Mathematical Model Of Process Research Articles

APPLICATION OF THE SAMUELSON EQUATION TO THE EVANS MODEL

Mathematical modeling of economic processes is an actual direction of research, because the well-being of citizens and the country as a whole depends on it. In the case of the market, the prices of most goods and services are not planned centrally, are not directly regulated by the state, but are freely set and changed by the market itself. The main factors that control the movement of prices in the market are the demand and supply of goods. In the economy, the most important are dynamic models, the parameters of which change over time. The Evans model (Walras-Evans-Samuelson model) is currently one of the basic concepts explaining the dynamic establishment of the equilibrium price in the market of one product under the influence of supply and demand. This is due to the fact that knowing the dynamics of the economic parameter we are interested in, we can try to build a forecast of its further evolution. The article examines the market of one product. For convenience, we will assume that the functions of the dependence of demand and supply on the price are given by linear relationships. The construction of the Evans model is that the change in price is directly proportional to the excess of demand over supply and the duration of this excess. The Samuelson differential equation with an initial condition, that is, the Cauchy problem, is obtained. Samuelson's equation has a stationary (equilibrium) point, which is a positive price at which supply and demand will be equal. The analysis of the obtained solution of the problem shows that over a long enough time (relatively speaking, at ) the price asymptotically approaches the equilibrium value. If we are not interested in the temporal dependence, but only in the equilibrium price, then it can be found from the differential equation immediately by setting the condition , this is the so-called limit stationary mode. The solution of the Cauchy problem is found by the method of variation of the constant. The parameterization of the model using the fractional differentiation operator in the sense of Gerasimov-Caputo is considered. The resulting solution was analyzed depending on the parameter . For this, the asymptotic representation of the function was used for large values of the argument. Conclusions are made regarding price dynamics in time relative to the equilibrium price when demand and supply are equal.

Mathematical Model of Oxygen Regime Analysis in a Bioreactor

Purpose. Today, aeration tanks are widely used to treat industrial and municipal wastewater. Assessing the efficiency of water treatment in aeration tanks under different operating conditions is an important task. To solve this problem, it is necessary to have mathematical models that allow to quickly obtain data on the effect of treatment in aeration tanks. The main objective of this work is to develop a numerical box model to determine the efficiency of wastewater treatment in an aeration tank. Methodology. For mathematical modeling of the biological wastewater treatment process in a bioreactor, zero-dimensional material balance equations are used, which are written in relation to the concentration of substrate, activated sludge, and dissolved oxygen in wastewater. The Monod model is used to calculate the substrate oxidation process. The Euler method is used to integrate the modeling equations. A simplified model of biological water treatment in a bioreactor is considered, which makes it possible to obtain an analytical solution to the problem. The obtained analytical expression makes it possible to quickly determine the change in the concentration of dissolved oxygen in wastewater depending on the change in the concentration of activated sludge in the bioreactor. The constructed numerical model makes it possible to determine the dynamics of changes in the concentration of contaminants, activated sludge and oxygen in wastewater during their stay in the bioreactor. Findings. A tool for theoretical evaluation of the efficiency of biological wastewater treatment in an aeration tank is proposed. A numerical model has been built that allows determining the concentration of dissolved oxygen, substrate and activated sludge at the outlet of the aeration tank. Originality. An effective numerical model has been developed that allows to quickly calculate the dynamics of wastewater treatment in a bioreactor and, based on this information, to evaluate the efficiency of the reactor. Practical value. The constructed mathematical model can be useful in the reconstruction of biological wastewater treatment facilities, as well as at the design stage of bioreactors to take into account their operation under different operating conditions. A computer program has been developed that implements the constructed numerical model. The results of a computer experiment are presented.

Comparative analysis of two-dimensional and three-dimensional modeling of heat transfer during operation of a gas infrared heater indoor

