Understanding production of fines in batch ball milling for mill scale-up design using the population balance model

The rate of production of fine material in the batch mode of grinding operation forms the basis for determination of the grindability parameter of the Bond approach and the breakage distribution function of the population balance model (PBM) approach to the mill scale-up design. For a given set of mill operating conditions, the rate of production of fines is determined by the breakage characteristics and production history of the material being ground. Another important aspect is the variation in the rate of production of fines with grinding time. With a view to developing a clear understanding of these aspects, a detailed analysis of variations in the rate of production of fines was carried out using the PBM framework and two well-known functional forms for the specific breakage rate and breakage distribution parameters. In this paper, it has been shown how the results of this analysis can be used for: (i) obtaining more accurate estimates of the breakage distribution parameters by performing just one short-duration batch grinding experiment, and (ii) explaining variation in the Bond Work index with the product size in terms of the exponent of particle size in the expression for the specific breakage rate function: $k_j=A^1{x}_j^{\alpha }$.     

Experimental identification of entropy model of comminution process

The entropy model based on population balance enables theoretical prediction of particle size distribution of comminution product. Selection and breakage functions occur in this model. The form of the selection function was determined experimentally. Informational entropy was used for the breakage function determination. Parameters of both functions can be estimated from experiments. The parameter identification of the entropy model was carried out on the basis of research on limestone comminution. Grinding tests were carried out in a laboratory fluidised bed jet mill. The results of the experimental identification confirm the accuracy of the entropy model.     

Development and validation of a two-dimensional population balance model for a supercritical CO2 antisolvent batch crystallization process

In this study, a two-dimensional population balance model with solvent removal kinetics has been developed to predict the dynamic behavior of carbamazepine form II crystals produced by a supercritical CO₂ antisolvent batch crystallization process. The model was simulated and validated using experimental crystal size distribution data (CSD). The model was able to accurately predict the behavior of CSD with a change in process operating conditions. The model was also applied to study the time evolution of aspect ratio, average crystal length, and solute concentration in the solution. Finally, solvent removal kinetics were modeled to evaluate the solvent content and drying temperature of the drying gas during the solvent removal process. The developed mathematical model and the presented results suggest the ability of the discussed approach to make suitable model predictions, which can significantly reduce the number of experimental trials required for process design, optimization, and control.     

A macro model for usage and recycling pattern of steel in Japan using the population balance model

Abstract

A macro model for evaluating the steel flow in Japan is proposed. The steels are classified into four types; virgin steel for machinery and for construction and recycled steel for machinery and for construction. The steel is assumed to be discharged from the society in accordance with the lifetime distribution of each usage. The amount of recycled steel and the accumulation in society are calculated using a population balance model. The comparison with the calculated results and statistics ensures the validity of the model. Since the amount of recycled steel mainly supplied for construction will increase and be oversupplied, recycled steel will have to be used for machinery. The required amount and the available amount to supply recycled steel for machinery are considered from the point of allowable copper concentration for machinery use. Copper contaminates steel during the recycling process of steel used for machinery and the contamination ratio is evaluated with the model. The copper concentration in the recycled steel and the amount of CO₂ emission are predicted for various scenarios. The relationship between recycling ratio and contamination ratio, which enables us to supply recycled steel for machinery, and the allowable CO₂ emission to decrease the contamination ratio are discussed. © 2000 Published by Elsevier Science Ltd.     

Scale-up procedure of parameter estimation in selection and breakage functions for impact pin milling

This paper presents a scale-up procedure of parameter estimation in the selection function and breakage function from single particle impact breakage to inform the predictions at the process scale of an impact pin mill. The selection and breakage functions used in population balance model (PBM) for particle breakage in the literature are briefly reviewed. Single particle breakage tests are conducted in a vertical impact tester subject to varying impact velocities. The single particle breakage results further serve to provide the database for the parameter estimation in Vogel and Peukert model (Vogel and Peukert, 2005). The estimated parameters in the particle level are upscaled in an impact pin mill using the population balance model, which is implemented in the software gPROMS (Process Systems Enterprise, UK) (gPROMS® 4.1 Release Notes, 2016). The impact milling tests were carried out in an impact pin mill UPZ100 subject to four feed rates, providing the dataset for model validation. The sensitivity analysis of the PBM parameters was conducted to help identify their leverage on the particle size distribution. The scale-up procedure by specifying the parameters from single particle level to the process level of PBM demonstrates an approach to help predict the size reduction process subject to the prevailing mechanism in an impact pin mill and other milling processes alike.     

