scholarly journals Method of integro-differential equations for interpreting the results of vertical electrical sounding of the soil

2021 ◽  
pp. 67-70
Author(s):  
D. G. Koliushko ◽  
S. S. Rudenko ◽  
A. N. Saliba
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Power Plants ◽  
Current Source ◽  
Soil Surface ◽  
Observation Point ◽  
Practical Significance ◽  
Maximum Relative Error ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper is devoted to the problem of determining the geoelectric structure of the soil within the procedure of testing the grounding arrangements of existing power plants and substations to the required depth in conditions of dense development. To solve the problem, it was proposed to use the Schlumbergers method , which has a greater sounding depth compared to the Wenner electrode array. The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure. Methodology. To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil. Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained for conducting VES (the point current source and the observation point are located only on the soil surface). Results. Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of approximation: the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. Originality. Based on the obtained expression, a test version of the program was implemented in Visual Basic for Applications to interpret the results of VES by the Schlumberger method. To check the obtained expressions, the interpretation of the VES results was carried out on the territory of a 150 kV substation of one of the mining and processing plants in the city of Kriviy Rih. Practical significance. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements of power stations and substations.

Development of a Predictive Equation for Ventilation in a Wall-Solar Chimney System

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
David Park ◽  
Francine Battaglia
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Natural Ventilation ◽  
Thermal Conditions ◽  
Ambient Air ◽  
Maximum Error ◽  
Scale Model ◽  
Small Scale ◽  
Maximum Relative Error ◽  
Solar Chimney

A solar chimney is a natural ventilation technique that has potential to save energy consumption as well as to maintain the air quality in a building. However, studies of buildings are often challenging due to their large sizes. The objective of this study was to determine the relationships between small- and full-scale solar chimney system models. Computational fluid dynamics (CFD) was employed to model different building sizes with a wall-solar chimney utilizing a validated model. The window, which controls entrainment of ambient air for ventilation, was also studied to determine the effects of window position. A set of nondimensional parameters were identified to describe the important features of the chimney configuration, window configuration, temperature changes, and solar radiation. Regression analysis was employed to develop a mathematical model to predict velocity and air changes per hour, where the model agreed well with CFD results yielding a maximum relative error of 1.2% and with experiments for a maximum error of 3.1%. Additional wall-solar chimney data were tested using the mathematical model based on random conditions (e.g., geometry, solar intensity), and the overall relative error was less than 6%. The study demonstrated that the flow and thermal conditions in larger buildings can be predicted from the small-scale model, and that the newly developed mathematical equation can be used to predict ventilation conditions for a wall-solar chimney.

scholarly journals Numerical simulation of the non-regular mode of cooling a high-temperature metal billet by the flow of a gas-liquid medium

2018 ◽  
Vol 224 ◽  
pp. 04003
Author(s):  
Sergey Makarov ◽  
Vyacheslav Dement’yev ◽  
Tat’yana Makhneva ◽  
Elena Makarova
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
High Temperature ◽  
Liquid Medium ◽  
Liquid Flow ◽  
Control Volume ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Gas Liquid Flow ◽  
Regular Mode

A mathematical model of heat transfer at cooling a high-temperature metal billet from structural steel by the flow of a gas-liquid medium in a vertical circular channel is presented. The model has been built with the use of the continuum mechanics approaches and the theory of heat-mass transfer. The non-regular mode of cooling is considered. The results of the numerical parametric investigations of the heat transfer at cooling a metal billet are obtained for a standard regime of thermomechanical strengthening on the basis of the mathematical model of conjugate heat transfer in a two-dimensional nonstationary formulation accounting for the symmetry of the cooling medium flow relative to the longitudinal axis of a cylinder. The control volume approach is used for solving the system of differential equations. The flow field parameters are computed by an algorithm SIMPLE. For the iterative solution of the systems of linear algebraic equations the Gauss-Seidel method with under-relaxation is used. Taking into account evaporation in the liquid, the intensity of the change of the rate of cooling the material of the metal cylindrical billet by the laminar gas-liquid flow is analyzed depending on the time of cooling and the velocity of the gas-liquid flow.

scholarly journals Modeling of the Free-Surface-Pressurized Flow of a Hydropower System with a Flat Ceiling Tail Tunnel

