Discrete models of the boltzmann equation for mixtures

To describe technological systems using models of Markov chains and discrete models of the Boltzmann equation it is necessary to determine the probabilities of transition of a system from one state to another. An urgent topic of a scientific research is to improve the accuracy of solving the Boltzmann equation by making a reasonable choice of probabilities of transition and admissible areas of their application. The strategy to model and determine the probabilities of transitions is based on the finite volume method, the ratios of the theory of probability and the joint analysis of material and energy balances. Considering the ratios of the theory of probability, the authors have obtained the refined formula for the probabilities of transitions over the cells of the computational space of discrete models of the Boltzmann equations in case of the description of technological systems. Recommendations to choose the area of application of the model are presented. The computational analysis has showed a significant improvement of the quality of forecasting when we implement the proposed dependencies and recommendations. The relative error of calculating the energy of the system is reduced from 8,4 to 2,8 %. The presented calculated dependencies to determine the probabilities of transition and recommendations for their application can be used to simulate various technological processes and improve the quality of their description.

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On the thermodynamics of the discrete models of the Boltzmann equation for gas-mixtures

Transport Theory and Statistical Physics ◽

10.1080/00411458808230873 ◽

1988 ◽

Vol 17 (4) ◽

pp. 423-442 ◽

Cited By ~ 11

Author(s):

Eugenia Longo ◽

Roberto Monaco

Keyword(s):

Boltzmann Equation ◽

Gas Mixtures ◽

Discrete Models ◽

The Boltzmann Equation

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MODELING AND CALCULATION OF GRINDING AND CLASSIFICATION OF MIXTURE OF DISSIMILAR COMPONENTS IN CLOSED MILLING CIRCUIT

IZVESTIYA VYSSHIKH UCHEBNYKH ZAVEDENIY KHIMIYA KHIMICHESKAYA TEKHNOLOGIYA ◽

10.6060/ivkkt.20196209.5934 ◽

2019 ◽

Vol 62 (9) ◽

pp. 102-109

Author(s):

Vladimir P. Zhukov ◽

Dmitriy А. Osipov ◽

Vadim E. Mizonov ◽

Tomasz Wyleciał

Keyword(s):

Boltzmann Equation ◽

Fluidized Bed ◽

Discrete Models ◽

Experimental Investigations ◽

Engineering Practice ◽

Actual Problem ◽

The Boltzmann Equation ◽

Particulate Solids ◽

The Impact

It is shown that the developed earlier model of joint grinding of dissimilar particulate solids based on the discrete models of the Boltzmann equation does not take into account the possible size and density classification of particles and the operating of technological equipment in a closed milling circuit. It is substantiated that the problem of improving the efficiency of separation of mixture components, based on the analysis of their treatment in a closed milling circuit is an actual problem in the field. Theoretical investigations were based on the discrete models of the Boltzmann equation and the methods of their solution related to the technology of treatment of dissimilar components mixture. A special program for measurements and experimental data treatment obtained at the laboratory fluidized bed jet mill and the impact mill was developed for the experimental investigations. In the course of computational and experimental study a mathematical model of classification of dissimilar components was proposed and tested as well as the algorithm of its implementation into the general model of a closed milling circuit. The computational investigations were carried out to estimate the model sensitivity to the input parameters disturbance. The directions of improving the efficiency of the objective component separation in the closed milling circuit with the fluidized bed jet mill are found. The developed model the closed milling circuit for grinding dissimilar particulate solids can be used in engineering practice with acceptable accuracy. It can be used to estimate the possible enrichment of particulate solids mixtures and to define the optimal technological conditions of their separation with taking into account the limitations on the objective product impurity. The paths are outlined and the order of selection of the parameters of the grinding processes and the classification for effective separation of the components of the mixture are shown.

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