An extended cell-average technique for a multi-dimensional population balance of granulation describing aggregation and breakage

This paper focuses on obtaining the numerical solution to a three-dimensional population balance model (PBM) of granulation using the cell-average technique first proposed by [22]. Conventionally, linear grids are used for the solution of PBMs, but the ability to incorporate non-linear grids would be more advantageous given that a larger size range can be covered using fewer number of grids, thus reducing computational overhead. Furthermore, the use of linear representation of grids in PBMs to represent industrial granulation processes that span a wide granule size range is computationally prohibitive and results show that a non-linear grid representation is computationally more efficient with comparable accuracy. Parallelization of the PBM via a multi-core strategy has also been incorporated in order to reduce the simulation time of the model. Incorporating the cell average technique along with parallelization of the overall model lends credence to the overall use of the model for effective granulation process design and analysis.     

A compartment based population balance model for the prediction of steady and induction granule growth behavior in high shear wet granulation

This paper presents a predictive modeling approach of the high shear wet granulation process, quantifying the difference between the steady and induction granule growth behavior. The spatial heterogeneity in liquid binder distribution and shear rate is simulated using a compartmental population balance model. The granulator is divided into two compartments based on particle motion, which consists of a circulation compartment, and an impeller compartment. In the circulation compartment, a viscous dissipation dependent coalescence kernel is adapted for the aggregation process. In the impeller compartment a shear rate dependent aggregation kernel is implemented. The model was calibrated and validated using the dynamic evolution of granule mean size (d₅₀). The granulation dynamics are studied with respect to change in impeller speed, liquid to solid ratio, wet massing time, initial porosity, and binder viscosity. The transition from induction growth to steady growth regime with changing process conditions is demonstrated using the model. It is observed that the model captures the effect of process parameters and spatial heterogeneity on the dynamic evolution of d₅₀.     

Stress- and process model for dispersing of nanoparticulate suspensions in laminar shear flow

This work presents a stress model and a process model for dispersing processes in laminar shear flow, which are performed in devices such as kneaders, extruders and three roller mills. Based on deep investigations of two kneaders (lab- and pilot-scale) a methodology is presented, which allows modelling the dispersing process by applying a stress model along with a population balance model to the given task. Stress models describe how often particles are stressed during the process and which energy is transferred to the particles. For dispersing in laminar shear flow, it is crucial to distinguish between dispersing-effective and ineffective stress events based on whether the stress intensity is high enough to overcome the particle strength. Results suggest, that the effective specific energy input determines the dispersing progress independent of the process scale. A process model in form of a population balance was required to obtain otherwise unknown parameters for the stress model. The applied population balance model accounts for the acting working principles during dispersing. For such a mechanistic depiction of the dispersing progress, the trilateral dependency between particle size, resulting viscosity, particle stressing and fragmentation must be modelled. It was found that based on the models, the process behaviour on both scales can be depicted, which offers the perspective of a knowledge-based scale-up of machines and processes, process control and process design based on these results.     

An analytical solution of the population balance equation for simultaneous Brownian and shear coagulation in the continuum regime

Brownian coagulation of aerosol particles can take place in both laminar and turbulent flows. Thus, the simultaneous Brownian and shear coagulation will always occur in practical applications. This study presents an analytical solution to describe the size evolution of polydisperse particles undergoing simultaneous Brownian and shear coagulation. The analytical solution is derived using the log-normal method of moments (LNMOM) with some approximations. Then, the analytical solution is validated by comparing with previous analytical solutions derived for limiting cases. The results show that the present analytical solution is consistent with previous analytical solutions for these limiting cases. Further, the time trajectories of the total particle number concentration, the geometric standard deviation and the geometric number mean particle volume predicted by the present analytical solution are compared with those of a numerical LNMOM model. The results show that the present analytical solution gives good predictions of the total particle number concentration but less accurate predictions of the geometric standard deviation and the geometric number mean particle volume. A dimensionless analysis shows that the coagulation rate ratio and the initial polydispersity are two important factors for characterizing the size evolution of simultaneous Brownian and shear coagulation.     

