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Multidimensional Particle Properties: Characterization, Description, Separation
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Multidimensional Particle Properties: Characterization, Description, Separation

A topical collection in Powders (ISSN 2674-0516).

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Editors

Prof. Dr. Urs Alexander Peuker

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Collection Editor
Institute of Mechanical Process Engineering and Mineral Processing, TU Bergakademie Freiberg, 09599 Freiberg, Germany
Interests: mechanical separation processes; solid liquid separation; recycling and mineral processing; particle-particle interactions; particle characterization
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alfred P. Weber

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Collection Editor
Institute of Particle Technology, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany
Interests: particle classification; surface modifications; mechanical properties; charging behavior; cold and hot plasmas
Prof. Dr. Einar Kruis

E-Mail Website
Collection Editor
Institute for Technology of Nanostructures, Universität Duisburg-Essen, 47057 Duisburg, Germany
Interests: nanoparticles; aerosols; nanomaterials and thin films; aerosol reactors; population balances; size classification; aerosol instrumentation; deep learning
Prof. Dr. Doris Segets

E-Mail Website
Collection Editor
Chair for Particle Science and Technology, University of Duisburg-Essen (UDE), 47057 Duisburg, Germany
Interests: functional electrochemical energy materials; nanoparticle processing; colloids; interface-dominated systems

Topical Collection Information

Dear Colleagues,

In 2016 the Senate of the German Science Foundation implemented the priority program 2045 “MehrDimPart - highly specific multidimensional fractionation of fine particles with technical relevance”. The joint research within the priority program aims at the research on and development of technological approaches which allow the separation/fractionation of fine particles below 10 μm. This separation has to meet more than one separation criteria (e.g., size and composition or size and shape), which ultimately allows the production of highly specific particle systems.

Highly specific particle systems form the basis for high-quality industrial products. Intermediate particle systems with defined multi-dimensional specifications are essential and quality-determining for the production of ceramic and powder metallurgical components, coatings, printed products, porous functional structures, particle-reinforced polymers, electrodes in electrochemical energy storage systems or electronic assemblies, for example, in the field of printed electronics.

Downstream of particle production via synthesis or comminution, the particle systems have to undergo further process steps to reach the specified quality (i.e., size distribution and distribution of further properties). These steps include fractionation, that is, the selective separation of the particle system according to particle characteristics. Comparable processes are also used in mineral processing technology to extract particles containing valuable substances from natural or secondary raw materials.

Since the characteristic lengths of both the technical structures and the primary and secondary raw materials are continuously decreasing, fine, highly specific particle systems will have to be processed or produced in the future. The technology required for this is reaching its limits since in this size range the physical separation principles used lose their effectiveness and selectivity.

This Topical Collection addresses the characterization and separation/fractionation of particle systems with multi-dimensional properties. The separation/fractionation is therefore following more than one particle property, and thus is multi-dimensional. Different technological approaches will be presented, as will fundamental methods for process control and multidimensional particle characterization. The processes, technologies, and approaches are grouped into three classes: investigations on fundamental mechanisms for separation, applications of new process concepts and applications of established separation/fractionation concepts adopted to the size scale.

Prof. Dr. Urs Alexander Peuker
Prof. Dr. Alfred P. Weber
Prof. Dr. Einar Kruis
Prof. Dr. Doris Segets
Collection Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multidimensional particle properties
  • separation features
  • multidimensional separation function
  • multidimensional tromp curve
  • selective agglomeration
  • flow field fractionation
  • electrostatic separation
  • forced triboelectric charging
  • separation at interfaces
  • particle characterization
  • aspect ratio/intergrowth
  • flotation
  • electrophoresis
  • multivariate transfer function

Published Papers (13 papers)

