A Markov chain model of mixing kinetics for ternary mixture of dissimilar particulate solids

International audienceThis paper presents a simple but informative mathematical model to describe the mixing of three dissimilar components of particulate solids that have the tendency to segregate within one another. A nonlinear Markov chain model is proposed to describe the process. At each time step, the exchange of particulate solids between the cells of the chain is divided into two virtual stages. The first is pure stochastic mixing accompanied by downward segregation. Upon the completion of this stage, some of the cells appear to be overfilled with the mixture, while others appear to have a void space. The second stage is related to upward segregation. Components from the overfilled cells fill the upper cells (those with the void space) according to the proposed algorithm. The degree of non-homogeneity in the mixture (the standard deviation) is calculated at each time step, which allows the mixing kinetics to be described. The optimum mixing time is found to provide the maximum homogeneity in the ternary mixture. However, this ``common'' time differs from the optimum mixing times for individual components. The model is verified using a lab-scale vibration vessel, and a reasonable correlation between the calculated and experimental data is obtained.(C) 2016 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved

Transitory powder flow dynamics during emptying of a continuous mixer

This article investigates the emptying process of a continuous powder mixer, from both experimental and modelling points of view. The apparatus used in this work is a pilot scale commercial mixer Gericke GCM500, for which a specific experimental protocol has been developed to determine the hold up in the mixer and the real outflow. We demonstrate that the dynamics of the process is governed by the rotational speed of the stirrer, as it fixes characteristic values of the hold-up weight, such as a threshold hold-up weight. This is integrated into a Markov chain matrix representation that can predict the evolution of the hold-up weight, as well as that of the outflow rate during emptying the mixer. Depending on the advancement of the process, the Markov chain must be considered as non-homogeneous. The comparison of model results with experimental data not used in the estimation procedure of the parameters contributes to validating the viability of this model. In particular, we report results obtained when emptying the mixer at variable rotational speed, through step changes

Intensification of vibration mixing of particulate solids by means of multi-layer loading of components

International audienceThe objective of the study is to show how initial distribution of dissimilar particulate components influences the mixing time and mixture quality. The dissimilar components have a tendency to segregate in one another, and it is impossible to achieve the perfect mixture of them in industrial settings. Nevertheless, the situation can be improved if the components are loaded as a sequence of several sandwiches, each of these sandwiches containing layers of components that are proportional to their share in the mixture. In this case, a sort of pre-mixing occurs while still at the loading stage – which allows reducing the optimum mixing time and increasing the homogeneity of the mixture. The theory of Markov chains was used to simulate the mixing kinetics. It is shown that the number of loaded sandwiches has a very strong influence on the process efficiency. A loading device that can effectively realize multi-layer loading is proposed. The mixing kinetics for ternary mixture of glass beads was investigated experimentally at a lab scale vibration mixer. A one-time loading and a two-sandwich loading were compared. It was shown that the optimum mixing time and non-homogeneity of the mixture were reduced by half in the latter case

ТЕОРЕТИЧЕСКИЙ ПОИСК ОПТИМАЛЬНОЙ ЗАГРУЗКИ ПЕРИОДИЧЕСКОГО СМЕСИТЕЛЯ ДИСПЕРСНЫХ МАТЕРИАЛОВ

International audienceThe objective of the study is to investigate how the hold-up of particulate solids to be mixed in a batch mixer influences the mixture quality and mixer capacity. It is known that a small amount of components (i.e., a small hold-up) allows reaching better quality of a mixer but leads to small capacity of a mixer. It is particularly appreciably when it is necessary to mix the components, which have a strong tendency to segregate into each other. In this case the perfect mixture cannot be reached, and there exists the optimum mixing time, at which the mixture homogeneity reaches maximum. This optimum time increases with the hold-up increase. Thus, from the mixing as such viewpoint, it is better to mix components not in big portions one time but in small portions several times. However, the total time of a mixing process consists of the loading time, mixing time and discharge time. The loading time depends on many factors such as a dosage device design, feeders design, and others, while the discharge time is usually much smaller. Thus, the mixer capacity is determined not only by the mixing time but also by the loading time at least. In order to estimate the mixer capacity at a required mixture quality, a cell model based on the theory of Markov chains is used. It is shown that the optimum hold-up exists that provides the maximum mixer capacity, and tЦель настоящего исследования – выявить, как загрузка предназначенных для смешивания в периодическом смесителе дисперсных материалов влияет на качество смеси и производительность смесителя. Известно, что небольшие количества компонентов (то есть малая загрузка) позволяют обеспечить лучшее качество смеси, но приводят к меньшей производительности смесителя. Особенно это проявляется, когда необходимо смешать компоненты, склонные к значительной сегрегации друг в друге. В этом случае полностью однородная смесь вообще недостижима, и существует оптимальное время смешивания, при котором качество смеси достигает максимума. Это оптимальное время возрастает с ростом загрузки. Таким образом, с точки зрения собственно смешивания, предпочтительно смешивать компоненты не один раз большими порциями, а несколько раз малыми порциями. Однако, полное время процесса смешивания состоит из времени загрузки смесителя, времени собственно перемешивания и времени разгрузки. Таким образом, производительность смесителя определяется не только временем собственно перемешивания, но также, по меньшей мере, и временем загрузки. Для того, чтобы оценить производительность смесителя при заданном качестве смеси, использована ячеечная модель, основанная на теории цепей Маркова. Показано, что существует оптимальная загрузка, которая обеспечивает максимальную производительность смесителя, и эта оптимальная загрузка существенно зависит от времени загрузки компонентов

