Estimation of particle size distributions from focused beam reflectance measurements based on an optical model

A popular in situ particle characterization technique, which can be applied without dilution, is the focused beam reflectance measurement (FBRM^®). The FBRM probe measures a chord length distribution (CLD) which is different from a particle size distribution (PSD). In order to compare results obtained by an FBRM probe with other measurement technologies such as laser diffraction, it is necessary to reconstruct the PSD from a measured CLD. For this reconstruction a measurement model and an inversion procedure are required. Most FBRM models presented in the literature assume that an FBRM records a geometric chord which can be deduced from a two-dimensional projection of the particle silhouette. In previous work [Kail, N., Briesen, H., Marquardt, W., 2008. Analysis of FBRM measurements by means of a 3D optical model. Powder Technology 185 (3), 211-222] it has been demonstrated that FBRM data show significant deviations from this geometric model. Consequently, an estimation of a PSD using such a geometric FBRM model will fail. A novel FBRM model is developed in this work. This model imitates the chord discrimination algorithm used in a Lasentec D600L FBRM system and takes the intensity profile of the laser beam and the optical aperture of the probe into account. The model is ideally suited for the estimation of a PSD from a measured CLD using a sequential, linear inversion routine, as proposed in this work. The novel FBRM model and the inversion procedure are evaluated using small, mono-disperse polystyrene beads, large ion-exchanger beads, and α-lactose-monohydrate particles. The applicability of the FBRM for PSD measurements is discussed on the basis of these results.     

Population balance modelling for high concentration nanoparticle sizing with ultrasound spectroscopy

Ultrasound particle sizing is attracting an increasing attention from academic research and industrial applications as it offers non-invasive, suitable for highly turbid and concentrated nanoparticle suspensions and potentially no sample dilution needed features. The main challenge to this technique is thought to be its capability of dealing with high concentration. Most ultrasound particle sizing techniques employ ECAH (Epstein, Carhart, Allegra and Hawley) theory based models for the inversion of ultrasound spectra to particle size distribution (PSD). However, this theory is based on “single particle scattering”, namely a single particle immersed in an infinite medium, it is therefore only valid when ultrasound attenuation and particle concentration are linearly related. With the increase of particle concentration, due to the interactions between particles, the relation between attenuation and concentration may become nonlinear for solid-liquid suspensions. This paper demonstrates a method using population balance (PB) modelling to deal with the high concentration PSD problem for silica suspensions. It concludes that with a de-aggregation model, it is possible to convert attenuation inverted PSDs (ECAH model based inversion) at high concentrations into the PSD that is thought to be the correct PSD at a critical low concentration by a PB simulation.     

Numerical study on particle size distribution in the process of preparing ultrafine particles by reactive precipitation

Based on the population balance and mass balance in a reactive precipitation process, a numerical simulation model was developed to predict the particle size distribution (PSD) in the reactive precipitation process. The precipitation system of BaCl₂ with Na₂SO₄ to prepare BaSO₄ in aqueous solution was adopted to obtain ultrafine particles in a stirred precipitation reactor and the particle size distribution and the morphology of the particle were observed under transmission electron microscope. It was illustrated by the experimental observation of the micrographs of BaSO₄ particles obtained that apparent agglomeration occurred between the particles, which phenomenon must be taken into consideration in PSD modeling. The population balance equation was calculated by discretization method to obtain particle number and particle size distribution. By implementing the model, the reactive precipitation process in a batch reactor including reaction, nucleation, growth and agglomeration was simulated. The simulation results were validated by the experimental data of BaSO₄ precipitation. Further analysis was endeavored to explore the effects of some important factors such as the supersaturation degree and agglomeration on the evolution of the volume-based characteristic particle size and the variance of volume-based characteristic size of the particles. It was depicted that particle size and particle size distribution are controlled by the supersaturation degree and agglomeration between the particles. Stemming from the analysis in the context, the disciplinarian of the influences of these factors and the method for controlling particle size distribution were presented for the reactive precipitation process.     

