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The sintering of vitreous nanoparticle doublets is investigated numerically by a volume of fluid method coupled to Hamaker summation and experimentally by a high-temperature sintering flow reactor as well as by doublet shape analysis in the transmission electron microscope. In particular, the characteristic differences between nanoparticulate and bulk sintering are studied. The sintering mechanism of vitreous nanoparticles is determined to be viscous flow with interparticle van der Waals interactions acting as additional driving force. The early stages of the nanoparticle sintering kinetics are inversely proportional to the square of the particle size, instead of an indirect proportionality to the first order of the particle size for the entire bulk process. The transition between nanoparticulate and bulk sintering is localised to primary particle diameters of approx. 200–300 nm.

This work investigates the mechanisms which lead to the formation of silicon nanoparticles with narrow size distributions by means of population balance modeling. The model accounts for the full aerosol process, including chemical reaction, nucleation from supersaturated vapor, growth and agglomeration. The results are in good agreement with experimental data. The effects of the process parameters temperature, silane concentration and reactor total pressure are systematically investigated. The simulation allows an in-depth insight into the particle formation mechanism and reveals the key requirements which are necessary for the generation of narrow particle size distributions. In this mechanism, only a short nucleation burst occurs, while surface growth plays the dominant role in silane precursor consumption. A key role is attributed to condensation, because the numerical calculations can only reflect the experimental observations, if the condensation mechanism is included in the model.

A study on the feasibility of aerosol processing of nearly monodisperse silicon nanoparticles via pyrolysis of monosilane in a hot wall reactor is presented. For optimal conditions silicon nanoparticles with a geometric standard deviation of 1.06 were synthesized at a production rate of 0.7 g/h. The size of the particles could be precisely controlled in the range of 20–40 nm, whilst maintaining a geometric standard deviation in the range of 1.06–1.08, by proper choice of the governing parameters temperature, residence time and precursor concentration. The results show that narrow particle size distributions can only be obtained in the temperature range between 900 and 1100 °C, as long as both the initial silane concentration (1 mbar silane partial pressure) and the reactor total pressure are low (25 mbar). This regime for the production of narrow particle size distributions has not been identified in prior work on the thermal decomposition of silane. Narrowly distributed particles can be obtained under conditions where nucleation and particle growth are separated and the agglomeration rates are negligible.

In this paper, we consider a model for precipitation experiments based on the population balance equation. The study revealed a high sensitivity of the system with respect to the modeling of intrinsic parameters, motivating a comprehensive validation of the estimates. In the forward simulation the impact of the influencing parameters including surface energy, nucleus size and distribution is investigated. Subsequently we construct a simplified model of the precipitation process in such a way that it is orbitally flat in terms of control theory, which enables the inverse calculation of the parameters. The numerical results of the inverse simulation for the interfacial energy have been compared to a physical model. The possibility of solving the inverse problem provides a promising way of estimating hardly measurable quantities for more complex molecules.

In this work, a new model for the simulation of nanostructured aggregates by simultaneous coagulation, sintering and surface growth is presented. Coagulation is treated as cluster–cluster agglomeration along the line connecting the center of mass of both agglomerates and is implemented using a Monte Carlo algorithm. Sintering is modeled as successive overlapping of spheres which cause reduction in the surface area based on a rate law for surface reduction. Surface growth is modeled as an increase in primary particle diameter, e.g. as a result of surface reactions. The evolved aggregates are analyzed by calculating their fractal dimension, radius of gyration, mobility diameter and mobility shape factor. It is found that the aggregates structure tends to be more compact when introducing the surface growth in shorter time comparing to the coagulation-sintering step only. Fractal dimension and the mobility shape factor of the resulting aggregates are correlated to an effective dimensionless time that combines the characteristic times of these three fundamental mechanisms. It is shown that the mobility diameter in the free molecular regime is not proportional to the radius of gyration. A power law relation that correlates the aggregates projected area and the equivalent number of primary particles is found to be in a very good agreement with estimates published in literature.