Structuring of Batch Mixer Loading to Improve Mixing Time and Mixture Quality of Solids

International audienceA simple yet informative model was built to estimate the influence of the initial distribution of dissimilar particulate solids to be mixed on the mixing time and mixture quality, and a way was searched to reduce the negative influence of segregation on the mixing process. The influence of segregation is particularly strong when it is necessary to mix a small amount of a key component with a large amount of a basic one. It is shown that the mixing time and mixture quality can be noticeably improved by introducing a premixing stage that consists in loading the key component in layers distributed over the basic component. A possible technical solution for such loading is also proposed. Experimental tests of such lab‐scale vibration mixer proved its efficiency

Markov-chain modelling and experimental investigation of powder-mixing kinetics in static revolving mixers

International audienceThis study aims to develop a general model that is able to describe powder flow and mixing in static mixers, regardless of the type of mixer or the mixing configurations. The process model is based oil a homogeneous Markov chain describing the flow of each component through the mixing zone by a series of interconnected cells. It accounts for the number of mixing elements and their disposition in the mixer. as well as particle segregation via different transition probabilities. Some simulations are given to emphasize this particular aspect. Other outcomes of the model include the number of passages to reach a required mixture quality. as well as the asymptotic distribution of components. A laboratory static mixer of revolving type was designed specially for this study. It comprises up to 10 mixing sections, and its high internal voidage favours free flow of the powder. Segregating and non-segregating mixtures have been used to test the model and adjust unknown parameters. The model gives very satisfying results. In particular, it is able to account for the oscillating character of mixing kinetics due to particle segregation. It is also suggested that these parameters could be linked separately to powder flowability and mixing element characterization

A 2D Markov chain for modelling powder mixing in alternately revolving static mixers of Sysmix (R) type

International audienceA two-dimensional model of the flow and mixing of particulate solids has been developed on the basis of the Markov chain theory for an alternately revolving static mixer of Sysmix (R) type. In such a system, mixing occurs in both vertical and horizontal directions. Simulations are presented here to investigate the effect of the initial loading of the components, as well as the effect of the values of the transition probabilities that constitute the main parameters of the model. It is shown that a horizontal arrangement of the components always leads to better mixture quality and improved mixing kinetics. This research is presented for non-segregating mixtures, as well as potentially segregating mixtures. for which the empirically well-known oscillations in variance are represented by the model. Results suggest that there is a rational way of approaching a static-mixing problem with regard to the initial loading of the component and the optimal number of revolutions. Comparison of model results with experimental data published previously for a Sysmix (R) apparatus contributes to validating the viability of the model

ТЕОРЕТИЧЕСКОЕ ИССЛЕДОВАНИЕ ВЛИЯНИЯ ПАРАМЕТРОВ СМЕШИВАНИЯ НА ВРЕМЯ СМЕШИВАНИЯ И КАЧЕСТВО СМЕСИ РАЗНОРОДНЫХ ДИСПЕРСНЫХ МАТЕРИАЛОВ

