established to estimate single particle breakage functions of the gold ore. For this purpose, t-family curves were modelled and variations of empirical breakage functions for the gold ore were estimated on different impact energy levels for comminution process modelling.

Analysis of cone crusher performance with changes in material properties and operating conditions using DEM

Discrete Element Method (DEM) simulation with non-round particles and including breakage has been used to understand the breakage behaviour and operating performance of an industrial scale cone crusher using a representative ore. The breakage model uses a replacement strategy and impact energy specific progeny size data from a Drop Weight Test (DWT). There is a strong variation in the breakage behaviour with height in the compression region as the differing profiles of the concave and mantle create five different regions with monotonically decreasing width and differing degrees of convergence between the surfaces. These control the rate of motion and the ability to load and break the particles, and determine whether high forces are generated via multi-particle stress chains or as single particle loading directly from the liner surfaces. The larger feed particles jam in the compression zone prior to breakage and cause observable obstruction to the flow of finer material and strong non-uniformity in the flow of product down the lower part of the mantle. Trends in the coarseness of product and changes in steady state throughput are identified with changes in material properties (rock breakage energy and friction coefficient) and crusher operating parameters (Closed Side Setting and crusher rotation rate).

Implicit integration of simple breakage constitutive model for crushable granular materials: A numerical test

In the context of the recently developed breakage mechanics that is based on thermodynamics principles, this paper presents a mathematical modelling procedure to implement the simple (i.e., linear elastoplastic) breakage constitutive model using finite element analysis (FEA) with illustrations by engineering applications. More informative mathematical derivation procedures of energy dissipations, plastic potential, yield function and non-associated flow rules are presented. In contrast, the existing relevant publications often lack sufficient elaboration, leaving knowledge gaps in the full understanding the model. This is followed by a series of numerical simulations in ABAQUS to test the model at the constitutive level. Various isotropic and triaxial shear tests in drained or undrained conditions are tested to illustrate the key features of the breakage model, which seem to be overlooked in the literature. Finally, a few numerical results are compared with experimental shear tests to demonstrate the ability of the simple breakage model in reflecting mechanical responses of crushable granular aggregates.

Eulerian–Lagrangian simulation of bubble coalescence in bubbly flow using the spring-dashpot model

The Eulerian–Lagrangian simulation of bubbly flow has the advantage of tracking the motion of bubbles in continuous fluid, and hence the position and velocity of each bubble could be accurately acquired. Previous simulation usually used the hard-sphere model for bubble–bubble interactions, assuming that bubbles are rigid spheres and the collisions between bubbles are instantaneous. The bubble contact time during collision processes is not directly taken into account in the collision model. However, the contact time is physically a prerequisite for bubbles to coalesce, and should be long enough for liquid film drainage. In this work we applied the spring-dashpot model to model the bubble collisions and the bubble contact time, and then integrated the spring-dashpot model with the film drainage model for coalescence and a bubble breakage model. The bubble contact time is therefore accurately recorded during the collisions. We investigated the performance of the spring-dashpot model and the effect of the normal stiffness coefficient on bubble coalescence in the simulation. The results indicate that the spring-dashpot model together with the bubble coalescence and breakage model could reasonably reproduce the two-phase flow field, bubble coalescence and bubble size distribution. The influence of normal stiffness coefficient on simulation is also discussed.