To improve the understanding of the poor dispersability of fumed silica nanoparticle agglomerates, the stability of highly defined agglomerated model particles was investigated. The high temperature synthesis conditions for fumed silica were simulated by tempering. Along with electron-microscopical analysis of the sintering necks, the interparticle forces were investigated by energy resolved fragmentation analysis based on low pressure impaction. At temperatures above 1,000 °C the fragmentability of the agglomerates rapidly decreased while the energy necessary for fragmentation increased. The development of sintering necks was observed for temperatures exceeding 1,300 °C. Comparison of the experimental data with the fragmentation behaviour of a commercially produced fumed silica indicated solid state contacts (sintering necks) as being most numerous in the agglomerates resulting in limited fragmentability.

The study presents first experimental results of the transfer of magnetite nanoparticles from an aqueous to a second non-miscible non-aqueous liquid phase. The transfer is based on the adsorption of macromolecular surfactants onto the particle surface at the liquid–liquid interface. For a successful direct phase transfer, it is essential to have cations, like ammonium ions, present in the aqueous phase as well as a threshold concentration of surfactant in the organic liquid phase. While penetrating the liquid–liquid interface, the particles are covered with the surfactant and therefore a partial de-agglomeration is initiated. Based on literature and experimental data a mechanism of surfactant adsorption is proposed. The competing adsorption of the surfactant molecules at the liquid–liquid interface leads to the formation of emulsions and therefore to a hindrance for particles passing the interface. Nevertheless a high efficiency of 100% yield can be reached using optimized process parameters for the phase transfer process.

Modeling of particle deposition on adjacent walls is a key issue in various applications like separation or transport processes. The present paper focuses on the modeling of turbophoretic deposition of particles in the micron size range. The first step is to evaluate the important range where turbophoresis plays an important role in comparison to other mechanisms e.g. gravity or electrostatic separation. The disadvantages of commonly used models will be analyzed and overcome by implementing a more sophisticated approach considering damping of turbulent fluctuations in the wall-boundary layer. In contrast to previous work, commonly used turbulence models are applied to solve the mean flow field of the examples under consideration. The results will show a good prediction of particle deposition in comparison to experimental values [B.Y.H. Liu, J.K. Agarwal, Experimental observation of aerosol deposition in turbulent flow, Aerosol. Sci. 5 (1974) 145–155.] by using the advanced model.

The drag force on aggregates and partially sintered agglomerates is assessed using the lattice Boltzmann method (LBM) and accelerated Stokesian dynamics (ASD). Both methods have been compared in terms of accuracy and computational effort. It is shown that they give comparable results if all numerical parameters are controlled carefully. LBM requires a much higher computational effort, however, in contrast to ASD it is able to simulate partially sintered agglomerates as well. The results show that even a very small amount of sintering leads to a significant reduction in the drag force. The analysis of the drag force on agglomerates as well as on aggregates shows that there is no simple geometric quantity which is uniquely related to the drag force. Moreover, there is a significant variation in drag force for single aggregates at different orientations or for the orientation averaged drag force of different aggregates of the same size. This is explained by the structural effects which may lead to a variation in the drag force up to +/-20%.

Die Modellierung der wandnahen Bereiche bei der Simulation von Zweiphasenstromungen spielt in der Praxis eine entscheidende Rolle, wenn erwunschter oder unerwunschter Transport der dispersen Phase zu einer Oberflache bedeutend ist. Zweiphasenstromungen mit einer dispersen Phase werden heutzutage haufig mit dem Euler-Lagrange-Verfahren in modernen CFD-Codes simuliert. Im Folgenden wird darauf eingegangen werden, wie solche Berechnungen in kommerziellen CFD-Codes durchgefuhrt werden.

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 simulationwith a Lagrangian particle trackingstrategy 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. 2005 Elsevier Ltd. All rights reserved.

Simulation results on the evolution of aggregate structure in aerosol processes with coagulation and sintering as the dominant mechanisms are presented. A model for simulation of the three-dimensional morphology of nano-structured aggregates formed by concurrent coagulation and sintering is applied. The model is based on a stochastic diffusion controlled cluster–cluster aggregation algorithm and sintering is modeled as a successive overlapping of spherical primary particles, which are allowed to grow in order to maintain mass conservation. This leads to computer simulated structured aggregates which are then subject to evaluation. Two different methods to determine the fractal dimension are presented which give comparable results. It is shown that even very small particles show the same fractal behavior. Furthermore, equilibrium structures assuming a constant ratio of the characteristic collision time to the characteristic fusion time are considered as well as the kinetics of structural changes due to a change in the ambient conditions.