International audienceBackground. Mixing of particulate solids is widely spread in many industries. Mixing can be realized as an independent process to obtain a homogeneous mixture or half-finished product, as well as an accompanying process in particulate solids treatment, for instance, in coal energetics. Any mixing is the combination of two sub-processes: pure diffusion mining that leads to equalizing of components distribution over a mixture volume, and segregation that leads to exfoliation of the mixture. Despite the models of mixing taking into account the both components of the process are known, their separate influence on mixture quality formation is not practically investigated that makes it difficult to choose the rational parameters of a mixture mechanical agitation to reach as much homogeneity as possible. It is obvious that, at present, such analysis deserves special attention.Materials and methods. The method of mathematical modeling and numerical experiments is used to solve the above problem. The model is based on the theory of Markov chains. The process of mixing is presented as a discrete one in space and time. The matrix of transition probabilities is presented as multiplication of two matrices: one for pure diffusion mixing, and another one for segregation mixing. The state of the mixture is presented as a column vector. Its homogeneity is characterized by the standard deviation. The recurrent calculations using the matrix equality allow estimating the evolution of the mixture state and the optimum mixing time corresponding to the maximum homogeneity of the mixture. Results. The dependence of the maximum reachable homogeneity and the time when it can be reached on the intensity of diffusion and segregation mixing is found. It is shown that one and the same maximum homogeneity can be reached at different combination of these intensities. It is found that the rate of segregation mixing is much higher than of diffusion one. However, the maximum value of homogeneity itself in this case is much worse in comparison to the case when diffusion mixing prevails.Conclusions. The proposed model allows finding the rational combination of diffusion and segregation intensity of mixing to reach a required mixture quality and, with orientation on them, choosing the rational way of mixture agitation to obtain it.Процессы смешивания дисперсных материалов широко присутствуют в различных отраслях промышленности. Они реализуются как самостоятельные процессы для получения однородных смесей и полуфабрикатов, а также как сопутствующие процессы при переработке дисперсных сред, например, в угольной энергетике. Любое перемешивание состоит из комбинации двух процессов: чисто диффузионного перемешивания, ведущего к выравниванию распределения компонентов по объему смеси, и сегрегации, ведущей к расслоению смеси. Несмотря на то, что модели смешивания с учетом обоих составляющих процесса известны, их отдельное влияние на формирование качества смеси практически не исследовано, что затрудняет выбор рационального механического воздействия на смесь для достижения ее максимально возможной однородности. Очевидно, что такое исследование в настоящее время требует специального внимания. Для решения поставленной задачи используется метод математического моделирования и численного эксперимента. Модель построена на основе теории цепей Маркова. Процесс смешивания представлен дискретным в пространстве и времени. Матрица переходных вероятностей представлена произведением двух матриц: для чисто диффузионного перемешивания и для сегрегационного перемешивания. Состояние смеси представлено вектор-столбцом. Ее однородность характеризуется среднеквадратичным отклонением. Расчеты по рекуррентному матричному равенству позволяют оценивать эволюцию состояния смеси и определять оптимальное время смешивания, соответствующее максимальной однородности смеси. Найдена зависимость максимально достижимой однородности смеси и времени ее достижения от интенсивности диффузионного и сегрегационного перемешивания. Показано, что одна и та же максимальная однородность может быть достигнута при разных комбинациях этих интенсивностей. Выявлено, что скорость сегрегационного перемешивания значительно выше диффузионного. Однако само значение максимальной неоднородности заметно хуже по сравнению со случаем, когда диффузионное перемешивание превалирует. Предложенная модель позволяет находить рациональную комбинацию интенсивности диффузионного и сегрегационного перемешивания для достижения требуемого качества смеси и, ориентируясь на этот выбор, подбирать рациональные способы воздействия на смесь для ее получения

Powder flow and mixing in a continuous mixer operating in either transitory or steady-state regimes: Mesoscopic Markov chain models

Continuous powder mixing is gaining interest in the industrial community concerned with more and more functional powder products. The understanding of powder flow and mixing/segregation of particles as well as their translation into models that can be used in process monitoring and control is a major issue. In the present work, we describe the development of different mesoscopic Markov chain models that are based on interconnected compartments or cells delimited in themixing chamber. The general structure of the chain allows the derivation of either homogeneous or non-homogeneous markovian models, for which transition probabilities are state-dependent. The models can be adapted to simulate variations of outflow rate, outlet mixture composition, hold-up weights and the distribution of these at the level of the compartments, during processing, including stationary and transitory phases. This is applied to a Gericke 500 GCM®continuousmixer for either pure powders or theirmixtures, in the latter case through the consideration of a Markov chain for each component. The models are fed by independent experiments that allow for the determination of the probabilities and the rules governing their change with the processing step, in particular during the transitory regimes. Agreement is found between model calculation and experimental data for a wide range of configurations. The models can catch the variations of hold-up weights and internal or outlet flowrates at any rotational stirrer's speed during mixer start and steady state. They can reproduce the variations of the outflow rate, and therefore mixture composition, when dealing with a mixture of two components. This is also presented for two nominal compositions. Conclusions are drawn in terms of processmonitoring and control. It gives insights for process intensification, in particular for mixer design and the feeding configuration

Un outil de modélisation systémique pour les procédés mettant en jeu des solides divisés : les chaînes de Markov

National audienc