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.     

Evaluation and comparison of a moist granulation technique to conventional methods

In the moist granulation technique (MGT), a minimum amount of liquid is used to activate a binder in a planetary mixer. Then, any excess moisture is absorbed by the addition of a moisture-absorbing substance. In the experiments described below, acetaminophen (APAP) was the model drug; polyvinylpyrrolidone (PVP) and microcrystalline cellulose (MCC) served as the binder and moisture-absorbing material, respectively. Water was used as the granulating fluid. Comparison of the MGT with direct compression (DC) and wet granulation (WG) methods was accomplished by sieve analysis (particle size) and density measurements. Moist granulation yielded an increase in particle size compared to direct compression; these results are comparable to those from the traditional wet granulation after drying and screening. Based only on the particle size, moist granulation appears comparable to conventional wet granulation for this formula. The moist granulation technique appears to have potential for the development of controlled-release formulations.     

A population balance model for the chocolate roller refining process

A study was performed to analyze the evolution of particle size distribution in the chocolate roller refining process. A mathematical model based on the continuity and population balance equations was developed to interpret industrial data. The industrial data were provided for chocolate pastes with powder/crystalline sugar particles at low/high shear rates. The parameters of the breakage frequency and fragment distribution function were estimated using industrial data. After parameter fitting, the model shows good agreement with the experimental results for varying conditions with a single consistently chosen set of parameters. This provides confidence that the general model structure is suitable for process evaluation. The population balance model was used to analyze the influence of changing some process variables on the efficiency of grinding. The results show that there might exist optimum values of the roll’s diameter and rotational velocity for a specific configuration of the equipment.     

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: $k_j=A^1{x}_j^{\alpha }$.     

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.     

Closed-form and finite difference solutions to a population balance model of grinding mills

The wear of steel balls in continuously operated grinding mills, used in mineral processing to comminute metalliferous rocks, can be described by a simple population-balance model. This model gives rise to a scalar transport equation with a singular source term for the number density of balls as a function of size and time. Exact solutions to this equation are determined under the assumption of a simple power-law type wear law. It is shown that a particular term proposed in the engineering literature that describes the removal of used balls from the mill leads to negative solutions (Model 1). An alternative, more realistic term for the sieve action, which admits nonnegative solutions only, is introduced (Model 2). A working first-order finite-difference scheme for Model 2 and a second-order TVD variant are introduced and applied for numerical simulations along with an error study. A weak solution concept for Model 2 is proposed, uniqueness of weak solutions is shown, and convergence of the first-order scheme to a weak solution is established. These results hold for a general class of wear laws, not just power-law type.     

Integrated Population Balance Model Development and Validation of a Granulation Process

This study is concerned with the development of an integrated three-dimensional population balance model (PBM) that describes the combined effect of key granulation mechanisms that occur during the course of a granulation process. Results demonstrate the importance of simulating the different mechanisms within a population balance model framework to elucidate realistic granulation dynamics. The incorporation of liquid addition in the model also aids in demarcating the dynamics in different regimes such as premixing, granulation (during liquid addition) and wet massing (after liquid addition). For the first time, the effect of primary particle size distributions and mode of binder addition on key granule properties was studied using an integrated PBM. Experimental data confirms the validity of the overall model as compared to traditional models in the literature that do not integrate the different granulation mechanisms.     

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 study of wet granule breakage in a breakage-only high-shear mixer

In high-shear wet granulation, breakage and attrition play an important role but are the least understood among all rate processes. To understand these phenomena, experiments were performed in a breakage-only granulator using pre-made granules. Non-granulating wet sand was used as the shearing medium in the granulator. A size analysis of the granules was done before and after the experiment for each trial in order to get a rough estimate of the size range of the granules after the breakage experiment. The number of granules broken and the granules mass change (before and after the experiment) were recorded after each experiment, in order to measure the extent of granule breakage in a granulator. Experimental results showed that formulation properties and operating parameters significantly influence the breakage behaviour. Population balance simulations were performed for breakage-only granulation and the simulation results were found to be in reasonable agreement with the experimental results.     

An original approximate method for estimating the invariant probability distribution of a large class of multi-dimensional nonlinear stochastic oscillators

This paper proposes an original method for obtaining analytical approximations of the invariant probability density function of multi-dimensional Hamiltonian dissipative dynamic systems under Gaussian white noise excitations, with linear non-conservative parts and nonlinear conservative parts. The method is based on an exact result and a heuristic argument. Its pertinence is attested by numerical tests.     

涡轴发动机细长柔性转子动力特性及高速动平衡技术研究

摘要：建立了涡轴发动机动力涡轮转子动力特性有限元分析模型,计算得到了转子的前三阶临界转速值,在高速旋转试验器上完成了全转速范围内的动力特性和高速动平衡试验,实测得到了转子在额定工作转速范围内高速动平衡前和高速动平衡后等状态下振动幅值随转速的变化曲线,研究结果表明：建立的计算模型反映了转子的真实情况,高速动平衡效果良好,解决了型号研制过程中的一项关键技术。     

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.     

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.     