Concentrated suspension-based additive manufacturing – viscosity,packing density,and segregation

This paper describes the influence of particle size distribution (PSD) of refractory silica on the suspension viscosity, packing density, and segregation in layers solidified by ceramic stereolithography (CerSLA). Using bimodal PSD displays most significant decrease of suspension viscosity than suspension made of mono-modal PSD. Given the Krieger-Dougherty model and packing density experiment, the lower viscosity results from the higher maximum volume fraction, φ_m, reached through the closely packed particles. Furthermore, from the differential sedimentation of coarser or denser particles in suspensions, particle size segregation in layers is detected. To determine the distribution of particle size within each layer, a linear intercept method is used, which demonstrates the vertical changes in PSD. Mono–modal PSD case show severe segregations in solidified layers in which the population of larger or denser particles is greater near the bottom. However, much less segregation occurs with bimodal PSD due to suppressed segregation.     

Determination of non-spherical particle size distribution from chord length measurements. Part 2: Experimental validation

In this paper, the theory on the translation of a measured chord length distribution (CLD) into its particle size distribution (PSD), which was developed in the first part of this study [Li and Wilkinson, 2005. Determination of non-spherical particle size distribution from chord length measurements. Part 1: theoretical analysis. Chemical Engineering Science 60, 3251-3265], has been validated using experimental results. CLDs were measured using the Lasentec focused beam reflectance measurement (FBRM) with three different materials, spherical ceramic beads and non-spherical plasma aluminium and zinc dust particles. Meanwhile, the particle shape and PSD of each material were also investigated by image analysis (IA). Comparison of the retrieved PSDs with the measured PSDs by IA shows that the PSD can be retrieved from a measured CLD successfully using the proposed iterative nonnegative least squares (NNLS) method based on the PSD-CLD model.     

丁二烯气相聚合产物分子量分布和粒径分布模型研究

对非均相催化的丁二烯气相聚合,基于聚合物多层模型,考虑催化剂颗粒间活性位初始浓度和粒径分布对聚合物分子量分布和粒径分布的影响,建立了聚合物分子量分布和粒径分布的数学模型。模拟了反应温度、催化剂颗粒间活性位初始浓度和粒径分布等因素的影响,结果表明。随着温度升高,聚合物颗粒平均粒径变小,粒径分布变窄,聚合物分子量变小,分子量分布变宽;催化剂颗粒间的活性组分负载越均匀,聚合物分子量越大,分子量分布和粒径分布越窄;随着催化剂平均粒径变大,聚合物分子量变小,分子量分布变宽,不存在催化剂颗粒粒径分布和聚合物颗粒粒径分布间的复制现象。模型模拟结果与实验结果吻合较好,可用于预测丁二烯气相聚合产物的分子量、分子量分布和粒径分布。     

Mechanism and Simulation of Removal Rate and Surface Roughness During Optical Polishing of Glasses

Glass optics with ultra‐low roughness surfaces (<2 Å rms) are strongly desired for high‐end optical applications (e.g., lasers, spectroscopy, etc.). The complex microscopic interactions that occur between slurry particles and the glass workpiece during optical polishing ultimately determine the removal rate and resulting surface roughness of the workpiece. In this study, a comprehensive set of 100 mm diameter glass samples (fused silica, phosphate, and borosilicate) were polished using various slurry particle size distributions (PSD), slurry concentrations, and pad treatments. The removal rate and surface roughness of the glasses were characterized using white light interferometry and atomic force microscopy. The material removal mechanism for a given slurry particle is proposed to occur via nano‐plastic deformation (plastic removal) or via chemical reaction (molecular removal) depending on the slurry particle load on the glass surface. Using an expanded Hertzian contact model, called the Ensemble Hertzian Multi‐gap (EHMG) model, a platform has been developed to understand the microscopic interface interactions and to predict trends of the removal rate and surface roughness for a variety of polishing parameters. The EHMG model is based on multiple Hertzian contacts of slurry particles at the workpiece–pad interface in which the pad deflection and the effective interface gap at each pad asperity height are determined. Using this, the interface contact area and each particle's penetration, load, and contact zone are determined which are used to calculate the material removal rate and simulate the surface roughness. Each of the key polishing variables investigated is shown to affect the material removal rate, whose changes are dominated by very different microscopic interactions. Slurry PSD impacts the load per particle distribution and the fraction of particles removing material by plastic removal. The slurry concentration impacts the areal number density of particles and fraction of load on particles versus pad. The pad topography impacts the fraction of pad area making contact with the workpiece. The glass composition predominantly impacts the depth of plastic removal. Also, the results show that the dominant factor controlling surface roughness is the slurry PSD followed by the glass material's removal function and the pad topography. The model compares well with the experimental data over a variety of polishing conditions for both removal rate and roughness and can be extended to provide insights and strategies to develop practical, economic processes for obtaining ultra‐low roughness surfaces while simultaneously maintaining high material removal rates.     