Relevance. It is proposed to heat local work areas with systems based on gas infrared heaters, capable of directing radiative heat flow to reduce heating costs in large premises. However, the widespread use of gas infrared heaters is hampered by the existing difficulties with the preliminary assessment of convective-radiative heat flows movement, on which the number and location of heating devices depends. The preliminary assessment is complicated by the need in some cases for 3D modeling of complex physical processes. It is necessary to evaluate the possibility of replacing labor-intensive 3D modeling with a method for calculating a heating system using gas infrared emitters based on a 2D approach to reduce the time spent on calculations. Aim. To prove that the use of a two-dimensional model of the processes under consideration makes it possible to obtain the main characteristics of the thermal regime of the premises, making it possible to replace spatial modeling. Objects. Heating system with a light-type gas infrared heater and an air exchange system. Methods. Two-dimensional and three-dimensional mathematical modeling of conjugate heat transfer processes using the finite element method. Mathematical modeling was carried out in the COMSOL Multiphysics software environment using the modules: “Heat Transfer Interface in Liquids”, “Radiation between Surfaces” and “Turbulent Flow, k-ε Interface”. Results. The article presents the results of mathematical modeling performed in three-dimensional and two-dimensional formulations. The distribution of temperatures in the air and enclosing structures, as well as the flow lines of heated air and air, which was heating, in the volume of the premise are presented. The results of two-dimensional and three-dimensional modeling were compared. Satisfactory similarity of the calculated average air temperatures in the local working area was established based on the results. The difference was less than 2℃ for different spatial modeling approaches.

Математическая модель процесса СВЧ и конвективной выпечки хлеба из сбивного теста

Whipped yeast-free bakery products require effective energy supply to dough in order to optimize energy consumption, baking time, and quality. This article introduces a verified mathematical model of microwave and convective baking for whipped bread based on heat and mass exchange equations.
 A full-scale experiment to verify the calculations involved dough samples with a humidity of 56 ± 1%. The samples underwent microwave and convective processing until the temperature in the crumb center reached 98 ± 1°C.
 The mathematical model was formalized as energy and mass conservation equations, which made it possible to consider baking as a non-stationary process of heat and mass transfer of moisture in an isotropic incompressible continuous medium in the diffusion approximation. The equation took into account the unstable phase transition boundary. The practical verification showed the mean error for microwave baking as 14.5% in temperature and 18.2% in moisture content. For convective baking, the results included 12.6% in temperature and 9.7% in moisture content. The mathematical model proved adequate to the real processes of heat and mass transfer. The error in calculating the temperature and moisture content fields was sufficient tooptimize the process.
 The physical and mathematical model of the baking process made it possible to evaluate the effect of technological variables on the temperature and moisture concentration fields in the dough samples. The mathematical model and the computational experiment can be used to identify static and dynamic characteristics of baking as an object of automatic control, i.e., to identify optimal control channels and actions, as well as to adjust the automatic control system to specific quality indicators.

FORMATION OF TWO-PHASE ZONES DURING INTERNAL OXIDATION OF BINARY ALLOYS

In the paper the formation patterns of two-phase regions during the internal oxidation process in low-alloyed binary alloys is studied. The goal of the research is the creation of a mathematical model for the internal oxidation process, which is to describe the kinetics of formation and evolution of internal oxidation zones, the study of the patterns governing the two-phase zone formation during internal oxidation, the study of the space-time dynamics of the structural parameters of the two-phase area, namely the number of the oxide particles, their average radius and the phase volume, at different stages of the internal oxidation zone evolution. The two-phase process of diffusion saturation with light components applied to binary alloys is analyzed theoretically. The conditions for formation of disperse oxide particles are formulated. A mathematical model for the formation of two-phase areas as well as the numerical solution for the system of equations describing the kinet- ics of formation and evolution of the two-phase area for different values of the process parameters is proposed. The modeling and experiments demonstrate that the distribution of structural parameters of the two-phase area has complex, non-monotonous nature. The influence of the main parameters of the process on the kinetics of the formation of the two-phase area in a lightly-alloyed binary nickel-based alloy plate is also determined. All stages of the plate’s oxidation process are studied – from the inception of the particles to the coalescence of second-stage particles. The results of the work could be used for developing new technologies in thermal and chemical-thermal treatments for disperse strengthening of materials using internal oxidation, as a physical basis for seeking ways of providing certain distributions of the internal oxidation zone structural parameters, which are required to give the al- loy a necessary set of physical and mechanical properties. The comparison of the modeling and experimental results shows a good correlation, which allows them to be recommended for developing new technologies in disperse strengthening, new heat-resistant steels, disperse-strengthened magnetic materials and electrotechnical alloys.