An aggregation method of Markov graphs for the reliability analysis of hybrid systems

To meet always increasing safety requirements in car industry, design and safety assessment methods are developed in order to fit the complexity of new embedded mecatronic systems. Hybrid (discrete/continuous) and dynamic features, specific to these systems, require choosing a suitable formalism. These features should also be considered in safety studies made all through the system design. The aim of this paper is to propose a quantitative analysis method based on the construction of an aggregated Markov graph, which allows a limitation of the combinatorial expansion. This graph is directly deducted from the Petri net modelling of the system. It is composed by a set of functional modes and a set of transitions to which statistical information regarding the system dynamics has been added.     

Influence of process variable and physicochemical properties on the granulation mechanism of mannitol in a fluid bed top spray granulator

This study investigated the influence of specific process variables, including the hydroxypropyl cellulose (HPC) binder solution atomization, on the fluidized bed top spray granulation of mannitol. Special attention was given to the relationship between wetting and the granule growth profile. The atomization of the HPC binder solution using a binary nozzle arrangement produced droplets of decreasing size as the atomization pressure was increased, while changes in the spray rate had little effect on the mean droplet size. Increasing the HPC binder concentration from 2 to 8% w/w increased the binder droplet size and was most likely attributed to higher solution viscosity. The top spray granulation of mannitol showed induction type growth behavior. Process conditions like high spray rate, low fluidizing air velocity and binder solution concentration that promote the availability of HPC binder solution at the surface of the particles appeared to be key in enhancing nucleation and growth of the granules. Increasing the bed moisture level, up to a certain value, reduced the contribution of attrition to the overall growth profile of the granule and, more significantly, produced less granule breakage on drying. It was observed that the mean granule size could be reduced as much as 40% between the end of granulation and the end of drying for lower initial bed moisture level despite a shorter drying phase. High atomization pressure, especially when maintained during the drying phase, contributed substantially to granule breakage.     

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.     

Development of a method for determining the maximum van der Waals force to analyze dispersion and aggregation of particles in a suspension

A new method for determining the maximum value of the van der Waals force, F_v,max, has been developed to analyze the dispersion and aggregation behavior of primary particles in a suspension. The simulation method considered the van der Waals force, the electrical double-layer force and the lubrication force as remote inter-primary-particle interaction forces. Distinct Element Method (DEM) was applied to tracking the motion of primary particles, and DEM was coupled with Computational Fluid Dynamics (CFD) by DEM-CFD coupling model to represent the motion in liquid. Aggregate size distributions (ASDs) were measured and simulated at different pHs and shear rates in order to determine the F_v,max and to validate the proposed method. The simulated ASDs qualitatively agreed with the experimental values in general. The result, therefore, indicates that the dispersion and aggregation behavior of primary particles could be analyzed by using the F_v,max determined by the proposed method.     

General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-harmonic analysis: Analytical and experimental approach

The effect of crack depth of a rotor-bearing-disk system on vibration amplitudes and whirl orbit shapes is investigated through a general harmonic balance technique and experimental verification. Two models of the crack, which are the breathing and the open crack models, are considered. Finite element models and general harmonic balance solutions are derived for breathing and open cracks which are valid for damped and undamped rotor systems. It is found via waterfall plots of the system with a breathing crack that there are large vibration amplitudes at critical values of crack depth and rotor speed for a slight unbalance in the system. The high vibration amplitudes at the backward whirl appear at earlier crack depths than those of the forward whirl for both crack models. Resonance peaks at the second, third and fourth subcritical speeds emerge as the crack depth increases. It is shown that the unique signature of orbits for the breathing crack model which have been verified experimentally can be used as an indication of a breathing crack in the shaft. In addition, the veering in the critical frequencies has been noticed in the open crack case.     

Using age and usage for prediction of reliability of an arbitrary system from a finite population

For single‐use non‐repairable systems, reliability is commonly estimated as a function of age and usage. For the effective management of individual systems or populations of systems, it is frequently important and necessary to predict the reliability in the future for age and usage values not yet observed. When predicting future system reliability, the age of the future system is easily predicted whereas future usage values will typically be unknown. In this paper we present the methodology for how to estimate both individual and population reliability summaries based on the currently known age and usage values. Projected usage values for future points in time can be obtained based on observed usage patterns or user‐specified patterns of usage rates. Individual system summaries can be used to answer the questions ‘For a given system of age A and usage U, what is its reliability with associated uncertainty?’ or ‘For a given system with known current age A and usage U, but unknown usage in the future, what is its reliability with associated uncertainty?’ The population summary of interest predicts the probability that a system randomly selected from the population of systems works. This summary takes into consideration the estimation of future usage, the estimated probability of individual systems working at their given ages and usage values, and the life cycle demographics of the population of interest. In this paper we discuss these questions for a given application. Published in 2010 by John Wiley & Sons, Ltd.     

An aggregation method of Markov graphs for the reliability analysis of hybrid systems

To meet always increasing safety requirements in car industry, design and safety assessment methods are developed in order to fit the complexity of new embedded mecatronic systems. Hybrid (discrete/continuous) and dynamic features, specific to these systems, require choosing a suitable formalism. These features should also be considered in safety studies made all through the system design. The aim of this paper is to propose a quantitative analysis method based on the construction of an aggregated Markov graph, which allows a limitation of the combinatorial expansion. This graph is directly deducted from the Petri net modelling of the system. It is composed by a set of functional modes and a set of transitions to which statistical information regarding the system dynamics has been added.