Cendrine Gatumel. A Markov chain model of mixing kinetics for ternary mixture of dissimilar particulate solids. Particuology , Elsevier, 2017, 31, p. a b s t r a c t This 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.
The 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.
Целью настоящей работы является демонстрация нескольких оптимизационных задач периодического смешивания склонных к сегрегации сыпучих материалов, которые могут быть поставлены и решены на основе моделей, основанных на теории цепей Маркова. Для того, чтобы повысить адекватность моделирования и исключить некоторые физические противоречия, которые возникают при использовании линейных моделей, предложена нелинейная Марковская модель перемешивания сегрегирующих компонентов. Исследовано влияние начального распределения компонентов после их загрузки в смеситель на оптимальное время перемешивания и качество смеси. Показано, что однократная загрузка компонентов как два вертикальных слоя значительно хуже, чем их многослойная как несколько сэндвичей. Принимая во внимание то, что полное время обработки смеси включает время собственно перемешивания и время загрузки, которое пропорционально числу сэндвичей, было определено оптимальное число сэндвичей. Приведены экспериментальные свидетельства того, что многослойная загрузка может быть реализована в практике смешивания.
A 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.
The 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 this optimum hold-up strongly depends on the loading time. Forcitation:Mizonov V.E., Balagurov I.A., Berthiaux H., Gatumel C. Theoretical search for optimum hold-up in a batch mixer of particulate solids. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018. V. 61. N 4-5. P. 93-97
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