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2024

Jump to: 2023

20 pages, 11056 KiB  
Project Report
Highly Permeable, Electrically Switchable Filter for Multidimensional Sorting of Suspended Particles
by Laura Weirauch, Jasper Giesler, Georg R. Pesch, Michael Baune and Jorg Thöming
Powders 2024, 3(4), 574-593; https://doi.org/10.3390/powders3040030 - 25 Nov 2024
Viewed by 509
Abstract
The creation of highly specific particle systems in the nano- and micrometer size range is a challenging task. The demand for particle systems with narrowly distributed properties is increasing in many applications, especially for use in high-tech products. Conventional separation techniques often reach [...] Read more.
The creation of highly specific particle systems in the nano- and micrometer size range is a challenging task. The demand for particle systems with narrowly distributed properties is increasing in many applications, especially for use in high-tech products. Conventional separation techniques often reach their limits in the micrometer size range or become (labor-)intensive, which makes them economically or ecologically unsustainable. In addition, sorting based on several properties is rarely feasible in just one separator. Dielectrophoretic processes can be a viable option for complex sorting tasks like this, given their ability to address several particle properties and their high degree of selectivity. In this paper, we summarize the progress of a project in which the capability of dielectrophoretic methods for multidimensional sorting of microparticles was investigated. We were able to develop an operation mode for multidimensional sorting of microparticles using dielectrophoresis as well as a scalable electrically switchable filter. This creates a basis for high-throughput and multi-target sorting of technical microparticles using dielectrophoretic processes. Full article
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24 pages, 13880 KiB  
Article
Size Classification and Material Sorting of Fine Powders with a Deflector Wheel Air Classifier and an Electrostatic Separator
by Mohamed Abohelwa, Annett Wollmann, Bernd Benker, Alexander Plack, Mehran Javadi and Alfred P. Weber
Powders 2024, 3(4), 550-573; https://doi.org/10.3390/powders3040029 - 12 Nov 2024
Viewed by 484
Abstract
In this study, a two-dimensional separation of microparticles based on their settling velocity and triboelectric charge ability is achieved using an air classifier for size fractionation and simultaneous charging, followed by an electrostatic separator. In the first part, considerations for enhancing particle classification [...] Read more.
In this study, a two-dimensional separation of microparticles based on their settling velocity and triboelectric charge ability is achieved using an air classifier for size fractionation and simultaneous charging, followed by an electrostatic separator. In the first part, considerations for enhancing particle classification with high sharpness and low-pressure drops are discussed through improvements in blade design investigated with CFD simulations and validated experimentally. Blades with extended lengths towards the center of the classifier prevent the formation of high-velocity vortices, thereby minimizing the back-mixing of particles and enhancing separation sharpness. This approach also reduces pressure drops associated with these flow vortices. In the second part of the study, the modified blades within the classifier are utilized for two-dimensional separation. Powders from two different materials are fed into the classification system, where particles become triboelectrically charged, mainly through collisions with the walls of the classification system components. Coarse particles are rejected at the wheel and exit the classifier, while differently charged fine particles of the two materials are directed into an electrostatic separator for material sorting. An enrichment of approximately 25–35% for both materials has been achieved on the electrodes of the separator. Full article
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19 pages, 9499 KiB  
Article
Correlative Multi-Scale Characterization of Nanoparticles Using Transmission Electron Microscopy
by Stefan Neumann and David Rafaja
Powders 2024, 3(4), 531-549; https://doi.org/10.3390/powders3040028 - 31 Oct 2024
Viewed by 599
Abstract
Chemical and physical properties of nanoparticles (NPs) are strongly influenced not only by the crystal structure of the respective material, including crystal structure defects but also by the NP size and shape. Contemporary transmission electron microscopy (TEM) can describe all these NP characteristics, [...] Read more.
Chemical and physical properties of nanoparticles (NPs) are strongly influenced not only by the crystal structure of the respective material, including crystal structure defects but also by the NP size and shape. Contemporary transmission electron microscopy (TEM) can describe all these NP characteristics, however typically with a different statistical relevance. While the size and shape of NPs are frequently determined on a large ensemble of NPs and thus with good statistics, the characteristics on the atomic scale are usually quantified for a small number of individual NPs and thus with low statistical relevance. In this contribution, we present a TEM-based characterization technique, which can determine relevant characteristics of NPs in a scale-bridging way—from the crystal structure and crystal structure defects up to the NP size and morphology—with sufficient statistical relevance. This technique is based on a correlative multi-scale TEM approach that combines information on atomic scale obtained from the high-resolution imaging with the results of the low-resolution imaging assisted by a semi-automatic segmentation routine. The capability of the technique is illustrated in several examples, including Au NPs with different shapes, Au nanorods with different facet configurations, and multi-core iron oxide nanoparticles with a hierarchical structure. Full article