This work investigates the effects of reduced accessible surface area of aggregate particles and of surface energy on relevant particle formation and growth mechanisms during titania formation from the vapour phase at industrial process conditions. Growth due to surface reaction and due to condensation is related to the fraction of the surface area that is exposed to the collision with single molecules. Surface shielding is found to hamper surface reaction and condensation once fractal aggregates start to form. It leads to significantly retarded precursor consumption and produces aggregate particles, which consist of more, but smaller primary particles. Surface energy data are varied within a range as proposed by available literature data. Moderate and high surface energy values result in a thermodynamic barrier to the formation of new particles and are shown to reduce the formation of seed particles by several orders of magnitude. This leads to the formation of aggregate particles which consist of a rather small number of primary particles and mainly grow by surface reaction. The primary contribution of condensation to growth of individual primary particles is shown to be very little. However, condensation should not be neglected as it has a strong impact on particle formation rates and hence on product characteristics such as the number of primary particles and primary particle size.

This work investigates effects of reduced accessible surface area of aggregate particles and surface energy on titania particle formation and growth. It is taken into consideration that surface-related growth mechanisms, i.e. surface reaction and condensation, are limited to the fraction of the surface area of primary particles which is exposed to the collision with single molecules. Surface energy data determine the critical particle size with respect to evaporation and values are varied within the published range. This implies to develop a model which considers “surface shielding” and accounts for the formation of stable clusters from a supersaturated vapour due to nucleation and condensation besides considering the generation of monomers due to chemical reaction, growth due to surface reaction, agglomeration and sintering. Taking the accessible surface area into account is found out to be especially important if agglomerates contain a large number of primary particles or if agglomerate structure is rather compact. In this case, precursor consumption and primary particle growth turn out to be significantly retarded. Surface energy data are shown to be decisive with respect to the thermodynamic barrier to the formation of particles, thus to active particle formation and growth mechanisms, besides affecting sintering kinetics. Elevated surface energy data typically retard precursor consumption and favour primary particle growth.

A high-temperature short residence time flow reactor system has been realized for the investigation of the gas phase sintering kinetics of nanoparticles separated from all other synthesis mechanisms, e.g., chemical reaction, nucleation, coagulation and condensation. Major components are a hot-wall tubular reactor for the production of unsintered aggregates consisting of spherical primary particles in the size range 10–80 nm and a sintering reactor for the investigation of the sintering kinetics at well-defined temperature and residence time history. Therefore, rapid heating and quenching of the particles at the beginning and at the end of the reaction zone, respectively, is maintained. Main parameters in the sintering reactor are: temperatures up to 1873 K and residence times in the range 8–1000 ms. The reactor conditions are characterized by temperature profile measurements with a newly constructed suction thermocouple probe and by computational fluid dynamics simulations of the residence time distributi...

A simulation method is applied to calculate particle dynamics in electrostatic precipitators as characterised by particle flux density and concentration profiles in arbitrary channel cross-sections and flux density profiles of dust precipitated at the collecting electrodes (CEs). A simple statistical model allows the determination of confidence intervals for flux profiles. First, a ‘standard case’ considering full coupling of all physical phenomena occurring in this problem, i.e., electric field, flow field and particle dynamics is simulated. Subsequently, this standard case is compared to simulations with one quantity (e.g., electric field strength, turbulence intensity) substituted by a mean value which is homogeneously distributed in the precipitation zone. This reveals the relevance of the various physical phenomena: It turned out that the secondary flows had only a minor influence on the overall particle precipitation although they cause some ‘patterning’ of local precipitation. Turbulence inhomogeneity shows a stronger effect on particle dynamics. However, the electric field appears to be by far the most important quantity in simulating particle dynamics. Consequently, in order to achieve most reasonable simulation results for a given numerical effort most attention has to be devoted to field calculations, including correct boundary conditions.