SIMULATION OF DYNAMIC PROCESSES IN A p - i - n DIODE BY THE METHODS OF PERTURBATION THEORY

In the paper a mathematical model of the dynamic process of the electron-hole plasma formation in the active region ( i - region) of a semiconductor p - i - n - diode in the direct bias mode with a harmonic signal fed on the diode is developed. The basis of the model is a nonlinear nonstationary sin- gularly perturbed boundary value problem for a system of continuity equations of electron-hole currents and Poisson's equation. The algorithm for searching the charge carrier concentration distributions and the potential distribution is based on the asymptotic method of boundary functions and the Fourier method. In the course of the research, a technique for decomposing a nonlinear problem based on the development of perturbation theory me- thods was proposed. The initial nonlinear problem is reduced to a recurrent sequence of linear stationary boundary value problems, solved by classical and partially original analytical-numerical methods. The identification of the boundary functions in the solution of the problem provides, in particular, in contrast to the classical approximation of ambipolar diffusion, an adequate description of the electric field strength behavior in the active region of p - i - n - diodes. The obtained research results shed light on the peculiarities of the formation of the impedance characteristics of p - i - n - structures. The results of the work reveal more deeply the nature of the physical processes in the studied technical system and are aimed at improving the methodology for modeling and designing control semiconductor devices.

Research on Prediction Model of Particulate Matter in Dalian Street Canyon

In urban areas where populations commonly reside, particle mass concentrations in street canyons can pose significant risks to human health. This study aimed to investigate the diffusion mechanism of particle mass concentrations in urban street canyons by developing and applying a prediction model based on the mathematical modeling of physical processes. The prediction model considered factors such as the influence of traffic wind, natural wind, traffic flow, and other relevant variables influencing particle mass concentrations in street canyons. Field measurements were conducted in Dalian, China, to verify the feasibility of the model. Particle mass concentrations, traffic flow, temperature, relative humidity, and wind speed were measured on Shichang Street (a two-lane one-way road), Tangshan Street (a four-lane two-way road), and Shengli Road (a six-lane two-way road). The results indicated that the majority of traffic peaks occurred around 19:00 on all road types. The PM1.0 mass concentration was well diluted on the four-lane two-way road, with the least dilution observed on the two-lane one-way road. A strong correlation between the particle mass concentrations and traffic flow was discovered. Furthermore, a prediction model was established, accurately predicting the particle mass concentrations when the prediction step was from 5 to 15 s. The coefficient of determination (R2) between the predicted and measured values on the two-lane one-way road, four-lane two-way road, and six-lane two-way road was 0.9319, 0.6582, and 0.9238, respectively. Additionally, the prediction model allowed for a detailed analysis of traffic flow limitations, corresponding to the recommended World Health Organization (WHO) PM2.5 values. Overall, the findings of this study offer valuable insights for forecasting particle exposure levels in street canyons.

Mathematical modelling of the emission characteristics of a field electron cathode in a scanning electron microscope under biosampling conditions

Currently, the use of electron microscopes in medicine is developing intensively, including scanning electron microscopes (SEM), which are designed to solve a huge number of problems in various fields with a wide range of electron accelerating voltages and electron beam energies. The development of an SEM with certain emission characteristics, with a range of lower beam energies for the study of biological samples, is an urgent task because modifying the SEM to solve problems in medicine, for example, would make it possible to obtain higher-quality images of biospecimens for diagnostics and monitoring the effectiveness of therapy. To develop new SEMs with certain characteristics, it is proposed to conduct less expensive research using numerical methods based on mathematical models of processes in electron-optical SEM systems. In this regard, this work sets the task of determining the size and shape of the beam, the main emission characteristics of the field electron cathode (FEC) of the SEM, which is under the influence of the electric field that excites electron emission and the external longitudinal magnetic field by studying the movement of the outermost electron of the beam, taking into account the influence of space charge beam electrons, external magnetic field. In the model, the FEC is approximated by a paraboloid of rotation, and the concept of a boundary “outermost” electron is introduced, the trajectory of which determines the shape and size of the beam. The problem of calculating the emission characteristics along the trajectory of the outermost electron of a FEC is solved using a mathematical model that includes the following equations: motion of the “outermost” electron, Maxwell outside and inside the beam, continuity of the current density, Fowler-Nordheim equation. As a result, a system of 18 first-order ordinary differential equations was obtained, the numerical calculation of which using the 4th order Runge-Kutta method allows us to obtain the emission characteristics of the FEC. As a result, it is suggested that it would be feasible to modify SEMs for more effective use in the medical field, taking into account their increasing use in disease diagnosis and the possible improvement of image quality through the development of FEC SEMs with more suitable characteristics.

Post-COVID breathlessness: a mathematical model of respiratory processing in the brain.