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31 pages, 6237 KiB  
Review
A Review of the Homogenized Lattice Boltzmann Method for Particulate Flow Simulations: From Fundamentals to Applications
by Jan E. Marquardt and Mathias J. Krause
Powders 2024, 3(4), 500-530; https://doi.org/10.3390/powders3040027 - 16 Oct 2024
Viewed by 701
Abstract
The homogenized lattice Boltzmann method (HLBM) has emerged as a flexible computational framework for studying particulate flows, providing a monolithic approach to modeling pure fluid flows and flows through porous media, including moving solid and porous particles, within a unified framework. This paper [...] Read more.
The homogenized lattice Boltzmann method (HLBM) has emerged as a flexible computational framework for studying particulate flows, providing a monolithic approach to modeling pure fluid flows and flows through porous media, including moving solid and porous particles, within a unified framework. This paper presents a thorough review of HLBM, elucidating its underlying principles and highlighting its diverse applications to particle-laden flows in various fields as reported in literature. These include studies leading to new fundamental knowledge on the settling of single arbitrarily shaped particles as well as application-oriented research on wall-flow filters, hindered settling, and evaluation of the damage potential during particle transport. Among the strengths of HLBM are its monolithic approach, which allows seamless simulation of different fluid-solid interactions, and its ability to handle arbitrary particle shapes, including irregular and concave geometries, while resolving surface interactions to capture local forces. In addition, its parallel scheme based on the lattice Boltzmann method (LBM) results in high computational efficiency, making it suitable for large-scale simulations, even though LBM requires small time steps. Important future development needs are identified, including the addition of a lubrication force correction model, performance enhancements, such as support for hybrid parallelization and GPU, and the extension of compatible contact models to accommodate concave shapes. These advances promise expanded capabilities for HLBM and broader applicability for solving complex real-world problems. Full article
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22 pages, 7318 KiB  
Article
Multidimensional Characterization and Separation of Ultrafine Particles: Insights and Advances by Means of Froth Flotation
by Johanna Sygusch and Martin Rudolph
Powders 2024, 3(3), 460-481; https://doi.org/10.3390/powders3030025 - 15 Sep 2024
Viewed by 975
Abstract
Particle systems and their efficient and precise separation are becoming increasingly complex. Therefore, instead of focusing on a single separation feature, a multidimensional approach is needed where more than one particle property is considered. This, however, requires the precise characterization of the particle [...] Read more.
Particle systems and their efficient and precise separation are becoming increasingly complex. Therefore, instead of focusing on a single separation feature, a multidimensional approach is needed where more than one particle property is considered. This, however, requires the precise characterization of the particle system, which is especially challenging for fine particles with sizes below 10 µm. This paper discusses the benefits and limitations of different characterization techniques, including optical contour analysis, inverse gas chromatography, flow cytometry, and SEM-based image analysis. The separation of ultrafine particles was investigated for a binary system using froth flotation, where a novel developed flotation apparatus is used. A special focus was placed on the multidimensional evaluation of the separation according to the particle properties of size, shape, and wettability, which was addressed via multivariate Tromp and entropy functions. The results emphasize the intricacy of the flotation process and the complex interaction of the individual particle properties and process parameters. The investigations contribute to the understanding of the characterization of particulate properties as well as the separation behavior of ultrafine particles via froth flotation, especially in the case of a multidimensional approach. Full article
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24 pages, 5058 KiB  
Article
Fractionation of Aerosols by Particle Size and Material Composition Using a Classifying Aerodynamic Lens
by Matthias Masuhr and Frank Einar Kruis
Powders 2024, 3(3), 392-415; https://doi.org/10.3390/powders3030022 - 22 Jul 2024
Cited by 1 | Viewed by 683
Abstract
The fractionation of airborne particles based on multiple characteristics is becoming increasingly significant in various industrial and research sectors, including mining and recycling. Recent developments aim to characterize and fractionate particles based on multiple properties simultaneously. This study investigates the fractionation of a [...] Read more.
The fractionation of airborne particles based on multiple characteristics is becoming increasingly significant in various industrial and research sectors, including mining and recycling. Recent developments aim to characterize and fractionate particles based on multiple properties simultaneously. This study investigates the fractionation of a technical aerosol composed of a mixture of micron-sized copper and silicon particles by size and material composition using a classifying aerodynamic lens (CAL) setup. Particle size distribution and material composition are analyzed using scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDX) for samples collected from the feed stream (upstream of CAL) and product stream (downstream of CAL) at varying operational pressures. The experimental findings generally agree with the predictions of an analytical fractionation model but also point to the importance of particle shape as a third fractionation property. Moreover, the results suggest that material-based fractionation is efficient at low operational pressures, even when the aerodynamic properties of the particle species are similar. This finding could have significant implications for industries where precise particle fractionation is crucial. Full article
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29 pages, 22681 KiB  
Article
Application of Multivariate Tromp Functions for Evaluating the Joint Impact of Particle Size, Shape and Wettability on the Separation of Ultrafine Particles via Flotation
by Johanna Sygusch, Thomas Wilhelm, Orkun Furat, Kai Bachmann, Volker Schmidt and Martin Rudolph
Powders 2024, 3(3), 338-366; https://doi.org/10.3390/powders3030020 - 2 Jul 2024
Cited by 1 | Viewed by 628
Abstract
Froth flotation predominantly separates particles according to their differences in wettability. However, other particle properties such as size, shape or density significantly influence the separation outcome as well. Froth flotation is most efficient for particles within a size range of about 20–200 μm, [...] Read more.
Froth flotation predominantly separates particles according to their differences in wettability. However, other particle properties such as size, shape or density significantly influence the separation outcome as well. Froth flotation is most efficient for particles within a size range of about 20–200 μm, but challenges arise for very fine or coarse particles that are accompanied by low recoveries and poor selectivity. While the impact of particle size on the separation behavior in flotation is well known by now, the effect of particle shape is less studied and varies based on the investigated zone (suspension or froth) and separation apparatus used. Beyond these complexities, many particle properties are correlated, making it challenging to analyze the isolated impact of individual properties on the separation behavior. Therefore, a multidimensional perspective on the separation process, considering multiple particle properties, enhances the understanding of their collective influence. In this paper, the two-dimensional case is studied; i.e., a parametric modeling approach is applied to determine bivariate Tromp functions from scanning electron microscopy-based image data of the feed and the separated fractions. With these functions it is possible to characterize the separation behavior of particle systems. Using a model system of ultrafine (<10 μm) particles, consisting of either glass spheres or glass fragments with different wettability states as the floatable fraction and magnetite as the non-floatable fraction, allows for the investigation of the influence of descriptor vectors consisting of size, shape and wettability, on the separation. In this way, the present paper contributes to a better understanding of the complex interplay between certain descriptor vectors for the case of ultrafine particles. Furthermore, it demonstrates the benefits of using multivariate Tromp functions for evaluating separation processes and points out the limitations of SEM-based image measurements by means of mineral liberation analysis (MLA) for the studied particle size fraction. Full article
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19 pages, 10249 KiB  
Article
Investigation of Multidimensional Fractionation in Microchannels Combining a Numerical DEM-LBM Approach with Optical Measurements
by Simon Raoul Reinecke, Zihao Zhang, Sebastian Blahout, Edgar Radecki-Mundinger, Jeanette Hussong and Harald Kruggel-Emden
Powders 2024, 3(2), 305-323; https://doi.org/10.3390/powders3020018 - 30 May 2024
Viewed by 567
Abstract
The fractionation in microchannels is a promising approach for the delivery of microparticles in narrow property distributions. The underlying mechanisms of the channels are however often not completely understood and are therefore subject to current research. These investigations are done using different numerical [...] Read more.
The fractionation in microchannels is a promising approach for the delivery of microparticles in narrow property distributions. The underlying mechanisms of the channels are however often not completely understood and are therefore subject to current research. These investigations are done using different numerical and experimental methods. In this work, we present and evaluate our method of combining a numerical Discrete Element Method (DEM)-Lattice Boltzmann Method (LBM) approach with experimental long-exposure fluorescence microscopy, micro-Particle Image Velocimetry (µPIV) and Astigmatism Particle Tracking Velocimetry (APTV) measurements. The suitability of the single approaches and their synergies are evaluated using the exemplary investigation of multidimensional fractionation in different channel geometries. It shows that both, numerical and experimental method are well suited to evaluate particle dynamics in microchannels. As they furthermore show strengths canceling out weaknesses of the respective other method, the combined method is very well suited for the comprehensive analysis of particle dynamics in microchannels. Full article
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25 pages, 6801 KiB  
Article
Size and Shape Selective Classification of Nanoparticles
by Cornelia Damm, Danny Long, Johannes Walter and Wolfgang Peukert
Powders 2024, 3(2), 255-279; https://doi.org/10.3390/powders3020016 - 17 May 2024
Cited by 1 | Viewed by 964
Abstract
As nanoparticle syntheses on a large scale usually yield products with broad size and shape distributions, the properties of nanoparticle-based products need to be tuned after synthesis by narrowing the size and shape distributions or via the removal of undesired fractions. The development [...] Read more.