A population balance model is presented, which tracks particle growth in the gas phase and accounts for simultaneous agglomeration and sintering: Simulations reveal the evolution of the full distribution of a volume equivalent diameter and, amongst others, the evolution of the agglomerate collision diameter, a mean primary particle size and the number of primary particles per agglomerate. Furthermore, assuming fractal behaviour of the growing agglomerate particles—for the 7rst time—a model for the evolution of a mean value of the fractal dimension based on physical and process parameters is proposed and incorporated into the simulation model. PARSIVAL, a commercial solver for integro-di;erential equations is employed to solve the equations involved. It is based on a generalised 7nite-element scheme with self-adaptive grid- and order construction. Calculations are performed to validate the model against monodisperse and sectional models published in literature for the exemplary case of Si production. The results are in good agreement if the same simplifying assumptions are made. However, results obtained from the new model for both—isothermal and non-isothermal process conditions—clearly show that it is important to consider the changing fractal dimension in many cases. ? 2003 Elsevier Science Ltd. All rights reserved.

Rasche technologische Entwicklungen haben fur Chemieingenieure bzw. Verfahrenstechniker neue Beschaftigungsmoglichkeiten eroffnet, wodurch sich zusatzliche Anforderungen an die Ausbildung ergeben. Der vorliegende Beitrag beschreibt die Losungsansatze im Fachgebiet der Partikeltechnik: Es findet eine noch starkere Betonung der Grundlagen statt, wobei Beispiele aus klassischen und neuen Anwendungen exemplarisch vertieft werden. Kernpunkt des Curriculums sind die Produkteigenschaften. Besonderes Gewicht wird auserdem auf die Betrachtung ganzer Prozesse gelegt. Die Vorlesungen sind in drei Ebenen strukturiert: Grundlagen, Grundoperationen und Prozesse/Anwendungen. Methodisch wird ein zweifacher Ansatz verfolgt: Die Lehre der grundlegenden Fakten wird erganzt durch eine aktive Teilnahme der Studenten, insbesondere bei der Erarbeitung von Beispielen und in Labor- bzw. Rechnerpraktika. Durch eine ganzheitliche Sicht der Universitatsausbildung sollen auch „soft skills“ gefordert werden. Mechanical Process Engineering as an Example for Educational Challenges Due to rapid technological developments new opportunities for chemical engineers evolved leading to additional demands in education. The present article describes how this challenge is faced in the subject particle technology: Fundamentals are emphasised even more, with exemplary treatment of classical as well as new applications. Key item of the curriculum is the product property approach. Furthermore, the view of complete processes is emphasised. The courses are structured in three levels: Fundamentals, Unit Operations and Processes/Applications. From the methodological point of view a twofold approach is used: Teaching the fundamentals is complemented by an active participation of the students in working on examples and application problems as well as lab and virtual experiments. By following an integrated view of university education also soft skills of students are promoted.

Powder ageing effects due to thermal load usually lead to a limited recyclability of unmolten polyamide 12 powders in the laser sintering process. In this work, the recycling optimized material PA 2221 is analyzed regarding its ageing behavior along a series of more than ten build/refresh cycles using a refresh ratio of about 30%. Thereby, the melt viscosity is measured before and after every cycle and the “steady-state” condition is determined accordingly. In addition, the material properties of parts manufactured with PA 2221 are discussed dependent on different process parameters and compared to parts made of standard PA 2200 material with a refresh rate of about 50%. These parameters include the mechanical properties as well as for example the impact strength and geometrical properties as function of layer thickness and testing temperature. As a result, the optimal process conditions and achievable material properties for PA 2221 can be derived from this work.

