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Progress in the Application of Multidimensional Particle Property Distributions: The separation function

KONA POWDER AND PARTICLE JOURNAL 2025年第42期2025卷 134-155页

作者： Schach, Edgar Buchwald, Thomas Furat, Orkun Tischer, Florentin Kaas, Alexandra Kuger, Laura Masuhr, Matthias Sygusch, Johanna Wilhelm, Thomas Ditscherlein, Ralf Peuker, Urs Alexander TU Bergakademie Freiberg Inst Mech Proc Engn & Mineral Proc Freiberg Germany Ulm Univ Inst Stochast Ulm Germany Friedrich Alexander Univ Erlangen Nurnberg Dept Chem & Biol Engn Erlangen Germany Karlsruhe Inst Technol Inst Funct Interfaces Karlsruhe Germany Univ Duisburg Essen Inst Technol Nanostruct Essen Germany Helmholtz Zentrum Dresden Rossendorf Helmholtz Inst Freiberg Resource Technol Dresden Germany

In KONA 2022, the fundamentals of two- and multidimensional particle size distributions were introduced. The next question in the field of two- and multidimensional distributions addresses their application to describe a particle process, e.g., agglomeration or separation. A multidimensional separation can be seen as retrieving only particles with a specific set of properties from a multidimensionally distributed system, e.g., retrieving only small particles (below a certain threshold in size) with a compact spherical shape (above a certain threshold in sphericity). The multidimensional separation allows the generation of functional particle systems with specific properties, e.g., semiconducting, optical, or electronic properties, which are required for high-technology applications. Starting from so-called particle-discrete information, i.e., an information vector for each particle containing its compositional, geometrical, and physical properties, it is possible to describe a multidimensional separation in full detail based on various properties. Each particle can be evaluated according to different separation properties, e.g., size, shape, and material composition. With this database, it is possible to define and work with separation functions to describe the multidimensional separation and quantify the separation results. For example, in the two-dimensional case, the median cut size becomes a median cut line, where the probability for a particle to belong to the concentrate is 0.5. Some case studies and examples show different approaches and possibilities to achieve a multidimensional separation in one or several connected process steps.

关键词： separation function Tromp curve partition curve multidimensional particle property distribution

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New separation function for the stationary screening process

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CANADIAN JOURNAL OF CHEMICAL ENGINEERING 2022年第9期100卷 2347-2354页

作者： Teipel, Ulrich TH Nurnberg Forsch Grp Partikeltechnol & Rohstoffinnovationen Fak Verfahrenstech Wassertorstr 10 D-90489 Nurnberg Germany Univ Ulm Inst Chem Ingn Mechan Verfahrenstech Ulm Germany

In the present study, a new separation function T(x,alpha',beta) for the steady-state screening process is presented. This new grade efficiency T(x,alpha',beta) presented here is a function of particle size x, separation sharpness alpha', and the newly introduced separation efficiency beta. With this new efficiency function, the screening classification process can be described exactly. Especially in the fine and coarse material ranges, a very good correlation of the calculated function with the measured values can be observed. A comparison of the grade efficiency function with separation sharpness alpha', separation efficiency beta, and only with the measure for separation efficiency has shown that the new grade efficiency T(x,alpha',beta) allows a significant improvement in the characterization of the stationary screen classification process. When compared with other models, the new model of the grade efficiency T(x,alpha',beta) shows a significantly higher correlation with the measured values and is therefore very well suited to describe a grade efficiency for the stationary screening process.

关键词： classification screening separation efficiency separation function steady-state screening process

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Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices - Application of perturbation theory for simple invariant subspaces

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LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期419卷 765-771页

作者： Wei, Yimin Li, Xiezhang Bu, Fanbin Zhang, Fuzhen Fudan Univ Key Lab Math Nonlinear Sci Minist Educ Shanghai 200433 Peoples R China Georgia So Univ Dept Math Sci Statesboro GA 30460 USA Fudan Univ Inst Math Shanghai 200433 Peoples R China Nova SE Univ Div Math Sci & Technol Ft Lauderdale FL 33314 USA Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China

Perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix. These upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices. (c) 2006 Elsevier Inc. All rights reserved.

关键词： diagonalizable and singular matrix simple invariant subspaces separation function relative perturbation bound

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A perturbation bound of the drazin inverse of a matrix by separation of simple invariant subspaces

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第1期27卷 72-81页

作者： Wei, YM Li, XZ Bu, FB Fudan Univ Dept Math Shanghai 200433 Peoples R China Georgia So Univ Dept Math Sci Statesboro GA 30460 USA Fudan Univ Inst Math Shanghai 200433 Peoples R China

A constructive perturbation bound of the Drazin inverse of a square matrix is derived using a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces. This is an improvement of the result published by the authors Wei and Li [ Numer. Linear Algebra Appl., 10 ( 2003), pp. 563 - 575]. It is a totally new approach to developing perturbation bounds for the Drazin inverse of a matrix. A numerical example which indicates the sharpness of the perturbation bound is presented.

关键词： Drazin inverse perturbations separation function simple invariant subspaces

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Which demands should and can meet a separation model for hydrocyclone classification?

