Simulation of the Bulk and Granular Materials Separation Process in the Scissor Type Gravity Separator

The separation of granular materials into specific fractions by size on sieve classifiers is a large and complex system both in terms of the separation process and hardware design. The separation process depends on many design and operating parameters, the shape and size of the sieve cells, the number of sieves, as well as the fractional composition, shape and particle size of the material to be separated, i.e. is a cyber-physical system (CPS). The key to CPS is the mathematical model of the separation process, which is used in the control system. A mathematical model of the process of separation of granular materials on sieve classifiers based on the theory of random processes is developed. As a random process, the linear particle density of the considered fractions on the sieve surface is considered and a system of stochastic differential equations is constructed for its determination. The obtained solutions allow us to determine the recovery coefficient and evaluate the separation efficiency. Based on the constructed mathematical model, the design and operational parameters of the classifier were optimized. The performance criteria and separation efficiency are considered as optimization criteria. All this allows us to control the process of separation of granular materials by determining the optimal values of the operating parameters of the classifier depending on the fractional composition, shape and size of the particles of the original material to be separated and its other characteristics.

The article provides a substantiation of the conceptual model of the production process of special forms of dumplings. It is the first stage in the development of a model of this process as a control object. The purpose of which is to form an objective basis for the development of an effective system for automatic process control. The development of the conceptual model presupposes the specification and description of the properties of control channels and acting disturbances to the level of their mathematical model, which can be implemented in a simulation environment. Problems of identification of the mathematical model of the process of the production of dumplings, i. e. obtaining a mathematical description of processes based on the results of its purposeful experimental research, due to its complexity as a control object.The experimental approach, in this case, gives much more reliable results on the properties of the process. An attempt to obtain such general properties on the basis of experimental data would inevitably lead to the need for very complex and lengthy multifactorial experiments and nontrivial procedures for their processing. But this will leave open the question of the adequacy of the model for those conditions of the process and types of raw materials that were not covered by the experiments. Fundamentally important is the fact that the mathematical model of the process is developed as a model of the control object.Model can be used in two ways. This is due to the fact that in the closed circuits of the SAC, the discrepancy between the models can be considered as a manifestation of uncontrolled coordinate and parametric disturbances. It is in conditions of this kind of disturbances that the SAC must fulfill its functional purpose. The developed mathematical model of the production process of special forms of dumplings will be used by us only in the direction, when it is of great importance not so much quantitative as its qualitative correspondence to the original object.

For a broad range of applications sieving/screening is well suited to separate bulk materials according to particle sizes. In the treated bulk materials particles frequently prevail in broad size distributions, with non-spherical shape and sometimes even under moist conditions, complicating the separation process. Therefore, it is inevitable to gain a deeper understanding of the subprocesses of screening (size based stratification, particle passage through the screen surface and possible transport along the screen) under the aforementioned conditions. To gain this knowledge, detailed particle-based simulation approaches like the discrete element method (DEM) are available. Based on the latter method, discontinuous and continuous screening as well as its subprocesses are investigated. Therein, different screen geometries and characteristics are considered along with various mechanical excitations applying model and real particle shapes first under dry conditions and later under the influence of various liquid amounts. In order to perform reliable DEM screening simulations, the exact determination of particle properties like size, shape, material and contact parameters is essential, which is required in advance of the simulations. Besides the DEM, the integral outcome of screening can be represented by various phenomenological process models. Usually, the material-, operating-, and apparatus-specific parameters of the latter process models are empirically determined by experiments, whereas, here, the parameters for screening process models are directly obtained from DEM simulations, which allows their benchmarking under defined conditions. Additionally, suitable process models are successfully extended to represent screening processes under the presence of moisture.

The paper deals with the problem of mathematical modeling of the process of thermochemical destruction using the theory of graphs. To synthesize a mathematical model, the Markov chain is used. For the formalization of the model a matrix-graph method of coding is used. It is proposed to consider the process of destruction as a random process, under which the state of the system changes, characterized by the proportion of macromolecules in each fraction of the molecular mass distribution. The intensities of transitions from state to state characterize the corresponding rates of destruction processes for each fraction of the molecular weight distribution (MWD). The processes of crosslinking and polymerization in this work have been neglected, and it is accepted that there is a probability of transition from any state with a lower order index (corresponding to fractions with higher molecular weights) to any state with a higher index (corresponding fractions with lower molecular weights). A computational formula is presented for estimating the number of arcs and model parameters from a given number of fractions of the molecular weight distribution of the polymer. An example of coding in a matrix form of a graph model of the process of degradation of polybutadiene in solution for the case of six fractions of the molecular weight distribution is shown. As the simulation environment, the interactive graphical simulation environment of MathWorks Simulink is used. To evaluate the parameters of the mathematical model, experimental studies of the degradation of polybutadiene in solution were carried out. The chromatography of the polybutadiene solution was used as the initial data for the estimation of the MWD polymer. The considered matrix-graph representation of the structure of the mathematical model of the polymer destruction process makes it possible to simplify the compilation of the model and its software implementation in the case of a large number of vertices of the graph describing the process of destruction