Water ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 699
Author(s):  
Jianxu Zhou ◽  
Yongfa Li
Keyword(s):  
Mathematical Model ◽  
Free Surface ◽  
Water Surface ◽  
Wave Speed ◽  
Water Levels ◽  
Water Diversion ◽  
Design Stage ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Hydropower System

For a water diversion hydropower system with a flat ceiling tail tunnel with high elevation, during transient states with relatively low tail water levels, free-surface-pressurized flow inevitably appears and its transient characteristics have obvious effects on the system’s operating stability. Using Newton–Raphson linearization in the characteristic implicit format for modeling of the free-surface-pressurized flow in the tail tunnel, the mathematical models for necessary boundary conditions were derived and linear algebraic equations with a band coefficient matrix were grouped for further transient simulation. Then, a unified mathematical model was established for hydraulic transient analysis of the hydropower system with free-surface-pressurized flow. Combined with experimental research and numerical simulation, the wave speed for the free-surface-pressurized flow was experimentally analyzed for further correctness in the unified model, and by comparative analysis the hydraulic characteristics of the free-surface-pressurized flow in the flat ceiling tail tunnel were investigated. It was found that the derived mathematical model can basically represent water behaviors in the water-surface-pressurized flow, the wave speed for the mixed water-surface-pressurized flow can be set to approximately 50m/s, and with this correctness the numerical results are in good agreement with the experimental results. Therefore, the obtained mathematical model combined with an experimental wave speed or a reference wave speed of 50 m/s for the free-surface-pressurized flow is preferable during the design stage of the hydropower system.

scholarly journals MATHEMATICAL MODELING IN SOCIAL AND ECONOMIC SYSTEMS

T-Comm ◽  
2021 ◽  
Vol 15 (5) ◽  
pp. 38-45
Author(s):  
Mikhail I. Grachev ◽  
◽  
Vyacheslav G. Burlov ◽  
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Synthesis Method ◽  
Control Process ◽  
Economic Systems ◽  
Algebraic Equations ◽  
Managerial Decision ◽  
Linear Algebraic Equations ◽  
The Social ◽  
New Variables

In the modern rapidly changing world, there are constant changes in all spheres of human activity, including in the social and economic systems, such as education, law, health care. All areas require study for improving management and decision-making mechanisms in order to increase the efficiency of their functioning and further logic of action improving the process of functioning of processes in social and economic systems. The introduction of the sites of educational organizations into educational institutions of higher education (UHE) has led to the need for the person in charge of the site (LOU) of the organization to have a mathematical model of management decisions to counter emerging threats. In this paper, we will consider the process of forming a mathematical model of an administrative decision, which is obtained on the basis of synthesis from the transition states of the system using the Kolmogorov differential equations, by further transforming them into a system of linear algebraic equations (SLAE) and solving them by the Gauss method. Mathematical modeling is based on synthesis processes using the law of preserving the integrity of the object (ZSCO) and the natural science approach (ESA). The use of the synthesis method allows you to achieve the control goal based on the required performance indicators. The proposed mathematical model helps in solving three processes aimed at monitoring the occurrence of a problem in a controlled system, the process of recognizing the problem and the process of implementing a managerial decision to eliminate the problem. The final obtained mathematical solution for modeling the situation in the social and economic system helps to establish a model of LOU behavior depending on the current situation, which will lead to saving time resources and the possibility of its redistribution for solving other problems. The resulting mathematical model can be further complicated by adding new variables and conditions for the implementation of the control process.

scholarly journals Thermistor Temperature Measurement Network with a Small Number of Wires

2019 ◽  
pp. 284-292
Author(s):  
Vladimir V. Shaydurov ◽  
Anna A. Korneeva
Keyword(s):  
Mathematical Model ◽  
Computational Experiment ◽  
Computational Algorithm ◽  
Model Problem ◽  
Algebraic Equations ◽  
Multiple Points ◽  
Power Sources ◽  
Linear Algebraic Equations ◽  
Sequential Solution ◽  
Thermistor Temperature

The article proposes a new measuring circuit for temperature control at multiple points of the product using a sequence of thermistors and diodes with three connecting wires and two voltage sources. A mathematical model and a computational algorithm for calculating the resistance of thermistors are presented which consist in the formation and sequential solution of systems of linear algebraic equations for different ratios of voltages of two power sources. A model problem and the results of a computational experiment are considered

scholarly journals Noncontact Measurement for Radius of Curvature of Unpolished Lens