The emphasis of the study presented is on a new process of particle extraction to transfer magnetite nanoparticles from an aqueous into an immiscible organic phase directly through the liquid-liquid interface. For the production of high-quality organosols, stabilized colloidal and functionalized particles are required in a liquid organic phase. The mechanism of phase transfer is initiated by adsorption and chemical binding of surfactants (fatty acids) at the particle surface. The resulting physico-chemical dispersion of the hydrophobically modified particles leads to the formation of the stabilized organic colloid, or organosol. The aim here is to demonstrate the entire chain of the transfer process in a continuous miniplant, which comprises particle synthesis, conditioning, and transfer, and which uses a drop column for extraction and as a transfer device. Based on the investigation of the governing principles and the material parameters, the results obtained for the transfer kinetics in the individual contact devices (centrifuge, single-drop column, and drop column for different operations) are used for the dimensioning of the entire process chain.

Thermal charging of submicron and nanometer particles has been studied for model aerosols of TiO2 and SiO2 as well as Al-Si (aluminosilicate) at 1 000 °C with a new quasi in-situ technique. The size dependence of the particle separation efficiency for electrostatic precipitation was determined. The charging state of the particles was obtained from evaluating the global Deutsch number for precipitation in an electric field applied to a laminar flow based on particle trajectory considerations.

Electro-hydrodynamic (EHD) flows are investigated theoretically and numerically in this paper and results are presented for the flow field in model electrostatic precipitators (EPs). The resulting flow fields are shown in various representations and explained qualitatively. Numerical calculations with different flow models (non-turbulent and RANS) were conducted to investigate the influence of the flow model on the resulting secondary flows. Furthermore, a perturbation analysis is presented, leading to a simple differential equation of the Helmholtz type. This allows a more detailed view of the important mechanisms forming the secondary flows as well as being able to obtain a very fast estimation of the resulting flow field. The calculations reveal a strong influence of a vortex formation at the beginning of the precipitation zone on the whole flow field. Furthermore, a strong effect of the boundary conditions of the electric field and the operating parameters is shown.

Polymer laser sintering (LS) is an important additive manufacturing (AM) technology. Individual and complex parts are directly produced from CAD data without the need of specific tools. The raw material is a polymer powder, which is deposited layerwise and melted selectively with a laser. Built parts are embedded in residual unmolten powder, the so-called part cake, which undergoes thermal ageing effects due to the exposure to high temperatures for long times during the manufacturing process. Hence, the recyclability of the unmolten powder is limited. This article focuses on a fundamental analysis of the ageing kinetics dependent on time, temperature, and oxygen content in the gas atmosphere. A model is developed and applied to measured, position-dependent process temperature histories to successfully predict the ageing distribution within a part cake. The results can be used to optimize the thermal process management in LS and to develop new efficient powder recycling methods. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017, 134, 45435.

The connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 yr. The article J. Radon published about 50 yr ago (J. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948) 286–300) has been very influential. In this text we describe some of the results that were obtained during the search for answers to questions raised by his article.

Laser Sintering is a powder based additive manufacturing technology. The amount of used powder, which is not sintered, can be refreshed and used for further building jobs. Due to cost reduction, better mechanical properties and a simpler processing it is reasonable to combine virgin and used powder for part manufacturing. During the building time the powder is thermally loaded and its structure is changed. State of the art in laser sintering is a defined powder ratio. Since the powder ageing depends on the individual history of sintering, this procedure cannot ensure a defined powder quality. However, for a serial production it is necessary to characterize the raw material, as well as all other steps along the process chain, in order to ensure a high reproducibility and reliability. Rheological, physical and particle properties are considered to correlate powder and chosen material properties. Different powder qualities adjusted by the Melt Volume Rate (MVR) are tested using promising rheological properties like viscosity or molecular weight. Investigations about powder bed density, bulk properties as well as thermal characterization complement the experimental setup. These parameters are correlated with mechanical, electrical, thermal and physical material properties of laser sintered parts. The most important influencing factors along the process chain are kept constant. At the end a procedure is shown to obtain a defined powder quality. This procedure will be transferred to other materials and will be tested using other laser sintering machines of the same type as well as other types.