A phenomenological model of single particle breakage as a multi-stage process

Understanding ore breakage is the key for the development of appropriate breakage characterisation experiments. Hence, there has been significant effort to understand a single breakage event, both from the experimental and the modelling standpoints. One particularly interesting approach is to consider breakage of a particle, for instance in a drop weight tester, as a sequence of primary and secondary breakage components. Secondary breakage is defined as subsequent breakage of progeny that takes place after the primary fracture of the parent particle due to surplus energy that is provided by the comminution or breakage testing device. This influences the spatial distribution of fragments as well as their selection. Due to the complex nature of secondary breakage, most of the researches in the past have either been dedicated to understanding primary fracture as a basic element of every comminution process or to describe the breakage process as a whole. However, the progression of breakage may be modelled properly by incorporating the sequential breakage of fragments into a breakage process model. In the paper, a novel approach is proposed to describe a single breakage event by modelling it as a process through several stages of primary fracture while appreciating the effect of sequential fragmentation. The model has three main components: the primary breakage, the selection, and the classification functions. The data that served as the basis for demonstrating this model were a number of impact experiments in an impact load cell on quartz and apatite particles. The results indicate a good correspondence between experiments and model fit. Additionally, a sensitivity analysis to changes in selection and classification functions was performed, which indicated a strong dependency of a breakage event on its sub-processes. It is envisaged that this approach will allow decoupling the contributions of the ore and of the stressing conditions on the outcome of the breakage event.

A bubble breakage model for finite Reynolds number flows

ABSTRACTThe breakage frequency of bubbles in turbulent liquid flows is modeled as the inverse of the breakage time by Martinez-Bazan et al. [J. Fluid Mech. 401: 157–182; 1999]. In this definition of the breakage frequency, it is assumed that the breakage probability is unity and hence all bubbles will break. This assumption is reasonable in turbulent flows at extremely high Reynolds numbers in which the turbulence energy dissipation is very high. For systems characterized by finite Reynolds numbers the energy dissipation rate decreases rapidly and the breakage probability is reduced significantly. In the present study, the breakage frequency model by Martinez-Bazan et al. has been extended to include the effect that only a fraction of the bubbles breaks at finite Reynolds numbers. For this model extension, an adjusted version of the breakage probability formula proposed by Coulaloglou and Tavlarides [Chem. Eng. Sci. 32: 1289–1297; 1977] was employed. The extended breakage frequency model for finite Reynolds number flows has been evaluated by comparison to recent experimental single bubble breakage data. It can be concluded that extensive experimental analyses are required to gather sufficient experimental data for improved understanding of the physical phenomena and for model validation. In particular, the bubble breakage analysis must be performed simultaneously with the characterization of the local turbulence properties in the flow.

Population Balance Modeling of Breakage and Coalescence of Dispersed Bubbles or Droplets in Multiphase Systems

Modeling of breakage and coalescence of dispersed phases such as bubbles or droplets in multiphase systems is of paramount importance to the control of the dispersed phase size distribution in process industry. Population balance model (PBM) has become a rountine tool to simulate the breakage, coalescence and size distribuiton of dispersed phase. However, the current kernel functions for breakage and coalescence in PBM are either derived from statistical models or based on some phenomenological models, empirical correlations or semi theoretical methods, since the physics of breakage and coalescence in multi-phase systems is complex. As a result, few models could completely considers all of the physical constraints relevant to the complex flow field and material properties, and it is still a challenging issue to accurately predict the breakage and coalescence for different operating conditions. This article gives a systematic overview of the mechanisms and models about the breakage and coalescence of bubbles or droplets, and the numerical algothirm for population balance equations as well as the application of PBM simulation in gas-liquid or liquid-liquid systems. Finally, the state-of-the-art and future development of PBM are analyzed.

Rearrangement hotspots in the sex chromosome of the Palearctic black fly Simulium bergi (Diptera, Simuliidae).

An extreme example of nonrandom rearrangements, especially inversion breaks, is described in the polytene chromosomes of the black fly Simulium bergi Rubtsov, 1956 from Armenia and Turkey. A total of 48 rearrangements was discovered, relative to the standard banding sequence for the subgenus Simulium Latreille, 1802. One rearrangement, an inversion (IIS-C) in the short arm of the second chromosome, was fixed. Six (12.5%) of the rearrangements were autosomal polymorphisms, and the remaining 41 (85.4%) were sex linked. More than 40 X- and Y-linked rearrangements, predominantly inversions, were clustered in the long arm of the second chromosome (IIL), representing about 15% of the total complement. The pattern conforms to a nonrandom model of chromosome breakage, perhaps associated with an underlying molecular mechanism.