scholarly journals Analysis of application and improvement of methods to solve discrete models of Boltzmann equation

Vestnik IGEU ◽  
2021 ◽  
pp. 62-69
Author(s):  
V.P. Zhukov ◽  
A.Ye. Barochkin ◽  
A.N. Belyakov ◽  
O.V. Sizova
Keyword(s):  
Boltzmann Equation ◽  
Discrete Models ◽  
Joint Analysis ◽  
Technological Systems ◽  
Boltzmann Equations ◽  
The Boltzmann Equation ◽  
Reasonable Choice ◽  
Volume Method ◽  
Energy Balances

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.

Model Boltzmann Equations

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Simplified Models ◽  
Boltzmann Equations ◽  
The Boltzmann Equation ◽  
Special Solutions ◽  
State Of Affairs ◽  
Physical Features

This chapter deals with simplified models of the Boltzmann equation, aimed at reducing its mathematical complexity, while still retaining the most salient physical features. As observed many times in this book, the Boltzmann equation is all but an easy equation to solve. The situation surely improves by moving to its linearized version, but even then, a lot of painstaking labor is usually involved in deriving special solutions for the problem at hand. In order to ease this state of affairs, in the mid-fifties, stylized models of the Boltzmann equations were formulated, with the main intent of providing facilitated access to the main qualitative aspects of the actual solutions of the Boltzmann equation, without facing head-on with its mathematical complexity. As it is always the case with models, the art is not to throw away the baby with the tub water.

Discrete models of the boltzmann equation for mixtures

2000 ◽  
Vol 36 (7) ◽  
pp. 1027-1032 ◽  
Author(s):  
V. V. Vedenyapin ◽  
S. A. Amosov
Keyword(s):  
Boltzmann Equation ◽  
Discrete Models ◽  
The Boltzmann Equation

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

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.

scholarly journals On the effects of the reactive terms in the Boltzmann equation

1997 ◽  
Vol 15 (1) ◽  
pp. 79-87 ◽  
Author(s):  
C. J. Zamlutti
Keyword(s):  
Boltzmann Equation ◽  
Theoretical Treatment ◽  
Many Body ◽  
Boltzmann Equations ◽  
Proper Form ◽  
Current Trends ◽  
The Boltzmann Equation ◽  
The Many ◽  
Loss Mechanisms

Abstract. The effects of the production and loss mechanisms that affect the Boltzmann equations are considered by the inclusion of a reactive term. The necessary elements to develop a proper form for this term are revised and the current trends analyzed. Although no accurate theoretical treatment of the problem is possible due to the many body nature of it, important relations can be derived which, besides being representative of the quantitative aspects of the matter, are illustrative of the qualitative features of the phenomenon. The overall procedure is detailed in this revision.

scholarly journals The $102$-velocity model and the related discrete models of the Boltzmann equation

1986 ◽  
Vol 62 (10) ◽  
pp. 367-370
Author(s):  
Yasushi Shizuta ◽  
Machi Maeji ◽  
Akemi Watanabe ◽  
Shuichi Kawashima
Keyword(s):  
Boltzmann Equation ◽  
Velocity Model ◽  
Discrete Models ◽  
The Boltzmann Equation
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