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INTERNATIONAL JOURNAL OF MINERAL PROCESSING 2010年第1-4期96卷 14-26页

作者： Schubert, Heinrich Tech Univ Bergakad Freiberg D-09596 Freiberg Germany

For technological design and operation of hydrocyclone classification a separation model should be available, which largely comprises the physical phenomena of the process, reflects their effects by appropriate process parameters and can be mathematically formulated as separation function. The most models, which are known so far, do not meet these requirements in their entirety. In the main they have been empirically formulated and/or do not reflect that the separation effect, caused by the centrifugal field, is superimposed by an intensive mixing effect caused by the macroturbulence of the hydrocyclone flow. Only the models of turbulent cross-flow classification, which are adjusted to the conditions of hydrocyclone classification in dilute suspensions, come in the form of the tapping model or partition model close to the satisfaction of the demands mentioned above. After outlines of the characteristics and the transport phenomena of turbulent flows as well as of the indispensable simplifications and adjustments to given materials, geometrical and hydrodynamical properties, the model derivation is described. In particular the wide range of the rheological properties of the feed suspensions as well as the phenomena of turbulence damping cause restrictions regarding the quantification of the model predictions. Further improvements of the model predictions are only attainable by means of empirical adjustment corrections. In order to ensure an adequate reliability of the correction factors, they should be determined using similar material systems and also similar hydrocyclones. Meanwhile several working teams have successfully tested the efficiency of such models for dilute flow classification. The first and so far only known model development for dense flow classification in hydrocyclones starts from the fact that the operation space must be divided into two subspaces and thus also two subprocesses, namely the sediment formation in the outer subspace and the formation

关键词： Hydrocyclone classification Turbulent transport phenomena Turbulent cross-flow classification separation models separation function Dilute flow classification Dense flow classification Fishhook effect

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Constrained Extremum Problems, Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第3期177卷 770-787页

作者： Zhu, Shengkun Southwestern Univ Finance & Econ Dept Econ & Math Chengdu 611130 Sichuan Peoples R China Univ Pisa Dept Math I-56127 Pisa Italy

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.

关键词： Image space analysis Regularity condition Constraint qualification separation function

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Constrained Extremum Problems, Regularity Conditions and Image Space Analysis. Part II: The Vector Finite-Dimensional Case

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第3期177卷 788-810页

作者： Pellegrini, Letizia Zhu, Shengkun Univ Verona Dept Econ I-37129 Verona Italy Southwestern Univ Finance & Econ Dept Econ & Math Chengdu 611130 Sichuan Peoples R China Univ Pisa Dept Math I-56127 Pisa Italy

The scalar finite-dimensional case has been discussed in the first part of this work series, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. Based on this preliminary result, the present paper dedicates itself to further study the regularity conditions for vector constrained extremum problems in a Euclidean space. The case of infinite-dimensional image will be the subject of a subsequent paper.

关键词： Image space analysis Vector optimization Regularity condition Constraint qualification separation function

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Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第3期161卷 738-762页

作者： Zhu, S. K. Li, S. J. Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

This paper is concerned with a unified duality theory for a constrained extremum problem. Following along with the image space analysis, a unified duality scheme for a constrained extremum problem is proposed by virtue of the class of regular weak separation functions in the image space. Some equivalent characterizations of the zero duality property are obtained under an appropriate assumption. Moreover, some necessary and sufficient conditions for the zero duality property are also established in terms of the perturbation function. In the accompanying paper, the Lagrange-type duality, Wolfe duality and Mond-Weir duality will be discussed as special duality schemes in a unified interpretation. Simultaneously, three practical classes of regular weak separation functions will be also considered.

关键词： Image space analysis Constrained extremum problem separation function Lagrange-type duality Perturbation function

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NUMERICAL MODELING OF A NONMONOTONIC separation HYDROCYCLONE CURVE

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JOURNAL OF ENGINEERING PHYSICS AND THERMOPHYSICS 2012年第6期85卷 1317-1326页

作者： Min'kov, L. L. Dueck, J. H. Tomsk State Univ 36 Lenin Ave Tomsk 634050 Russia Friedrich Alexander Univ Erlangen Nurnberg FAU D-91052 Erlangen Germany

In the context of the mechanics of interpenetrating continua, numerical modeling of separation of a polydisperse suspension in a hydrocyclone is carried out. The so-called "mixture model" valid for a low volume fraction of particles and low Stokes numbers is used for description of the suspension and particle motion. It is shown that account taken of the interaction between large and small particles can explain the nonmonotonic behavior of the separation curve.

关键词： numerical modeling hydrocyclone separation function "fish-hook" effect accelerated settling of small particles polydisperse suspension

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Constrained Extremum Problems and Image Space Analysis-Part III: Generalized Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第3期177卷 660-678页

作者： Li, Shengjie Xu, Yangdong You, Manxue Zhu, Shengkun Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China Chongqing Univ Posts & Telecommun Coll Sci Chongqing 400065 Peoples R China Southwestern Univ Finance & Econ Dept Econ & Math Chengdu 611130 Sichuan Peoples R China

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area.

关键词： Image space analysis Generalized system separation function Gap function Error bound

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