Radioactive particle tracking in a lab‐scale conical fluidized bed dryer containing pharmaceutical granule

Radioactive particle tracking (RPT) has been used to study the motion of the particulate phase in a bench‐scale conical fluidized bed containing dried pharmaceutical granule. RPT revealed that there is a distinct circulation pattern of the granule with particles moving upwards at high velocities near the centre of the bed and falling slowly near the walls. There was also a localized region near the centre of the bed where particles moved downward rapidly. The particle size distribution (PSD) of the granule had an appreciable impact on particle motion with a wide PSD leading to larger fluctuations in particle velocity as well as poorer granule mixing.     

Tomographic imaging of a conical fluidized bed of dry pharmaceutical granule

Hydrodynamics in a conical fluidized bed were studied using electrical capacitance tomography (ECT) for a bimodal and mono-disperse particle size distribution (PSD) of dry pharmaceutical granule. The bimodal PSD exhibited a continuous distribution with modes at 168 and 1288 μm and contained approximately 46% Geldart A, 32% Geldart B and 22% Geldart D particles by mass. The mono-disperse PSD had a mean particle size of 237 μm and contained approximately 71% Geldart A, 27% Geldart B, and 2% Geldart C particles by mass. The granule particle density was 830 kg/m³. Experiments were conducted at a static bed height of 0.16 m for gas superficial velocities ranging from 0.25 to 2.50 m/s for the mono-disperse PSD, and from 0.50 to 3.00 m/s for the bimodal PSD. These gas velocities covered both the bubbling and turbulent fluidization regimes. An ‘M’-shaped time-averaged radial voidage profile appeared upon transition from bubbling to turbulent fluidization. The ‘M’-shaped voidage profile was characterized by a dense region near the wall of the fluidized bed with decreasing solids concentration towards the centre. An increased solids concentration was observed in the middle of the bed. Frame-by-frame analysis of the images showed two predominant bubble types: spherical bubbles with particle penetration in the nose which created a core of particles that extended into, but not through, the bubble; and spherical bubbles. Penetrated bubbles, responsible for the ‘M’ profile, were a precursor to bubble splitting; which became increasingly prevalent in the turbulent regime.     

Local size distributions of particles deposited by inertial impaction on a cylindrical target in dust-laden streams

We exploit recent developments on impinging single particle capture laws and rational correlations for inertial impaction on a circular cylinder in high Reynolds number crossflow [Israel and Rosner (1983) Aerosol Sci. Technol. 2, 45–51; Wessel and Righi (1988) Aerosol Sci. Technol. 9, 26–60] to predict the local size distribution of particles deposited by impaction on a cylindrical target when the mainstream particle suspension is “log-normal”. Because of both the aerodynamics of selective impingement, and the nature of the sticking/rebound law, we show that the granular deposit particle size distribution (hereafter abbreviated (PSD)_w) is generally quite different from mainstream particle size distribution (PSD)_∞, by so much that (PSD)_w generally cannot be characterized accurately by single-mode log-normal distribution parameters. Apart from its relevance in correcting for systematic errors in aerosol sampling from high-speed streams, this local variation of the “granular deposit” PSD along with information on deposit morphology, must be known (in addition to the total mass accumulated per unit area) to predict, say, the loss in convective heat transfer rate associated with the growth of a fouling layer. Three distinct classes of single solid particle capture laws are considered: constant capture fraction (independent of impinging particle velocity and angle of incidence), “on-off” capture behavior expected for impaction on a clean, particle-free, smooth solid surface, and particle capture on a dry, sufficiently thick, granular deposit. Our (PSD)_w results are cast in terms of following accessible dimensionless parameters: sensitivity of capture fraction to particle incident velocity and angle, ratio of mainstream velocity to the critical (threshold) velocity for particle rebound (at, say, normal incidence), ratio of mean particle size in the mainstream to the critical size required for impaction on a cylindrical target in crossflow, spread of log-normal mainstream particle size distribution, and the characteristic “slip” Reynolds number for the critical size particle in the mainstream.     

