On the analytical solution of a nonstationary kinetic equation for a point heat source

2003 ◽  
Vol 29 (1) ◽  
pp. 35-38
Author(s):  
S. A. Savkov
Keyword(s):  
Kinetic Equation ◽  
Analytical Solution ◽  
Heat Source ◽  
Point Heat Source

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.

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]

Analytical Formulation for the Temperature Profile by Duhamel’s Theorem in Bodies Subjected to an Oscillatory Heat Source

2005 ◽  
Vol 129 (2) ◽  
pp. 236-240 ◽  
Author(s):  
Jun Wen ◽  
M. M. Khonsari
Keyword(s):  
Finite Element Method ◽  
Heat Flux ◽  
Heat Source ◽  
Temperature Profile ