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.     

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.     

Determining particle-size distributions from chord length measurements for different particle morphologies

A recently proposed model to determine particle-size distributions (PSDs) from chord length measurements has been applied to different particle morphologies, namely compact, platelet- and rod-shaped particles. To study these systems, chord length distributions (CLDs) were measured at varying particle size and solids concentration for each compound and were subsequently utilized to determine the system-specific parameters. Each model was successfully applied to its respective compound such that the experimental PSDs and model predictions were in good agreement. Moreover, the effect of other variables such as agitation rate and solvent composition was investigated and found to be negligible for the specific systems tested. Finally, potential model optimizations of the general model construct have been studied. Two variants of the CLD compression step, namely principal component analysis and a geometric model have been considered as surrogate models. However, neither of these approaches yielded superior results than the previously proposed approach.     

Effect of solvent type on preparation of ethyl cellulose microparticles by solvent evaporation method with double emulsion system using focused beam reflectance measurement

The purpose of this study was to investigate the effect of solvent type on the solidification rate of ethyl cellulose (EC) microparticles and particle size/distribution of emulsion droplets/hardened microparticles during the solvent evaporation process using focused beam reflectance measurement (FBRM). EC microparticles were prepared with a water‐in‐oil‐in‐water solvent evaporation method using various solvents, including dichloromethane, dichloromethane–methanol (1:1), ethyl acetate and chloroform. The particle size/distribution of the emulsion droplets/hardened microparticles was monitored using FBRM. The morphology of EC microparticles was characterized using scanning electron microscopy (SEM). The transformation of the emulsion droplets into solid microparticles for all solvents occurred within the first 10–90 min. The square weighted mean chord length of EC microparticles prepared using chloroform was smallest, but the chord count was not the highest. The chord length distribution (CLD) measured by FBRM showed that a larger mean particle size gave longer CLD and a lower peak of particle number. SEM data revealed that the morphology of microparticles was influenced by the type of solvent. FBRM can be employed for online monitoring of the shift in the microparticle CLD and detect transformation of emulsion droplets into solid microparticles during the solvent evaporation process. The microparticle CLD and transformation process were strongly influenced by solvent type. © 2017 Society of Chemical Industry     

DEM Study on the effect of particle-size distribution on jamming in a 3D conical hopper

Despite the advancement on the understanding on the unwelcome phenomenon of hopper jamming, the influence of the particle-size distribution (PSD) width of the ubiquitous continuous PSDs remains unknown. Accordingly, this study investigated using discrete element method a range of PSD widths (10%≤σ/μ≤50% ) of lognormal PSDs with a constant mean particle diameter of 10 mm in a three-dimensional hopper. Results indicate that, although the monodisperse particles of 10 mm diameters do not jam, the wider PSDs (σ/μ≥ 30%) jam. The flatter arches and higher mass-averaged force that are linked to the denser packing of wider PSDs underlie the propensity to jam. To reduce the jamming probability, the friction coefficient and initial fill height can be reduced, but not changes in particle density. The results here highlight that the mean particle diameter alone is an insufficient parameter to predict jamming and the impact of PSD width is non-negligible. © 2018 American Institute of Chemical Engineers AIChE J, 65: 512–519, 2019     

DEM study on the discharge characteristics of lognormal particle size distributions from a conical hopper

This study employs the discrete element method (DEM) to investigate the impact of the widths of lognormal particle size distributions (PSDs) with the same mean particle diameter on hopper discharge behaviors, namely, discharge rate, particle velocities, and size‐segregation. Results reveal that (i) the hopper discharge rate decreases as PSD width increases; (ii) the mean discharge rates are constant with time, but the fluctuations increase as the PSD width increases; (iii) the overall size‐segregation increases with PSD width; (iv) the overall mean particle diameters of the narrower PSDs do not exceed the initial mean of 5 mm, whereas that of wider ones do; (v) the relationship between PSD width and particle velocities is non‐monotonic with no consistent trends; and (vi) no direct correlation exists between particle velocity and size‐segregation. The results here provide valuable insights on the behavior of the prevalent polydisperse mixtures in hoppers. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1174–1190, 2018     

Measurement of size and shape distributions of particles through image analysis

The size and shape of particles can be described using a 2D particle size distribution (PSD) where two characteristic lengths define each particle in the population. The determination of 2D PSDs based on microscopic pictures of particles in suspension is studied. The experimental data are represented as an axis length distribution (ALD) that can be extracted from a series of microscopic pictures by a fully automated image analysis. The problem of finding the underlying bi-dimensional PSD is stated as an optimization problem. For the solution a genetic algorithm is used. The approach is tested on simulated ALDs, as well as on an experimentally measured ALD obtained from carbon fiber particles.     