As nanoparticle syntheses on a large scale usually yield products with broad size and shape distributions, the properties of nanoparticle-based products need to be tuned after synthesis by narrowing the size and shape distributions or via the removal of undesired fractions. The development of property-selective classification processes requires a universal framework for the quantitative evaluation of multi-dimensional particle fractionation processes. This framework must be applicable to any property and any particle classification process. We extended the well-known one-dimensional methodology commonly used for describing particle size distributions and fractionation processes to the multi-dimensional case to account for the higher complexity of the property distribution and separation functions. In particular, multi-dimensional lognormal distributions are introduced and applied to diameter and length distributions of gold nanorods. The fractionation of nanorods via centrifugation and by orthogonal centrifugal and electric forces is modeled. Moreover, we demonstrate that analytical ultracentrifugation with a multi-wavelength detector (MWL-AUC) is a fast and very accurate method for the measurement of two-dimensional particle size distributions in suspension. The MWL-AUC method is widely applicable to any class of nanoparticles with size-, shape- or composition-dependent optical properties. In addition, we obtained distributions of the lateral diameter and the number of layers of molybdenum disulfide nanosheets via stepwise centrifugation and spectroscopic evaluation of the size fractions. Full article
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16 pages, 1750 KiB  
Article
Multidimensional Separation by Magnetic Seeded Filtration: Theoretical Study
by Frank Rhein, Haoran Ji and Hermann Nirschl
Powders 2024, 3(2), 217-232; https://doi.org/10.3390/powders3020014 - 22 Apr 2024
Viewed by 1149
Abstract
Magnetic seeded filtration (MSF) is a multidimensional solid–liquid separation process capable of fractionating a multimaterial suspension based on particle size and surface properties. It relies on the selective hetero-agglomeration between nonmagnetic target and magnetic seed particles followed by a magnetic separation. Experimental investigations [...] Read more.
Magnetic seeded filtration (MSF) is a multidimensional solid–liquid separation process capable of fractionating a multimaterial suspension based on particle size and surface properties. It relies on the selective hetero-agglomeration between nonmagnetic target and magnetic seed particles followed by a magnetic separation. Experimental investigations of multimaterial suspensions are challenging and limited. Therefore, a Monte Carlo model for the simulation of hetero-agglomeration processes is developed, validated, and compared to a discrete population balance model. The numerical investigation of both charge-based and hydrophobicity-based separation in an 11-material system, using synthetic agglomeration kernels based on real-world observations, yields results consistent with prior experimental studies and expectations: Although a multidimensional separation is indeed possible, unwanted hetero-agglomeration between target particles results in a reduced selectivity. This effect is more pronounced when separation is based on a dissimilarity rather than a similarity in the separation criterion and emphasizes the advantages of hydrophobicity-based systems. For the first time, 2D grade efficiency functions T(φ,d) are presented for MSF. However, it is shown that these functions strongly depend on the initial state of the suspension, which casts doubt on their general definition for agglomeration-based processes and underlines the importance of a simulation tool like the developed MC model. Full article
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11 pages, 9413 KiB  
Project Report
Dielectrophoretic Particle Chromatography: From Batch Processing to Semi-Continuous High-Throughput Separation
by Jasper Giesler, Laura Weirauch, Jorg Thöming, Georg R. Pesch and Michael Baune
Powders 2024, 3(1), 54-64; https://doi.org/10.3390/powders3010005 - 6 Feb 2024
Viewed by 1327
Abstract
The development of highly selective separation processes is a focus of current research. In 2016, the German Science Foundation funded a priority program SPP 2045 “MehrDimPart—highly specific multidimensional fractionation of fine particles with technical relevance” that aims to develop new or enhance existing [...] Read more.
The development of highly selective separation processes is a focus of current research. In 2016, the German Science Foundation funded a priority program SPP 2045 “MehrDimPart—highly specific multidimensional fractionation of fine particles with technical relevance” that aims to develop new or enhance existing approaches for the separation of nano- and micrometer-sized particles. Dielectrophoretic separators achieve highly selective separations of (bio-)particles in microfluidic devices or can handle large quantities when non-selective separation is sufficient. Recently, separator designs were developed that aim to combine a high throughput and high selectivity. Here, we summarize the development from a microfluidic fast chromatographic separation via frequency modulated dielectrophoretic particle chromatography (DPC) toward a macrofluidic high throughput separation. Further, we provide a starting point for future work by providing new experimental data demonstrating for the first time the trapping of 200 nm polystyrene particles in a dielectrophoretic high-throughput separator that uses printed circuit boards as alternatives for expensive electrode arrays. Full article
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2023