超临界辅助雾化制备适于气溶胶给药的药物微粒

超临界流体辅助雾化法（SAA）是一种新型的以超临界流体为基础的固体微粒制备工艺,既能用于水溶性,也可用于脂溶性的溶质.该工艺能制备出适用于气溶胶给药要求的微粒,预期可在药物行业内得到应用.综述了SAA过程的形成和特色、工艺流程和操作条件以及相应的造粒结果.评述了6种药物微粒的粒度、粒度分布及其形貌.经SAA加工后的药物质量未见变化,目前该工艺已进入中试阶段,展示出较快的发展速度.比较了SAA和现有的主要以超临界流体为基础的微粒制造工艺.展望了加强应用基础研究的必要性,进一步分析了SAA的机理和过程实质.     

Model development for the description of silica particles dispersion in silicone polymer

In order to improve the mechanical characteristics of silicone polymer, silica particulate filler can be dispersed in the continuous phase. During the dispersion process, silicone is adsorbed on the particle surfaces, and at the same time, the particles are submitted to breakage and erosion phenomena, modifying therefore their particle size distribution (PSD). A numerical tool, that can simulate the evolution of the silica PSD in polydimethylsiloxane (PDMS) during dispersion, and allows an a priori knowledge of the evolution of the suspension properties, is developed. The comparison between experimental and calculated PSD as well as process viscosity values is presented.The advantage of such a modelization lies in the prediction of the final product properties, or in the design and optimization of a process for obtaining a desired product. This approach also shows the effect of the particle porosity on the product characteristics evolution during the dispersion process. The influence of parameters such as the filler quantity, the stirring speed or the number of silanols sites on the silica surface is also investigated.     

Predictive simulation of nanoparticle precipitation based on the population balance equation

Nanoparticle precipitation is an interesting process to generate particles with tailored properties. In this study we investigate the impact of various process steps such as solid formation, mixing and agglomeration on the resulting particle size distribution (PSD) as representative property using barium sulfate as exemplary material. Besides the experimental investigation, process simulations were carried out by solving the full 1D population balance equation coupled to a model describing the micromixing kinetics based on a finite-element Galerkin h-p-method. This combination of population balance and micromixing model was applied successfully to predict the influence of mixing on mean sizes (good quantitative agreement between experimental data and simulation results are obtained) and gain insights into nanoparticle precipitation: The interfacial energy was identified to be a critical parameter in predicting the particle size, poor mixing results in larger particles and the impact of agglomeration was found to increase with supersaturation due to larger particle numbers. Shear-induced agglomeration was found to be controllable through the residence time in turbulent regions and the intensity of turbulence, necessary for intense mixing but undesired due to agglomeration. By this approach, however, the distribution width is underestimated which is attributed to the large spectrum of mixing histories of fluid elements on their way through the mixer. Therefore, an improved computational fluid dynamics-based approach using direct numerical simulation with a Lagrangian particle tracking strategy is applied in combination with the coupled population balance-micromixing approach. We found that the full DNS-approach, coupled to the population balance and micromixing model is capable of predicting not only the mean sizes but the full PSD in nanoparticle precipitation.     

气相流化床聚乙烯颗粒粒径分布的定制

基于气相法聚乙烯流化床反应器颗粒粒径分布的预测[1],提出了颗粒粒径分布定制模型.通过模型的优化计算,可得到催化剂粒径及其分布、操作气速、反应温度、乙烯浓度和丁烯浓度等生产操作参数,由此进行生产可获得具有良好流态化特性的聚乙烯颗粒粒径分布,能为生产具有特定粒径分布的聚乙烯颗粒提供理论指导.模型由工业装置的生产数据分析了计算结果的合理性.最后,以三种粒径分布的聚乙烯颗粒为例讨论了模型的可行性.同时,运用粒子群优化算法求解模型的非线性规划问题,算法具有调整参数少、收敛速度快和全局优化等优点.     