The viscous-flow sintering of different agglomerate particle morphologies is studied by three-dimensional computer simulations based on the concept of fractional volume of fluid. For a fundamental understanding of particle sintering characteristics, the neck growth kinetics in agglomerate chains and in doublets consisting of differently sized primary particles is investigated. Results show that different sintering contacts in agglomerates even during the first stages are not completely independent from each other, even though differences are small. The neck growth kinetics of differently sized primary particles is determined by the smaller one up to a size difference by a factor of approximately 2, whereas for larger size differences, the kinetics becomes faster. In particular, the agglomerate sintering kinetics is investigated for particle chains of different lengths and for different particle morphologies each having ten primary particles and nine initial sintering contacts. For agglomerate chains, the kinetics approximately can be normalized by using the radius of the fully coalesced sphere. In general, different agglomerate morphologies show equal kinetics during the first sintering stages, whereas during advanced stages, compact morphologies show significantly faster sintering progress than more open morphologies. Hence, the overall kinetics cannot be described by simply using constant morphology correction factors such as fractal dimension or mean coordination number which are used in common sintering models. However, for the first stages of viscous-flow agglomerate sintering, which are the most important for many particle processes, a sintering equation is presented. Although we use agglomerates consisting of spherical primary particles, our methodology can be applied to other aggregate geometries as well.

A model for simulation of the three-dimensional morphology of nano-structured aggregates formed by concurrent coagulation and sintering is presented. Diffusion controlled cluster-cluster aggregation is assumed to be the prevailing coagulation mechanism which is implemented using a Monte-Carlo algorithm. Sintering is modeled as a successive overlapping of spherical primary particles, which are allowed to grow as to preserve overall mass. Simulations are characterized by individual ratios tau of characteristic collision to fusion time. A number of resulting aggregate-structures is displayed and reveals structure formation by coagulation and sintering for different values of tau. These aggregates are described qualitatively and quantitatively by their mass fractal dimension D-f and radius of gyration. The fractal dimension increases from 1.86 for pure aggregation to approximate to 2.75 for equal characteristic time scales. As sintering turns out to be more and more relevant, increasingly compact aggregates start to form and the radius of gyration decreases significantly. The simulation results clearly reveal a strong dependence of the fractal dimension on the kinetics of the concurrent coagulation and sintering processes. Considering appropriate values of D-f in aerosol process simulations may therefore be important in many cases.

For two classes of integrals it will be shown that the number of nodes of cubature formulae of degree 4n + 1, n > 1, will not attain Moller’s lower bound. Thus in these cases that bound has to be increased by 1.

An optical measuring technique is presented allowing the exact in-situ measurement of local particle flux densities in a confined channel flow by counting single particles penetrating an optically well defined measuring volume. This enables a precise flux determination up to the direct vicinity of planar walls. The measurement set-up and its calibration as well as the whole test facility are described in detail. This measurement technique is used to study the particle transport in electrostatic precipitators. Exemplarily, results of particle flux profiles as well as precipitation, as gained from balances of parts of the precipitator channel, are presented. Furthermore, the possibility to determine particle velocity fluctuations is demonstrated.

In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree ?m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree.

In diesem Artikel werden Ergebnisse einer Untersuchung des geometrischen Vorgehens der Ungarischen Methode zusammengefaBt. Das Hauptresultat besteht in folgender Deutung: Die Ungarische Methode bewegt sich mit jedem Schritt in Richtung des steilsten Abstiegs aller m6glichen, auf 1 normierten Fortschreitungsrichtungen bis zum Rand des zul~issigen Bereichs. Darin erweist sich die Ungarische Methode dem Simplexverfahren iiberlegen, das sich nur in Richtung der steilsten aller auf 1 normierten Kantenrichtungen bewegen kann.

In this note an interactive program is presented which allows the visualization of the common roots of two-dimensional polynomials. The program is useful when constructing cubature formulas by use of orthogonal polynomials.

In this note cubature formulae of degree 5 are studied for n-dimensional integrals over the ball with constant weight function. We apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is equivalent to the existence of tight spherical 5-designs. The known results concerning spherical 5-designs show that the lower bound for the integral under consideration will not be attained in general. The bound will be attained for n = 2,3,7,23 and possibly for n = (2p+1)2-2, p ≥ 5. In all other cases the bound must be increased at least by 1, in particular, Stroud's formulae for n = 4,5,6,7 are minimal.