Experimental study by online measurement of the precipitation of nickel hydroxide:Effects of operating conditions

The objective of this work is using the online measurement method to study the process of precipitation of nickel hydroxide in a single-feed semi-batch stirred reactor with an internal diameter of D=240 mm. The effects of impeller speed, impeller type, impeller diameter and feed location on the mean particle size d43 and particle size distribution (PSD) were investigated. d43 and PSD were measured online using a Malvern Insitec Liquid Pro-cess Sizer every 20 s. It was found that d43 varied between 13μm and 26μm under different operating conditions, and it decreased with increasing impel er diameter. When feeding at the off-bottom distance of D/2 under lower impeller speeds, d43 was significantly smaller than that at D/3. PSDs were slightly influenced by operating conditions.     

Modeling of chord length distributions

A procedure for the calculation of chord length distributions (CLD) of populations of rigid and opaque particles of any size and shape distribution is given. It combines the capabilities of a virtual reality renderer to create 2D projections of particles and of an image analysis software which determine their chord lengths. The procedure has been validated on simple shapes (spheres, ellipsoids, parallelepipeds, cuboids, uniform polyhedra) that can be combined to simulate agglomerates or twinned crystals. The procedure has been used to discuss the experimental results obtained on gibbsite particles in different size ranges and to compare the mean chord length to the average particle size.     

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     

Monitoring of cell growth using the focused beam reflectance method

In this paper the use of a focused beam reflectance measurement (FBRM) particle characterisation probe was investigated as a tool for monitoring the growth of the filamentous bacteria Streptomyces natalensis. The optimum operating parameters of the probe were identified, and it was found that the sample agitation rate had the largest effect on the mean value of a number of statistics of the chord length distribution (CLD) measured by the FBRM probe. A series of fractions of the particle size distribution was generated by sieving. It was found that the mean chord length measured by the probe responded to changes in the size distribution examined, increasing from 57.2 µm to 69.9 µm with an increase in the upper particle size from 180 µm to 300 µm. It was also found that the average total counts measured by the probe also increased with increasing sample concentration. This relationship was further investigated and it was found that in the normal range of biomass levels experienced in a fermentation (up to 4.0 g dm^?3) there was an apparently linear relationship between counts and concentration. However, as concentration was increased further the relationship became increasingly non‐linear. Copyright © 2004 Society of Chemical Industry     

逆体积排阻层析法测定层析介质的孔径分布

针对层析介质的孔径分析,基于刚性球状分子进入柱状孔的假设,分别采用高斯正态分布和对数正态分布描述孔径分布,利用简化的单孔分配因子模型,建立了孔径分布函数和分配因子Kd的关联,通过体积排阻层析实验测定系列标准物的Kd,从而拟合得到孔径分布信息,建立了逆体积排阻层析法。以葡聚糖作为分子大小的标准物,测定了5种典型层析介质(SP Sepharose FAST FLOW、Q Sepharose FAST FLOW、ToyopearlDEAE-650M、Streamline DEAE和Sephadex G-150)的Kd,计算和比较了不同介质的孔径分布,分析了介质的可吸附孔表面积等结构参数,证实了逆体积排阻层析法分析层析介质孔径分布的可行性和实用性。     

An investigation of the droplet size distributions in a cyclone reactor for liquid-liquid heterogeneous reactions using FBRM and PVM

A multi-element process coupled cyclone reactor for the liquid-liquid heterogeneous reaction was proposed. This paper presents the investigation of dispersed phase droplet size distributions near the wall surface using FBRM and PVM. To translate a chord length distribution (CLD) measured by the FBRM instrument into its droplet size distribution (DSD), a CLD-DSD transformation model was proposed. Moreover, the effects of operational parameters on the droplet size distributions were studied. The results show that the total inlet flow plays a decisive role on the dispersed phase droplet size distributions, while the influence of feed ratio is mainly reflected in inhibiting the probability of droplet coalescence. In comparison, the influence of overflow ratio on the droplet size distributions can be neglected.     

Batch emulsion polymerization of vinyl chloride: Application of experimental design to investigate the effects of operating variables on particle size and particle size distribution

In this article, the influences of operating variables on the particle size (PS) and particle size distribution (PSD) of emulsion poly(vinyl chloride) in batch reactor were investigated using Taguchi experimental design approach. The variables were temperature (T), water to monomer weight ratio (R), concentrations of initiator ([I]) and emulsifier ([E]), and agitation speed (S). Scanning electron microscope was used together with image analysis software to determine the PS and PSD. Statistical analysis of results revealed that the PS of emulsion poly(vinyl chloride) strongly depends on emulsifier and initiator concentrations, respectively, whereas the other factors have no significant effects in the range of levels investigated in this study. Except initiator concentration, all factors have important influence on the PSD (significance sequence: S > R > T > [E]). It is implied from the greater influence of agitation speed relative to temperature on PSD that the shear coagulation predominates the Brownian coagulation in this system. The relative optimum condition for a typical paste application was also determined using overall evaluation criteria. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Critical evaluation of coupling particle size distribution with the shrinking core model