Modeling and simulation of polypropylene particle size distribution in industrial horizontal stirred bed reactors

This work aims at developing a steady-state particle size distribution (PSD) model for predicting the size distribution of polypropylene particles in the outflow streams of propylene gas-phase horizontal stirred bed reactors (HSBR), on the one hand and investigating the effect of the catalyst residence time distribution (RTD) on the polymer PSD, on the other hand. The polymer multilayer model (PMLM) is used to describe the growth of a single particle. Knowing the PSD and RTD of a Ziegler–Natta type of catalyst and polymerization kinetics, this model allows calculating the polymer PSD of propylene polymerization in the HSBRs. The calculated polypropylene PSDs agree well with those obtained from the industrial reactors. The results reveal that both the PSD and the RTD of the catalyst affect the polymer PSD but in different manners. The effect of RTD on the PSD is less significant in the case of a nonuniform size catalyst feed. This model also allows investigating the effects of other process parameters on the polymer PSD under steady-state conditions, including intraparticle mass- and heat-transfer limitations, initial catalyst size, and polymer crystallinity. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Determination of non-spherical particle size distribution from chord length measurements. Part 1: Theoretical analysis

In this paper, extensive theoretical studies are described on two important issues in translating a chord length distribution (CLD) measured by FBRM instrument into its particle size distribution (PSD) including PSD-CLD and CLD-PSD translation models for general non-spherical particles. Analytical solutions to calculate the PSD-CLD models for spherical and ellipsoidal particles are developed. For non-spherical particles, a numerical method is given to calculate the PSD-CLD model. The iterative non-negative least squares (NNLS) method is proposed in the CLD-PSD model, because of its many advantages converting measured CLD into its PSD, such as insensitivity to measurement noise and particle shape. The effectiveness of the proposed methods is validated by extensive simulations.     

Model of fragmentation of limestone particles during thermal shock and calcination in fluidised beds

Fragmentation of limestone due to thermal shock and calcination in a fluidised bed was studied through experiments and modelling. The time for heating was estimated by model calculations and the time for calcination by measurements. Fragmentation due to thermal shock was carried out by experiments in a CO₂ atmosphere in order to prevent the effect of calcination. It was found to be much less than fragmentation due to calcination. Average particle sizes before and after fragmentation are presented for several types of limestone. The effects of particle size and gas composition on the primary fragmentation were studied through experiments. Increasing the fluidisation velocity increased the tendency to fragment. The evolution of the particle size distribution (PSD) of limestone particles due to thermal shock and during calcination (or simultaneous calcination and sulphation) were calculated using a population balance model. Fragmentation due to thermal shock is treated as an instantaneous process. The fragmentation frequency during calcination is presented as exponentially decaying over time. In addition to the final PSD, this model also predicts the PSD during the calcination process. The fragmentation was practically found to end after 10 min. Furthermore, a population balance method to calculate the particle size distribution and amount of limestone in fluidised beds in dynamic and steady state, when feeding history is known, is presented.     

On-line investigation of anatase precipitation from titanyl sulphate solution

Precipitation of concentrated titanyl sulphate is an important step in the Sulphate Process used for manufacturing titanium white in which titanium dioxide particles are formed and the particles size as well as its distribution is determined. This study investigated the transient changes of TiO₂ precipitation process by using two on-line particle measurement techniques, turbidimeter and focused beam reflectance measurement (FBRM), in order to explore the mechanism of particles formation. Induction period as well as primary nuclei formation of the precipitation process was calculated by turbidity value changes and the influence of feeding rate on primary nucleation, primary crystal sizes and final product particle size distribution (PSD) were also investigated. FBRM was employed for detecting the transient variations of PSD during the process. It was found that the process was divided into four stages: induction, rapid hydrolysis and aggregation, aggregation balance with further reaction, and ripening. The Sigmoidal–Boltzmann model is used to describe these transient phenomena and describes well the formation of TiO₂ particles.     

高温烟气颗粒物在线检测装置性能评价

为评价高温工况下颗粒物在线检测装置的性能,在催化裂化高温烟气过滤实验装置上使用光学在线颗粒物检测装置测定了过滤器下游烟气中的催化剂浓度及粒径分布,同时用离线式粉尘等动采样装置和Coulter粒径分析仪测量过滤器上游与下游的催化剂浓度和粒径分布,对在线式的测量结果进行验证。考虑了温度对在线式测量结果的影响。实验结果表明:在线式检测装置可以在操作温度320~465℃,操作压力0.21 MPa工况下实现稳定的测量,在线式测量结果和离线式测量结果吻合很好,实验和计算模拟结果表明操作温度对实验测量的影响可以忽略。在此基础上,针对煤化工行业的实际工况,提出了利用迭代方式来获得在线测量结果修正值的方法,该方法对煤化工工艺中高温气体管道内颗粒物的在线测定同样具有指导意义。