Application of TUSSIM with a variable Tromp curve for predicting optimal operation of multi-compartment mills with various ball size distributions

A true unsteady-state simulator (TUSSIM) for ball milling was integrated with a variable Tromp curve for classification to simulate and optimize closed-circuit, multi-compartment cement ball milling. Using representative model–operational parameters from available literature, we first investigated the system dynamics for a two-compartment mill. Then, various simulations examined the impacts of closed-circuit vs. open-circuit operation, number of compartments, and various ball size distributions. Our results suggest that integrating an air classifier into an open-circuit ball mill can increase the production rate by 15% or increase the cement-specific surface area by 13%. A single-compartment mill entails a pre-milled feed for proper operation, whereas a two-compartment mill yields a finer cement product than a three-compartment mill. Uniform mass distribution of balls led to slightly finer product than uniform surface area or number distributions, while the impact of a classifying liner was negligibly small. Finally, we identified optimal ball mixtures in a two-compartment mill using a combined global optimizer–DAE solver, which suggests 14% capacity increase with desirable cement quality. Overall, TUSSIM’s results are not only in line with limited, full-scale experimental studies and industry best practices, but also provide fundamental process insights, while enabling process optimization with tailored ball mixtures in different compartments.     

A stochastic model for the optimal batch size in multi-step operations with process and product variability

Virtually all manufacturing processes are subject to variability, an inherent characteristic of most production processes. No two parts can ever be exactly the same in terms of their dimensions. For machining processes such as drilling, milling, and lathing, overall variability is caused in part by machine tools, tooling, fixtures and workpiece material. Since variability, which can be accumulated from tolerance stacking, can result in defective parts the number of parts produced in a batch is limited. When there are too many parts in a batch, the likelihood of producing all acceptable parts in a batch decreases due to the increased tolerances. On the other hand, too small a batch size incurs an increase of manufacturing costs due to frequent setups and tool replacements, whereas the likelihood of acceptable parts increases. To address this challenge, we present a stochastic model for determining the optimal batch size where we consider part-to-part variation in terms of tool wear, which tends to be proportional to batch size. In this paper, a mathematical model is constructed based on the assumption that the process used for producing preceding parts affects the state of subsequent parts in a probabilistic manner.     

X射线小角散射测定粒度分布的一种数据处理解析程序

本文提出了一种确定材料中微不均匀区大小分布 X 射线小角散射数据处理的解析方法。用 n 个分立的粒度分布解对应的散射强度叠加来逼近实验强度。以尝试法修正结果使误差达到最小。用铁红釉试样作为实例,对液相分离球滴粒度分布作了测定,结果表明该方法简便实用。     

A modified Wenzel model for hydrophobic behavior of nanostructured surfaces

A modified Wenzel model was proposed to explore the influence of pore size distributions (PSDs) on water repellency of nanostructured surfaces. Rough surfaces with different porous structures, including surface areas and PSDs, were fabricated by stacking different solid ratios of TiO₂ nanoparticles. These fluorinated surfaces exhibited an excellent hydrophobic performance with the highest value of contact angle ∼ 165°. The PSDs of these surfaces, determined from Dubinin-Stoeckli equation, were found to vary with the solid ratios. The modified Wenzel model incorporated with the PSDs gave a fairly good prediction in describing the variation of contact angle with surface roughness, which is very close to the experimental data. These results demonstrated that the heterogeneity of surfaces caused by different PSDs would induce the hydrophobic behavior.     

A sequential hypothesis testing procedure for the process capability index Cpk

In this study, we propose a sequential method for hypothesis testing on the C_pk process capability index. We compare the properties of the sequential test with the performances of nonsequential tests by performing an extensive simulation study. The results show that the proposed sequential test makes it possible to save a large amount of sample size as compared with the fixed sample size tests while maintaining the desired α‐level and power.     

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

Population balance systems are models for processes in nature and industry that lead to a coupled system of equations (Navier–Stokes equations, transport equations, etc.) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three‐dimensional (3D) equation of a precipitation process in a two‐dimensional flow domain. The numerical schemes for the 3D equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time‐ dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of the cabling process for Rutherford cables: An advanced finite element model

In all existing large particle accelerators (Tevatron, HERA, RHIC, LHC) the main superconducting magnets are based on Rutherford cables, which are characterized by having: strands fully transposed with respect to the magnetic field, a significant compaction that assures a large engineering critical current density and a geometry that allows efficient winding of the coils. The Nb₃Sn magnets developed in the framework of the HL-LHC project for improving the luminosity of the Large Hadron Collider (LHC) are also based on Rutherford cables. Due to the characteristics of Nb₃Sn wires, the cabling process has become a crucial step in the magnet manufacturing. During cabling the wires experience large plastic deformations that strongly modify the geometrical dimensions of the sub-elements constituting the superconducting strand. These deformations are particularly severe on the cable edges and can result in a significant reduction of the cable critical current as well as of the Residual Resistivity Ratio (RRR) of the stabilizing copper. In order to understand the main parameters that rule the cabling process and their impact on the cable performance, CERN has developed a 3D Finite Element (FE) model based on the LS-Dyna® software that simulates the whole cabling process. In the paper the model is presented together with a comparison between experimental and numerical results for a copper cable produced at CERN.     