The underground coal gasification is a continually evolving technology, which converts coal to calorific gas. There are many important parameters in this technology, which are difficult to measure. These parameters include the underground cavity growth, amount gasified coal, and the leakage of input and output gaseous components into the surrounding layers during the coal gasification process. Mathematical modeling of this process is one of the possible alternatives for determining these unknown parameters. In this paper, the structure of the mathematical model of laboratory underground coal gasification process from the material balance aspect is presented. The material balance consists of mass components entering and leaving from the UCG process. The paper shows a material balance in the form of a general mass balance and atomic species balance. The material balance was testing by six UCG laboratory experiments, which were realized in two ex-situ reactors.

Introduction Chapter I. General ideas of mass transfer processes in critical regimes 1. Granulometric characteristics of bulk material 2. Distribution of different fractions in the process of separation 3. Fractional separation curves and their properties Chapter II. Principles of modeling processes in moving media 1. Correlation between a full-scale process and its model 2. Mathematical models construction 3. Similarity criteria determination Chapter III. System of particles of the same size class in a critical flow. 1. Dynamics of mass motion of particles in a flow 2. Definition of a statistical system 3. Estimation of the state of a statistical system 4. Principal statistical characteristics of the separation factor Chapter IV. System of particles of several size classes 1. Interaction of particles in a flow 2. Forces caused by interactions of particles of various classes 3. Two-phase flow entropy in critical flow regimes 4. Main features of entropy in critical regimes 5. Mobility factor 6. Statistical identities Chapter V. Principal statistical relationships of mass transfer in critical flow 1. Mass exchange between the zone and the apparatus 2. Determination of average values 3. Cell and apparatus, entropy 4. Separation at low concentrations 5. General regularities for the zone Chapter VI. Correlation between the apparatus and the cell. 1. Coarse particles separation 2. Fine particles separation 3. Definition of mass transfer parameters 4. Cellular model of separation 5. Physical meaning of separation factors 6. Extraction from a cell located in the zone Chapter VII. Structural model of mass transfer in critical regimes of two-phase flows. 1. Validation of the distribution coefficient 2. Physical meaning of the distribution coefficient 3. Analysis of distribution coefficient 4. Analysis of experimental dependencies from the standpoint of structural models 5. Check of the structural model adequacy 6. Correlation between the structural and cellular models of the process Chapter VIII. Correlation between statistical and empirical results 1. Approximation of universal separation curve 2. Principal separation parameters depending on the apparatus height 3. Equal extractability of various size classes Chapter IX. Entropy of composition. Optimization criterion. 1. Entropy and particles stratification 2. Evaluation of heterogeneity of powder composition 3. Binary separation 4. Multi-product separation 5. Algorithms of optimization of separation into n components 6. Mathematical model of separation into n components 7. Optimum conditions for binary separation 8. Optimum conditions for multi-product separation Chapter X. Stability and kinetic aspects of mass distribution in critical regimes 1. Entropy stability 2. Particles distribution over the channel height 3. Velocity distribution of particles of a narrow size class 4. Kinetic aspect of the material distribution Chapter XI. Critical regimes of two-phase flows in complicated systems 1. Problem setting 2. Mathematical model of a duplex cascade 3. Mathematical model of a cascade process allowing control of the effect of the material feed site on separation results 4. Cascade model with two or more material inputs into the apparatus 5. Combined cascade classifiers 6. Quality criterion for combined cascades 7. Fractal principle of the construction of schemes of combined classifiers 8. Some methods of combined schemes optimization Chapter XII. Stochastic model of critical regimes of two-phase flows 1. Principal definitions 2. Statistical description of gravitational separation in turbulent flows 3. Equations of particles motion taking into account their rotation around the center of mass in a turbulent flow 4. Description of one-dimensional stationary process of gravitation separation in a turbulent flow 5. One-dimensional model of a non-stationary process 6. Statistical equations of random process of gra