Breathlessness is among the most common post-COVID symptoms. In a considerable number of patients, severe breathlessness cannot be explained by peripheral organ impairment. Recent concepts have described how such persistent breathlessness could arise from dysfunctional processing of respiratory information in the brain. In this paper, we present a first quantitative and testable mathematical model of how processing of respiratory-related signals could lead to breathlessness perception. The model is based on recent theories that the brain holds an adaptive and dynamic internal representation of a respiratory state that is based on previous experiences and comprises gas exchange between environment, lung and tissue cells. Perceived breathlessness reflects the brain's estimate of this respiratory state signaling a potentially hazardous disequilibrium in gas exchange. The internal respiratory state evolves from the respiratory state of the last breath, is updated by a sensory measurement of CO2 concentration, and is dependent on the current activity context. To evaluate our model and thus test the assumed mechanism, we used data from an ongoing rebreathing experiment investigating breathlessness in patients with post-COVID without peripheral organ dysfunction (N = 5) and healthy control participants without complaints after COVID-19 (N = 5). Although the observed breathlessness patterns varied extensively between individual participants in the rebreathing experiment, our model shows good performance in replicating these individual, heterogeneous time courses. The model assumes the same underlying processes in the central nervous system in all individuals, i.e., also between patients and healthy control participants, and we hypothesize that differences in breathlessness are explained by different weighting and thus influence of these processes on the final percept. Our model could thus be applied in future studies to provide insight into where in the processing cascade of respiratory signals a deficit is located that leads to (post-COVID) breathlessness. A potential clinical application could be, e.g., the monitoring of effects of pulmonary rehabilitation on respiratory processing in the brain to improve the therapeutic strategies.

Mathematical modelling of transient processes in a HVAC transmission line considering the effect lightning protection cables in the state of two-phase short-circuiting

Mathematical modelling of transient processes in a HVAC transmission line considering the effect lightning protection cables in the state of two-phase short-circuiting

Extremal Problems for Hyperbolic Systems with Boundary Conditions Involving Integral Time Lags

Extremal problems for integral time lag hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which integral time lags appear in the Neumann boundary conditions are solved. Such systems constitute, in a linear approximation, a universal mathematical model for many processes in which transmission signals at a certain distance with electric, hydraulic and other long lines take place. The time horizon is fixed. Making use of Dubovicki-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

A generalized method aiming at predicting the polymer melt flow field in the metering zone of large-scale single-screw extruders

Single-screw extruders (SSE) are commonly used in a wide variety of applications, ranging from polymer-extrusion to pellet additive manufacturing (PAM). Existing mathematical models focus on Newtonian and power-law rheologies to model melt flow in the last screw vanes. However, molten polymers usually follow more complex rheological patterns, and a generalized extrusion model is still lacking. Therefore, a semi-analytical model aiming at describing the flow of molten polymers in SSE is presented, to encompass a wide range of non-Newtonian fluids, including generalized non-Newtonian fluids (GNF). The aim is to evaluate the molten polymer flow field under the minimum set of dimensionless parameters. The effect of dimensionless extrusion temperature, flow rate, channel width, and height on the flow field has been investigated. A full factorial plane has been chosen, and it was found that the impact of dimensionless flow rate is the most prominent. The results were initially compared to numerical computations, revealing a strong agreement between the simulations and the proposed GNF method. However, significant deviations emerged when employing the traditional power-law model. This is particularly true at high values of flow rate and extrusion temperature: the mean error on overall flow speed is reduced from 12.91% (traditional power-law method) to 1.04% (proposed GNF method), while keeping a reasonable computational time (time reduction: 96.70%, if compared to fully numerical solutions). Then, the predicted pressure drop in the metering section was benchmarked against established literature data for industrial-scale extruders, to show the model’s accuracy and reliability. The relative errors of the traditional model range between 34.33 and 62%. The proposed method reduces this gap (errors ranging between 5.34% and 10.97%). The low computational time and high accuracy of the GNF method will pave the way for its integration in more complex mathematical models of large-scale additive manufacturing processes.