The shrinking core model (SCM) is widely used to model fluid-solid reactions such as the leaching of metals from minerals. In most cases, however, the particle size distribution (PSD) of the solid material was disregarded. In this paper the erroneous shift in the control regime when neglecting PSD was quantified and the dependence of the shift on the coefficients of variation (CV) and the type of PSD was analysed. By coupling the SCM with a Gamma PSD, it was found that neglecting the PSD would shift the control regime from chemical reaction to inert/ash layer diffusion, when the CV was between 0.7 and 1.2. For a system controlled by liquid film diffusion, neglect of the PSD, would shift the control regime to chemical reaction when CV is between 0.3 and 0.7 or to inert/ash layer diffusion when CV is greater (0.9-1.5). It was therefore postulated that some researchers had unknowingly made invalid conclusions about the control regime due to the neglect of PSD. However, an inert/ash layer diffusion-controlled process was insensitive to the neglect of PSD. When CV<0.3, neglect of the PSD would not cause any erroneous shifts, irrespective of the control regime. Experimental data confirmed the observation. For a given CV, the deviation in the fraction reacted from the mono-PSD increases with CV and decreases with time. The maximum deviation, which occurs at the beginning, is about 10% with a gamma PSD of CV=0.3. The percent deviation is dependent of the type of PSDs. Gamma PSD gives the lowest deviation while Gaudin-Schuhmann results in the largest deviation (maxi. ∼19%, with CV=0.3) in the first half of dissolution process. Log-normal distribution gives a larger deviation than gamma but quickly approaches the latter with time. The deviation for Rosin-Rammler is between log-normal and Gaudin-Schuhmann. For systems with CV less than 0.3, the SCM can be fairly used without considering PSD. When CV is greater than 0.3, particularly in the early stage of a dissolution process with a PSD other than gamma, PSD should be included to avoid substantial errors.     

Link between bubbling and segregation patterns in gas‐fluidized beds with continuous size distributions

Experiments involving a bubbling, gas‐fluidized bed with Gaussian and lognormal particle‐size distributions (PSDs) of Geldart Group B particles have been carried out, with a focus on bubble measurements. Previous work in the same systems indicated the degree of axial species segregation varies non‐monotonically with respect to the width of lognormal distributions. Given the widely accepted view of bubbles as “mixing agents,” the initial expectation was that bubble characteristics would be similarly non‐monotonic. Surprisingly, results show that measured bubble parameters (frequency, velocity, and chord length) increase monotonically with increasing width for all PSDs investigated. Closer inspection reveals a bubble‐less bottom region for the segregated systems, despite the bed being fully fluidized. More specifically, results indicate that, the larger the bubble‐less layer is, the more segregated the system becomes. The direct comparison between bubbling and segregation patterns performed provides a more complete physical picture of the link between the two phenomena. © 2011 American Institute of Chemical Engineers AIChE J, 2011     

Population balance modeling of particle size distribution in monomer‐starved semibatch emulsion polymerization

The evolution of particle size distribution (PSD) in the monomer‐starved semibatch emulsion polymerization of styrene with a neat monomer feed is investigated using a population balance model. The system under study ranges from conventional batch emulsion to semicontinuous (micro)emulsion polymerization depending on the rate of monomer addition. It is shown that, contrary to what is often believed, the broadness of PSD is not necessarily associated with the length of nucleation period. The PSDs at the end of nucleation are found to be independent of surfactant concentration. Simulation results indicate that at the completion of nucleation the particle size is reduced and the PSD narrows with decreasing rate of monomer addition despite nucleation time increasing. The broad distribution of particles frequently encountered in semibatch emulsion polymerizations is therefore attributed to stochastic broadening during the growth stage. The zero‐one‐two‐three model developed in this article allows perceiving that the dominant kinetic mechanism may be different for particles with different sizes. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Control of fines suspension density in the fines loop of a continuous KCl crystallizer using transmittance measurement and an FBRM® probe

The control of crystal size distribution (CSD) is investigated in a 1.5 L laboratory cooling KCl crystallizer using fines dissolution rate as the manipulated variable. The controlled variable was either the fines suspension density in the fines withdrawal loop, measured by an innovative double‐sensor turbidity meter manufactured in‐house, or the chord length distribution (CLD) measured by the Focused Beam Reflectance Measurement (FBRM®) probe (model: Par‐Tec® 100, Lasentec®, Redmond, WA). It was shown that effective control of mean crystal size and fines suspension density in the presence of setpoint and disturbance changes is feasible with both control schemes. The double sensor turbidity sensor proved to be very rugged even in the presence of insoluble clay background particles. The FBRM® probe was more sensitive and capable of detecting particle breakage and flocculation.