A texture component model for anisotropic polycrystal plasticity

There are many crystallographic textures which can be approximated by a small number of texture components [see, e.g., Int. J. Mech. Sci. 31(7) (1989) 549]. In some cases, such texture components can be described by central distributions. Central distributions are characterized by a mean orientation and a half width. The classical Taylor model for viscoplastic polycrystals assumes that a discrete set of single crystals deforms homogeneously. If the viscoplastic version of the Taylor model is numerically implemented then the crystallite orientation distribution function (codf) is usually discretized by a set of Dirac distributions, where each of the Dirac distributions represents a single crystal. Due to the specific discretization of the codf this approach requires usually a large number of discrete crystal orientations even if the texture can be described by a small number of texture components. In the present work, we consider face-centered cubic (fcc) polycrystals and compare the classical upper bound model with an approach based on texture components. The texture components are modeled by Mises–Fischer distributions, which are central distributions. The stress of the polycrystal is obtained by a numerical integration of the single crystal stress state over the orientation space.     

A new approach for the testing method of coal grindability

In this study, a total of 26 hard coal sample (either clean or original) from Zonguldak Coal Basin (17 from Zonguldak, 9 from Amasra) and a total of 17 low grade Turkish coals from various locations in Turkey were collected. So, a total of 69 samples were analyzed initially with standard method of coal grindability, i.e. HGI values were determined. In addition, an alternative method (does not require a standard HGI mill) for grindability measurement was proposed in this study. The abovementioned alternative method includes a ring mill with specified conditions and Malvern Mastersizer. As regards to the procedure of the new proposed method for the determination of coal grindability, samples were prepared in the size range of −1.7 + 1.18 mm size group and they were ground in ring mill. Here, ring mill was chosen because of the fact that it is very widely used for sample preparation and it is very commonly available in every laboratory. Procedure proposed includes placing abovementioned specified samples in ring mill (See Material and Method) and having the samples ground in a previously determined time period. After this grinding process (with ring mill) they have rather different size distribution at the end depending on their nature of grindability. For better understanding, having the ground samples of (−1.7 + 1.18 size group for 20 s for 50 g samples) and analyzing their size distributions with Malvern Mastersizer, ground samples have different D₁₀, D₅₀, D₉₀, D₃₂ and D₄₃ at the end. Comparing these size parameter results with previously determined HGI values, it can be claimed that coal grindability can be easily determined with this method, since evaluated HGI values with the method proposed are ±0.05% different than the determined HGI values.     

A mathematical model of subpopulation kinetics for the deconvolution of leukaemia heterogeneity

Acute myeloid leukaemia is characterized by marked inter- and intra-patient heterogeneity, the identification of which is critical for the design of personalized treatments. Heterogeneity of leukaemic cells is determined by mutations which ultimately affect the cell cycle. We have developed and validated a biologically relevant, mathematical model of the cell cycle based on unique cell-cycle signatures, defined by duration of cell-cycle phases and cyclin profiles as determined by flow cytometry, for three leukaemia cell lines. The model was discretized for the different phases in their respective progress variables (cyclins and DNA), resulting in a set of time-dependent ordinary differential equations. Cell-cycle phase distribution and cyclin concentration profiles were validated against population chase experiments. Heterogeneity was simulated in culture by combining the three cell lines in a blinded experimental set-up. Based on individual kinetics, the model was capable of identifying and quantifying cellular heterogeneity. When supplying the initial conditions only, the model predicted future cell population dynamics and estimated the previous heterogeneous composition of cells. Identification of heterogeneous leukaemia clones at diagnosis and post-treatment using such a mathematical platform has the potential to predict multiple future outcomes in response to induction and consolidation chemotherapy as well as relapse kinetics.     

A Cox model for radioactive counting measure: Inference on the intensity process

A solution to the problem of calibrating a counting device from observed data, is developed in this paper by means of a Cox process model. The stochastic intensity of the process for counting emitted particles is estimated by functional principal components analysis and confidence bands are provided for two radioactive isotopes, ²²⁶Ra and ¹³⁷Cs. A hypothesis test to assess the coherence of the new observed data with the estimated model is also included.