Jump to: 2024

19 pages, 1164 KiB  
Article
Multidimensional Separation by Magnetic Seeded Filtration: Experimental Studies
by Frank Rhein, Ouwen Zhai, Eric Schmid and Hermann Nirschl
Powders 2023, 2(3), 588-606; https://doi.org/10.3390/powders2030037 - 1 Aug 2023
Cited by 7 | Viewed by 1699
Abstract
The current state of separation technology often neglects the multidimensional nature of real particle systems, which are distributed not only in terms of size, but also in terms of other properties, such as surface charge. Therefore, the aim of this study is to [...] Read more.
The current state of separation technology often neglects the multidimensional nature of real particle systems, which are distributed not only in terms of size, but also in terms of other properties, such as surface charge. Therefore, the aim of this study is to experimentally investigate the applicability of magnetic seeded filtration as a multidimensional separation process. Magnetic seed particles are added to a multisubstance suspension, and a selective heteroagglomeration with the nonmagnetic target particles is induced, allowing for an easy subsequent magnetic separation. The results show that high separation efficiencies can be achieved and that the parameters pH and ionic strength govern the agglomeration process. Selective separation based on surface charge was observed, but undesirable heteroagglomeration processes between the target particles lead to a loss of selectivity. Particle size was clearly identified as a second relevant separation feature, and its partially opposite influence on collision frequency and collision efficiency was discussed. Finally, experimental data of multidimensional separation are presented, in which a size-distributed two-substance suspension is separated into defined size and material fractions in a single process step. This study highlights the need for multidimensional evaluation in general and the potential of magnetic seeded filtration as a promising separation technique. Full article
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19 pages, 8777 KiB  
Article
Parametric Stochastic Modeling of Particle Descriptor Vectors for Studying the Influence of Ultrafine Particle Wettability and Morphology on Flotation-Based Separation Behavior
by Thomas Wilhelm, Johanna Sygusch, Orkun Furat, Kai Bachmann, Martin Rudolph and Volker Schmidt
Powders 2023, 2(2), 353-371; https://doi.org/10.3390/powders2020021 - 11 May 2023
Cited by 4 | Viewed by 1575
Abstract
Practically all particle separation processes depend on more than one particulate property. In the case of the industrially important froth flotation separation, these properties concern wettability, composition, size and shape. Therefore, it is useful to analyze different particle descriptors when studying the influence [...] Read more.
Practically all particle separation processes depend on more than one particulate property. In the case of the industrially important froth flotation separation, these properties concern wettability, composition, size and shape. Therefore, it is useful to analyze different particle descriptors when studying the influence of particle wettability and morphology on the separation behavior of particle systems. A common tool for classifying particle separation processes are Tromp functions. Recently, multivariate Tromp functions, computed by means of non-parametric kernel density estimation, have emerged which characterize the separation behavior with respect to multidimensional vectors of particle descriptors. In the present paper, an alternative parametric approach based on copulas is proposed in order to compute multivariate Tromp functions and, in this way, to characterize the separation behavior of particle systems. In particular, bivariate Tromp functions for the area-equivalent diameter and aspect ratio of glass particles with different morphologies and surface modification have been computed, based on image characterization by means of mineral liberation analysis (MLA). Comparing the obtained Tromp functions with one another reveals the combined influence of multiple factors, in this case particle wettability, morphology and size, on the separation behavior and introduces an innovative approach for evaluating multidimensional separation. In addition, we extend the parametric copula-based method for the computation of multivariate Tromp functions, in order to characterize separation processes, also in the case when image measurements are not available for all separated fractions. Full article
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