Optimizing defect rework for a software start-up's schedule variation: a Six-Sigma-based approach

PurposeManaging project completion within the stipulated time is significant to all firms' sustainability. Especially for software start-up firms, it is of utmost importance. For any schedule variation, these firms must spend 25 to 40 percent of the development cost reworking quality defects. Significantly, the existing literature does not support defect rework opportunities under quality aspects among Indian IT start-ups. The present study aims to fill this niche by proposing a unique mathematical model of the defect rework aligned with the Six Sigma quality approach.Design/methodology/approachAn optimization model was formulated, comprising the two objectives: rework “time” and rework “cost.” A case study was developed in relevance, and for the model solution, we used MATLAB and an elitist, Nondominated Sorting Genetic Algorithm (NSGA-II).FindingsThe output of the proposed approach reduced the “time” by 31 percent at a minimum “cost”. The derived “Pareto Optimal” front can be used to estimate the “cost” for a pre-determined rework “time” and vice versa, thus adding value to the existing literature.Research limitations/implicationsThis work has deployed a decision tree for defect prediction, but it is often criticized for overfitting. This is one of the limitations of this paper. Apart from this, comparing the predicted defect count with other prediction models hasn’t been attempted. NSGA-II has been applied to solve the optimization problem; however, the optimal results obtained have yet to be compared with other algorithms. Further study is envisaged.Practical implicationsThe Pareto front provides an effective visual aid for managers to compare multiple strategies to decide the best possible rework “cost” and “time” for their projects. It is beneficial for cost-sensitive start-ups to estimate the rework “cost” and “time” to negotiate with their customers effectively.Originality/valueThis paper proposes a novel quality management framework under the Six Sigma approach, which integrates optimization of critical metrics. As part of this study, a unique mathematical model of the software defect rework process was developed (combined with the proposed framework) to obtain the optimal solution for the perennial problem of schedule slippage in the rework process of software development.

Mathematical modeling of non-linear reaction-diffusion process in autocatalytic reaction: Akbari-Ganji method

A theoretical model of the electroreduction of halogen oxoanions via autocatalytic redox mediation by halide anions is discussed. This autocatalytic process depends upon a system reaction-diffusion equation including a nonlinear term for homogeneous reactions. This study solves these equations using the robust and user-friendly analytical technique, the Akbari-Ganji method. The current and the redox component concentration are presented in terms of the rate constant. The influence of variables on current is examined using sensitivity analysis. The most important factor affecting current is the ratio between the bulk concentrations of halogen molecules and non-electroactive halogen oxoanions. The theoretically predicted results are compared with the numerical results. These approximate results enhance our understanding of system behavior.

Моделирование работы газотурбинной установки SGT5-4000F с исследованием влияния режимных и климатических факторов на устойчивость горения в камере сгорания

When regulating gas turbine units, it is necessary to be confident in the stable combustion process in the combustion chambers, considering the dynamics of the processes and changes of external climatic factors. Many scientific studies are devoted to the problems of experimental research and mathematical modeling of processes in combustion chambers. The possibilities of flame failure, expansion of the lower limit of combustion, determination of the stable position of the flame front and other issues have been assessed. Even though the problem of design of mathematical models and their use remains relevant. The purpose of this research is to model and study the influence of operation and climatic factors on the stability of combustion in the combustion chamber of the SGT5-4000F gas turbine unit. The subject of the research is the SGT5-4000F gas turbine unit with a low-emission combustion chamber. The simulation model of the gas turbine unit has been developed in the environment of dynamic modeling of technical systems SimInTech. The research is carried out using theoretical methods of calculating fuel combustion, thermodynamic foundations of the theory of gas turbine engines, methods of mathematical and simulation modeling, as well as data from the archive of automated process control systems. A simulation model of SGT5-4000F gas turbine unit has been developed. It is distinguished by the ability to assess the proximity of the compressor operating mode to the stability limit and determine the boundaries of stable combustion procedure in the combustion chamber, considering environmental climatic factors. The authors have studied the stability of the compressor and combustion chamber. The results of the experimental studies of a gas turbine unit in the operating load range from 113 to 282 MW and in the range of outside air temperature from –12 to 30 °C have shown that the combustion process in the combustion chamber is unvarying over the entire operational stress spectrum. The results obtained are verified by comparing model values with technological parameters of a gas turbine unit taken from the archive of the power plant control system. The authors have assessed the influence of electrical load on the performance indicators of a gas turbine unit at outdoor temperatures of +30, +15 and –12 °C. According to the results of simulation modeling, the model of SGT5-4000F gas turbine unit is found to be adequate. The developed model can be used as a tool at the stage of functional and technological design of automated control systems for the development of effective automatic control systems.

