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.

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 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.

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.

scholarly journals Investigation of Thermodynamic Properties of the Slurry of the Molding Process Beryllium Ceramics

Proceedings ◽  
2018 ◽  
Vol 2 (22) ◽  
pp. 1391
Author(s):  
Zamira Sattinova ◽  
Gaukhar Ramazanova ◽  
Bakhytzan Assilbekov ◽  
Elmira Mussenova
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Thermal Properties ◽  
Heat Exchange ◽  
Thermodynamic Properties ◽  
Physical Properties ◽  
Beryllium Oxide ◽  
Molding Process ◽  
The Mathematical Model

Obtaining of ceramic fabrications by hot molding from dispersion materials with anomalous physical properties, such as BeO is particularly complicated. In this case, the difficulties of obtaining of quality products were caused firstly by thermal properties of beryllium oxide, in particular, its unique thermal conductivity. Results of experiments and calculations of the mathematical model of the motion and heat exchange of the slurry mass in the annular cavity are presented. The results of experiments and calculations show the process of molding of the slurry in the annular cavity.

Conduction Characteristics of Frosting Circumferential Fin of Rectangular Profile

2011 ◽  
Vol 354-355 ◽  
pp. 779-783
Author(s):  
Kun Feng Sun ◽  
Xiao Lu Wang ◽  
Heng Liang Zhang
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Analytical Solution ◽  
High Thermal Conductivity ◽  
Engineering Practice ◽  
Low Thermal Conductivity ◽  
Rectangular Profile ◽  
Alternative Approaches ◽  
Frost Layer ◽  
Form Of Expression

Through the appropriate assumptions, a mathematical model was derived on circumferential fin of rectangular profile coated with frost layer, and a new analytical solution is presented and compared to alternative approaches for calculating the fin temperature and efficiency. The analytical solution enhances the accuracy ,and have a simpler form of expression. It can be easily used in engineering practice. The solution is useful for other cases of a low-thermal-conductivity coating on a high-thermal-conductivity substrate .

On the General Solutions for Annular Problems With a Point Heat Source

1999 ◽  
Vol 67 (3) ◽  
pp. 511-518 ◽  
Author(s):  
C. K. Chao ◽  
C. J. Tan
Keyword(s):  
Fourier Series ◽  
Analytical Solution ◽  
Heat Source ◽  
Analytical Continuation ◽  
Series Expansions ◽  
Series Solutions ◽  
Point Heat Source ◽  
Free Boundary Conditions ◽  
Complex Functions ◽  
Stress Functions

A general analytical solution for the annular problem with a point heat source is provided in this paper. Based upon the method of analytical continuation and the technique of Fourier series expansions, the series solutions of the temperature and stress functions are expressed in complex explicit form. Single-valuedness of complex functions in the doubly connected region has been examined for both the stress-free and displacement-free boundary conditions. The dilatation stress in the annulus due to the application of a point heat source is discussed and shown in graphic form. [S0021-8936(00)02803-8]

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