Recently, how the desert lizards run, hide or swim in the sand has attracted much attention of many scientists in granular matter field, and many valuable results have been published, except for the Phrynocephalus mystaceus, a type of the desert lizard, which can embeds itself into the sand through a motion mode which is completely different from other types of desert lizards. To illuminate the roles played by the spinning-mode in the Phrynocephalus mystaceus' motion in the sand, the three-dimentional (3D) numerical simulation using the Hertz model on the system, in which one sphere is spinning in the granular matter, is carried out with the open-source code LIGGGHTS released by the Sandia National Laboratory in USA. In the numerical simulations for all the cases, the initial conditions are the same and the sphere spins around X-axis while the X-Y plane is the horizontal plan and the Z axis is the vertical direction. According to the numerical results and analyses, for the spinning sphere deeply embedded in the granular matter we can draw some conclusions. 1) The X-axis spinning motion can cause the sphere embedded in the granular to notably displace along the Z-axis and Y-axis, but the displacement along the spinning direction is smaller than the sphere diameter. 2) The friction coefficient between the sphere and the granular matter has a notable influence on the motion of the sphere in granular matter, the spinning sphere can move vertically and horizontally only when the friction coefficient between the sphere and the granular matter is larger than that of the granular matter; and the bigger the , the more violent the movement of the sphere is. This can be used to explain why most of the desert creatures each have a coarse skin. 3) On the premise that the friction coefficient between the sphere and the granular matter is larger than that of the granular matter, the spinning velocity of the sphere also has a great influence on the movement of the sphere in the granular matter. In a spinning velocity range between 10 rad/s and 640 rad/s, the larger the , the more obvious the movement of the sphere is. When the spinning velocity reaches 1280 rad/s, the movement of the sphere slightly decreases compared with when the spinning velocity is 640 rad/s. 4) For the spining sphere in granular matter, the sphere always moves upward in the Z direction, but in the Y direction the sphere may move in a positive or negative direction depending on the and . The sphere moves in the positive direction of Y axis if the and are relatively small, while it moves in the negative direction if the and are larger.

The patterns of changes in the parameters of the movement of a part of a tank after separation from the bottom depending on its design parameters and the conditions of its filling were studied. Based on the conducted studies, a mathematical model of the separation process of a part of a tank was formed, which takes into account both geometric and physical nonlinearities of the material. The methods of mathematical modeling, numerical analysis and computer simulation were used for the selected study. Mathematical models allow for an accurate description of the stress-strain state, and numerical methods, in particular finite element and smoothed particle (SPH) methods, provide accurate modeling of the interaction between structural elements and the liquid. Numerical experiments were conducted using the LS-DYNA software package, which made it possible to determine the main parameters of the separation process, such as the maximum stresses in the welded joint zone and the distribution of deformations in the tank material. The results of numerical modeling showed that the main destruction occurs precisely in welded joints, due to the lower strength of the seam compared to the main tank material. This is consistent with the experimental data obtained in laboratory conditions. For a more detailed analysis, a full factorial experiment and regression analysis were conducted, which allowed us to identify the key factors that affect the separation process and establish empirical relationships between the load, material and geometry parameters of the samples. The constructed regression model demonstrates a high correlation between the selected factors and the separation parameters of the tank. The results obtained can be used to predict the behavior of steel tanks during a fire and develop recommendations to improve their safety and strength, taking into account the conditions of real operational loads.

Gas separation via membrane has been recognized as the main technology that is used for hydrogen recovery, air separation, natural gas sweetening, helium recovery, natural gas dehydration, and so on. The successful application of a membrane-based separation process depends significantly on the appropriate chemical, physical, mechanical, and permeation properties of the membranes. Mathematical modeling of the membrane-based gas separation process can be useful to predict the performance under different conditions. Gas separation through membranes can take place by solution-diffusion mechanisms, depending mainly on the chemical and physical structures of the membrane.Gas transport through a mixed-matrix membrane (MMM) is a complicated problem. Different modeling attempts have been developed for the prediction of the performance of MMMs by various theoretical expressions depending on the MMM’s morphology and chemistry, including ideal and nonideal MMMs. Attempts have been made to predict the effective permeability of a gaseous penetrant through the MMMs as a function of continuous phase (polymer matrix) and dispersed phase (porous or nonporous particles) permeabilities, as well as volume fraction of the dispersed phase. Also various models and mechanisms for the solvent and solute transport through reverse osmosis membrane have been proposed by a number of investigators.