Selecting a stirrer and optimizing the process of mixing the substrate in the fermenter of biogas unit

The relevance of the work is related to the problem of soil degradation as a result of pollution by livestock waste and the need for timely disposal of waste into effective organic fertilizer and biogas. Increasing the share of the use of organic fertilizers makes it possible to develop organic farming technologies, and accordingly leads to the production of healthy crop products. The work investigated the process of substrate mixing in the fermenters of an operating pilot biogas plant. The technological line is designed for processing liquid organic waste, mainly manure or litter, by anaerobic digestion to produce innovative environmentally friendly organic fertilizers on an industrial scale. The purpose of the study was to select a stirrer and optimize the conditions for mixing the substrate in the fermenters of a biogas plant. The physicochemical properties of the substrate were studied by laboratory methods. The mathematical model of the mixing process is based on a semi-empirical description of the hydrodynamics of flows in the reactor, modernized taking into account the current state of the issue under the conditions of a fermenter with specific geometric characteristics. As a result of the work carried out, the optimal options for the geometric dimensions of the stirrers were determined (three variants for three-blade stirrers intermittent with a blade angle of 30°, differing in the size of the impeller and rotation speed) and the parameters of the mixing process were calculated. Based on a laboratory experiment and the results of calculating the sedimentation rate, the frequency of turning on the stirrer (once a day) and the time of its continuous operation to homogenize the suspension were determined. The results of the work can be used in the design of mixing devices and substrate mixing modes in bioreactors.

Mathematical modeling of dynamic processes of agricultural mobile energy vehicles on an electric drive

The article considers the issues of modeling the processes of agricultural mobile energy vehicles (MEV) with electric drive (ED). A review of modern literature on the problem under study as well as questions on modeling the functional properties of mobile machines and improving the quality indicators of MEV work are presented. A mathematical model of the motion of the MEV with an electric motor is presented, as well as a description of the method used and a sequence of actions for conducting research. Preliminary theoretical studies of motion in different modes of operation have been carried out. The proposed model is convenient for implementation and calculation in any of the applied software products that support the modeling of dynamic systems with an electromechanical drive. The proposed model, the solution of which is based on the methods of numerical integration of systems in the Matlab Simulink software environment, made it possible to simulate the dynamic processes of electromechanical power transmission of MEV during various agricultural operations. With the help of this model, the analysis of electromechanical processes in transient and steady-state operating modes, as well as dynamic processes in the power transmission were carried out. Graphs of changes in the studied parameters of the MEV power transmission are obtained and elastic moments in the joints of the 5-mass design scheme are determined. The use of the model allows you to track the change in statistical characteristics when the conditions of the experiment change. The model has shown its operability when performing simulation of agricultural operations: fertilization, cultivation and sowing, and it can be used at the design stage to study the characteristics of dynamic processes of small-class traction MEV power transmissions with an electromechanical drive. With different parameters of the model, there is a change in the mathematical expectation of the angular velocity of the ED from 147.89 to 156.87 rad/s, a change in the mathematical expectation of the speed of the MES from 4.51 to 4.79 m/s.

Использование преобразованного уравнения электронейтральности при проведении технологических расчетов водоподготовки на ТЭС и АЭС

When performing technological calculations of water treatment units for thermal power plants (TPP) and nuclear power plants (NPP), it is necessary to determine the quality indicators of process water after the processing steps. The main difficulty is to calculate the concentrations of weak electrolytes since they are in chemical equilibrium with all forms of dissipation. Currently, for each type of processing, either simplified models or methods that solve a complex computational problem are used for calculations. It is necessary to develop a universal mathematical model that allows high accuracy to calculate the concentrations of weak electrolytes and the pH value for various types of water treatment. To develop the model the authors have applied the method of mathematical modeling of chemical-technological processes at thermal power plants and nuclear power plants. A mathematical model based on the transformed equation of electrical neutrality is proposed. It can be used to determine the equilibrium concentrations of dissociation forms of weak electrolytes (primarily water itself and carbonic acid) after various stages of water treatment. The proposed model allows, when conducting technological calculations of water treatment plants, to more accurately and universally calculate the concentrations of weak electrolytes and determine the pH value in process waters for various types of water treatment. The model can be used to determine the required doses of chemical reagents to ensure the required values of technological indicators (pH and Langelier index) in treated water.

DISCRETE GROUP METHOD FOR A MATHEMATICAL MODEL OF THE DIFFUSION IN SWELLING GELATIN

In this article, we will introduce the mathematical model of the diffusion process in a swelling medium that has already been modeled. The main purpose of this paper is introducing a new class of group transformations for solving the nonlinear emulsion equation. Thus, we introduce this method as the discrete group methods based on solvable class orbits. One of the crucial advantages of this method is that a transformation is sought, which reduces the equation being investigated to some standard form for which the methods of integration are known. In developing this approach, one may construct all canonical forms of solvable equations.