scholarly journals Temperature modes in a heat-sensitive plate with local heating

2021 ◽  
Vol 3 (2) ◽  
pp. 21-26
Author(s):  
V. I. Havrysh ◽  
◽  
R. R. Shkrab ◽  
Keyword(s):  
Thermal Conductivity ◽  
Fourier Transform ◽  
Boundary Conditions ◽  
Mathematical Models ◽  
Analytical Solutions ◽  
Local Heating ◽  
Specific Power ◽  
Heat Sources ◽  
Integral Fourier Transform ◽  
Spatial Coordinates

Nonlinear mathematical models for the analysis of temperature regimes in a thermosensitive isotropic plate heated by locally concentrated heat sources have been developed. For this purpose, the heat-active zones of the plate are described using the theory of generalized functions. Given this, the equation of thermal conductivity and boundary conditions contain discontinuous and singular right parts. The original nonlinear equations of thermal conductivity and nonlinear boundary conditions are linearized by Kirchhoff transformation. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, their analytical solutions in the images were determined. The inverse integral Fourier transform was applied to these solutions, which made it possible to obtain analytical expressions for determining the Kirchhoff variable. As an example, the linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, analytical relations were obtained to determine the temperature in the heat-sensitive plate. The given analytical solutions are presented in the form of improper convergent integrals. According to Newtons method (three-eighths), numerical values of these integrals are obtained with a certain accuracy for given values of plate thickness, spatial coordinates, specific power of heat sources, the thermal conductivity of structural materials of the plate, and geometric parameters of the heat-active zone. The material of the plate is silicon and germanium. To determine the numerical values of temperature in the structure, as well as the analysis of heat transfer processes in the middle of the plate due to local heating, developed software, using which geometric mapping of temperature distribution depending on spatial coordinates, thermal conductivity, specific heat flux density. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in the thermosensitive plate with local heating to the real physical process. The software also makes it possible to analyze such environments that are exposed to local heat loads in terms of their heat resistance. As a result, it becomes possible to increase it and to protect it from overheating, which can cause the destruction not only of individual elements but also of the entire structure. Keywords: temperature field; isotropic thermosensitive plate; thermal conductivity; heat-insulated surface; perfect thermal contact; local heating.

Mathematical Model of Determination and Analysis of Heat Exchange in Elements of Digital Technology Devices

2021 ◽  
Vol 43 (4) ◽  
pp. 37-50
Author(s):  
V.I. Havrysh ◽  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Fourier Transform ◽  
Heat Exchange ◽  
Analytical Solution ◽  
Heat Source ◽  
Point Heat Source ◽  
Integral Fourier Transform ◽  
Spatial Coordinates ◽  
Boundary Surfaces

A mathematical model of heat exchange analysis between an isotropic two-layer plate heated by a point heat source concentrated on the conjugation surfaces of layers and the environment has been developed. To do this, using the theory of generalized functions, the coefficient of thermal conductivity of the materials of the plate layers is shown as a whole for the whole system. Given this, instead of two equations of thermal conductivity for each of the plate layers and the conditions of ideal thermal contact, one equation of thermal conductivity in generalized derivatives with singular coefficients is obtained between them. To solve the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the plate, the integral Fourier transform was used and as a result an analytical solution of the problem in images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The obtained analytical solution is presented in the form of an improper convergent integral. According to Simpson's method, numerical values of this integral are obtained with a certain accuracy for given values of layer thickness, spatial coordinates, specific power of a point heat source, thermal conductivity of structural materials of the plate and heat transfer coefficient from the boundary surfaces of the plate. The material of the first layer of the plate is copper, and the second is aluminum. Computational programs have been developed to determine the numerical values of temperature in the given structure, as well as to analyze the heat exchange between the plate and the environment due to different temperature regimes due to heating the plate by a point heat source concentrated on the conjugation surfaces. Using these programs, graphs are shown that show the behavior of curves constructed using numerical values of the temperature distribution depending on the spatial coordinates. The obtained numerical values of temperature indicate the correspondence of the developed mathematical model of heat exchange analysis between a two-layer plate with a point heat source focused on the conjugation surfaces of the layersand the environment, the real physical process.

scholarly journals MATHEMATICAL MODEL OF HEAT EXCHANGE IN ELEMENTS OF DIGITAL DEVICES