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Haifeng Liang
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Prediction Error ◽  
Convex Surface ◽  
Separation Distance ◽  
Radius Of Curvature ◽  
Maximum Relative Error ◽  
Spherical Lens ◽  
Parallel Lines ◽  
Lens Surface

A noncontact mathematical model to measure radius of curvature (ROC) of an unpolished spherical lens was proposed and also proved by experiments. This model gives ROC as a function of arcs radii and their separation distance, where the radii of the corresponding arcs could be acquired by taking coordinates of points on the arcs when two parallel lines of light project onto a lens surface. Our experiments demonstrated that the measured maximum relative error was 0.027% for a concave surface with a 38.19 mm ROC and 0.021% for a convex surface with a 97.75 mm ROC, which were all in agreement with those of theory prediction error. The suggested method presented a fast noncontact method for testing ROC of lens during coarse grinding and fine grinding.

scholarly journals Two-level mathematical models for determining the stress state and life plate with a hole

2021 ◽  
pp. 55-59
Author(s):  
Mariya Ihorivna Shapovalova ◽  
Oleksii Oleksandrovich Vodka
Keyword(s):  
Stress State ◽  
Yield Surface ◽  
Structural Damage ◽  
Theory Of Elasticity ◽  
Practical Significance ◽  
Material Structure ◽  
Algebraic Equations ◽  
Mathematical Methods ◽  
Plastic Deformations ◽  
Linear Algebraic Equations

Modern trends in the development of mechanical engineering and other industries related to the production of materials and structures with a given set of physical, mechanical, and technological properties are aimed at reducing material consumption, energy consumption, increasing accuracy, reliability, and competitiveness of the manufactured product. Therefore, the creation of mathematical methods for assessing the stress state of structural elements based on the analysis of the elastic characteristics of a material, taking into account the peculiarities of its internal microstructure, is an actual task. The considered algorithm includes the following stages: identification of strength parameters using data obtained from images of the material microstructure; study of the stress-strain state of the model based on the variational-difference finite element method; formation of a system of linear algebraic equations for solving the problem of analyzing the elastic properties of a material using the plane problem of the theory of elasticity; construction of the material yield surface for a series of tests based on the strength criteria of composite materials, taking into account the different resistance of the material under tensile and compressive loads. Based on the developed mathematical model, the SSS and the yield surface of the plate with a hole are estimated. Structural analysis is performed at the macro and micro levels. The occurrence of plastic deformations at the micro-level can lead to the development of cracks and structural damage at the macro level. As a result of the study, the probability of plastic deformation in the plate is determined, and the critical zones of the model are established. The practical significance of the results obtained is to create an approach to assessing the mechanical properties of a material, such as elastic modulus, shear modulus, Poisson's ratio, and their probabilistic characteristics following the internal material structure. The proposed approach contributes to the expansion of knowledge about the material and allows to increase the valuable information obtained by modeling. To assess the probability of plastic deformations, the generated method uses the entire set of probabilistic characteristics of the yield surface.

A boundary element method for 2-D dc resistivity modeling with a point current source

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 399-404 ◽  
Author(s):  
Shi‐Zhe Xu ◽  
Shengkai Zhao ◽  
Yi Ni
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Relative Error ◽  
Current Source ◽  
Normal Derivative ◽  
Relative Difference ◽  
Dc Resistivity ◽  
Maximum Relative Error ◽  
Resistivity Modeling ◽  
Element Method

A new boundary element method (BEM) is presented for 2-D dc resistivity modeling with a point source. When compared with previously published techniques, the new method has two main features: (1) The normal derivative of potential has been eliminated from the integral equation. The formulation of the present method is simpler and requires less memory and time than the previous published methods. (2) Multiple subsurface inhomogeneous bodies can be modeled. For a simple testing model, the maximum relative error of reciprocity test is 0.24% and the average relative error is 0.05%. For the same model, the maximum relative difference between the BEM solution and the finite‐element method solution is 1.14% and the average relative difference is 0.73%. For a field geoelectrical profile, the responses of the constructed model agree with the observed data quite well.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

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