The paper deals with the problem of mathematical modeling of thermochemical destruction process. The apparatus of Markov's chains is used to synthesize a mathematical model. The authors of the study suggest to consider the destruction process as a random one, where the system state changes, which is characterized by the proportion of macromolecules in each fraction of the molecular- and weight distribution. The intensities of transitions from one state to another characterize the corresponding rates of destruction processes for each fraction of the molecular- and weight distribution. The processes of crosslinking and polymerization in this work were neglected, and it was accepted that there is a probability of transition from any state with a lower order index (corresponding to fractions with higher molecular weights) to any state with a higher index (corresponding to fractions with lower molecular weights). Markov's chain with discrete states and continuous time was taken as the mathematical model basis. Interactive graphical simulation environment MathWorksSimulink was used as a simulation environment. Experimental studies of polybutadiene destruction in solution were carried out to evaluate the mathematical model parameters. The GPC (gel-penetration chromatography) data of the polybutadiene solution were used as the initial (starting) data for estimating the polymer WMD (molecular weight distribution). Mean-square deviation of the calculated data from the experimental data for each fraction and at specified times was minimized for the numerical search of parameter values. The results of comparison of experimental and calculated on mathematical model data showed an error of calculations on the average about 5%, which indicates an acceptable error in estimating of polymer fractions proportions change during the process of destruction for the process under consideration and conditions.

This monograph reviews and examines the theoretical background and literature published so far for the mathematical modeling and simulation of the rubber vulcanization process. This work has been organized in the following way. First, an introductory section is given, which provides a brief overview of the topic and the aim and scope of the paper. Then, it goes on to give a comprehensive overview of the historical background up to present state. It is tried here not only to consider rubber vulcanization process, but the curing of other thermoset polymers are examined. The third section presents the mathematical modeling of the curing process and focuses on four key items including heat transfer governing equations, thermal properties of rubbers and their measurements, kinetics of the curing reaction and solution methodology. In this part, the details of the solution methods for the governing equations and also different types of kinetics of curing reactions are studied. The fourth section presents the findings of some research works carried out in this field, which is divided into two subsections as tires and non-tires applications. A brief summary on the presented work and some suggestions for future research are introduced in the fifth section, and finally the bibliography of this review is given.

The article discusses the option of modelling a business process using a logistics concept to optimise costs. The issues of influence on the formation of the model, the concept of optimising transport costs, and the costs of inventory management, in terms of production and economic efficiency, are considered. The mathematical and economic model of a typical business process is shown as a formed economic paradigm that materialises the process of conceptual integration of technological and logistic approaches in the process of implementing critical business processes of the company's current activities.The article also touches upon the issues of calculating logistics costs, both in the context of their points of origin, that is, by functional areas, and general accounting in the context of each business process. The options for calculating this category of costs are considered, in the general flow of both material and financial resources of a separate business process. From the standpoint of a logistic approach to managing the total operating costs of key business processes, options for a comprehensive study of the impact of logistics costs on the general operating activities of the company are analysed.The methodology for constructing a mathematical-economic model of a typical business process is described, taking into account the factors of influence of logistics concepts. Practical use of this methodology in real economic conditions is considered, and recommendations are given for adapting the methodology to specific parameters of the subject of the economic system. The proposed model outlines the fundamental approaches to the mathematically verified analysis of vital economic parameters that characterise the fluid nature of the economic components of the modern free market. The stated methodology, while not claiming to be complete coverage of the issues presented, nevertheless assumes the possibility of practical implementation of simulation modelling, economically crucial for current operational activities, fundamental business processes.

The most important features of the self-lubricating bearings are the antifriction properties such as friction coefficient and wear resistence and some mechanical properties such as hardness, tensile strength and radial crushing strength. In order to improve these properties new antifriction materials based on iron-copper powders with several additional components (tin, lead and molybdenum disulphide) have been developed by PM techniques. To find the optimal relationship between chemical compositions, antifriction and mechanical properties, in this paper a mathematical model of the sintering process is developed, which highlighted the accordance of the model with data by regression analysis. For the statistical processing of the experimental data the VH5 hardness values of the studied materials were considered. The development of mathematical model includes the enunciation of the model, the establishment of the performance function (optimization) and the establishment of the model equations and verifying. The accordance of the model with experimental data has been highlighted by regression analysis

A complete mathematical modeling of the technological process of manufacturing vinyl acetate monomer by vapor-phase method has been implemented. The model is partial differential equations of the material and energy balance and catalyst aging. An algorithm for an integrated equation based on the second-order Gregg-Bulirsh-Shter rectangle symbol has been developed. The error in modeling the process is equal (CH3COOCH=CH2) ± 5.4% rel. for the target matter, and (CO2) ± 6.4% rel. for the secondary substance The kinetic coefficients of differential equations, namely the pre-exponential factors and the activation energy have been refined. The obtained data of the modeling of the vapor-phase manufacturing process of vinyl acetate are shown in the form of tables and graphic three-dimensional dependencies of quality indicators on technological parameters.

Chapter 3 - Mathematical and Physical Modelling of the Flat Rolling Process