2021 ◽  
Vol 3 (1) ◽  
pp. 15-21
Author(s):  
Havrysh Havrysh ◽  
◽  
W. Yu. W. Yu. ◽  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Fourier Transform ◽  
Heat Exchange ◽  
Analytical Solution ◽  
Heat Source ◽  
Point Heat Source ◽  
Integral Fourier Transform ◽  
Spatial Coordinates ◽  
Boundary Surfaces

A mathematical model of heat exchange analysis between an isotropic two-layer plate heated ba point heat source concentrated on the conjugation surfaces of layers and the environment has been developed. To do this, using the theory of generalized functions, the coefficient of thermal conductivity of the materials of the plate layers is shown as a whole for the wholesystem.Given this, instead of two equations of thermal conductivity for each of the plate layers and the conditions of ideal thermal contact, one equation of thermal conductivity ingeneralized derivatives with singular coefficients is obtained between them. To solve the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the plate, the integral Fourier transform was used and as a result an analytical solution of the problem in images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The obtained analytical solution is presented in the form of an improper convergent integral. According to Simpsons method, numerical values of this integral are obtained with a certain accuracy for given values of layer thickness, spatial coordinates, specific power of a point heat source, thermal conductivity of structural materials of the plate and heat transfer coefficient from the boundary surfaces of the plate. The material of the first layer of the plate is copper, and the second is aluminum. Computational programs have been developed to determine the numerical values of temperature in the given structure, as well as to analyze the heat exchange between the plate and the environment due to different temperature regimes due to heating the plate by a point heat source concentrated on the conjugation surfaces. Using these programs, graphs are shown that show the behavior of curves constructed using numerical values of the temperature distribution depending on the spatial coordinates. The obtained numerical values of temperature indicate the correspondence of the developed mathematical model of heat exchange analysis between a two-layer plate with a point heatsource focused on the conjugation surfaces of the layersand the environment, the real physical process.

scholarly journals MATHEMATICAL MODELS OF HEAT TRANSFER IN ELEMENTS OF TURBO GENERATORS (CONTINUED)

2020 ◽  
Vol 2 (1) ◽  
pp. 21-28
Author(s):  
V. I. Havrysh ◽  
◽  
B. O. Bilinskyi ◽  
O. S. Korol ◽  
R. R. Shkrab ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Temperature Distribution ◽  
Mathematical Models ◽  
Half Space ◽  
Analytical Solutions ◽  
Internal Heat ◽  
Normal Operation ◽  
Cylindrical Shape ◽  
Temperature Regimes

Previously developed [8] and presented new mathematical models for the analysis of temperature regimes in individual elements of turbo generators, which are geometrically described by isotropic half-space and space with an internal heat source of cylindrical shape. Cases are also considered for half-space, when the fuel-releasing cylinder is thin, and for space, when it is heat-sensitive. For this purpose, using the theory of generalized functions, the initial differential equations of thermal conductivity with boundary conditions are written in a convenient form. To solve the obtained boundary value problems of thermal conductivity, the integral Hankel transformation was used, and as a result, analytical solutions in the images were obtained. The inverse Hankel integral transformation was applied to these solutions, which made it possible to obtain the final analytical solutions of the initial problems. The obtained analytical solutions are presented in the form of improper convergent integrals. Computational programs have been developed to determine the numerical values ​​of temperature in the above structures, as well as to analyze the heat transfer in the elements of turbo generators due to different temperature regimes due to heating by internal heat sources concentrated in the cylinder volume. Using these programs, graphs are presented that show the behavior of curves constructed using numerical values ​​of the temperature distribution depending on the spatial radial and axial coordinates. The obtained numerical values ​​of temperature indicate the correspondence of the given mathematical models for determining the temperature distribution to the real physical process. The software also allows you to analyze media with internal heating, concentrated in the spatial figures of the correct geometric shape, in terms of their heat resistance. As a result, it becomes possible to increase it, to determine the allowable temperatures of normal operation of turbo generators, to protect them from overheating, which can cause the destruction of not only individual elements but also the entire structure.

scholarly journals Visualization of Contact Interaction of Nanobeams

2019 ◽  
Author(s):  
Максим Жигалов ◽  
Maxim Zhigalov ◽  
Виктор Апрыскин ◽  
Victor Apryskin ◽  
Вадим Крысько ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Boundary Conditions ◽  
Mathematical Models ◽  
Phase Portrait ◽  
Contact Interaction ◽  
Deformation Theory

The paper presents a visualization of the contact interaction of two Bernoulli-Euler nanobeams connected through boundary conditions. Mathematical models of beams are based on the gradient deformation theory and the theory of contact interaction of B. Y. Cantor. The visualization is based on Fourier transform and wavelet transform, phase portrait.

scholarly journals CONCEPTION QUASIDERIVATIVES IN THE PROBLEMS OF MATHEMATICAL MODELING OF HEAT TRANSFER

2020 ◽  
Vol 21 ◽  
pp. 79-85
Author(s):  
R. Tatsii ◽  
M. Stasiyk ◽  
O. Pazen ◽  
L. Shypot
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Closed Form ◽  
Critical Radius ◽  
Internal Heat ◽  
Heat Sources ◽  
Internal Heat Sources

In this paper, in closed form, the problems of determining stationary temperature fields in multilayer (flat, cylindrical and spherical) structures in the presence of discrete-continuous internal and point heat sources are solved. The one-dimensional differential equation of thermal conductivity in different coordinate systems is given through one parametric family of quasi-differential equations. It is assumed that the coefficients of the differential equation of thermal conductivity are piecewise constant functions. A system of two linearly independent boundary conditions is added to the equation, which in the general case are nonlocal. The solutions of such problems are constructive and are expressed exclusively through their initial data. The basic provisions of the concept of quasi-derivatives, the provisions of the theory of heat transfer, the theory of generalized systems of linear differential equations, elements of the theory of generalized functions are used. For the mathematical model of stationary thermal conductivity, the practical use of the concept of quasi-derivatives is illustrated, for the efficient construction, in a closed form, of solutions of boundary value problems with the most general boundary conditions. As an example, the problem of finding the critical radii of thermal insulation of multilayer hollow cylinders and spheres, taking into account the internal heat sources in the layers. Boundary conditions of the first and third kind. It is established that the value of the critical radius does not depend on the number of layers and the intensity of internal heat sources, but only on the thermal conductivity of the outer layer of the structure and the heat transfer coefficient between the structure and the environment. The formula for determining the critical radius of thermal insulation for a multilayer cylindrical and spherical structure is derived. The methods developed in this work have the prospect of further development and can be used in engineering calculations.

scholarly journals Thermal 3D model for non-homogeneous elements in mobile devices

2019 ◽  
Vol 29 (8) ◽  
pp. 130-133
Author(s):  
V. I. Havrysh ◽  
O. S. Korol ◽  
R. R. Shkrab ◽  
B. O. Kviatkovskyi
Keyword(s):  
Thermal Conductivity ◽  
Fourier Transform ◽  
Heat Exchange ◽  
Analytical Solution ◽  
Generalized Derivatives ◽  
Inhomogeneous Layer ◽  
Conductivity Equation ◽  
Integral Fourier Transform ◽  
Thermal Conductivity Equation ◽  
Boundary Surfaces

A mathematical model for the analysis of heat exchange between the environment and an isotropic space layer with an alien inclusions is developed, which is heated by a heat flux centered on one of the boundary surfaces. For this purpose, using the theory of generalized functions, the coefficient of thermal conductivity of this structure is depicted as one unit for the whole system. In view of this, instead of two equations of thermal conductivity for the layer and the inclusion and conditions of perfect thermal contact on the surfaces of the junction between them, one equation of thermal conductivity was obtained in the generalized derivatives with breaking coefficients. We consider the case when the inclusion sizes are small compared to the distances from the inclusion surfaces to the boundary surfaces of the layer. In this connection, the combined thermophysical parameters were introduced and the thermal coefficients of the thermal conductivity equation were transformed into singular ones. For the solution of the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the layer, an integral Fourier transform was used and, as a result, an analytical solution of the problem in the images was obtained. The inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The analytical solution obtained is presented as a non-native double convergent integral. To determine the numerical values ​​of the temperature in the above design, as well as to analyze the heat exchange between the layer and the environment caused by different temperature regimes due to the heating of the inhomogeneous layer by a heat source concentrated in the area of ​​inclusion, computational programs have been developed. Using these programs, graphs are displayed showing the behavior of curves constructed using numerical values ​​of the temperature distribution depending on the spatial coordinates for different inclusion materials. The obtained numerical values ​​of temperature indicate a significant influence of the inclusion on its distribution in the design "layer-inclusion". The software also makes it possible to analyze these inhomogeneous media with respect to their heat resistance during heating. As a consequence, it becomes possible to raise and protect it from overheating, which can cause destruction not only of individual elements, but also of the whole structure.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

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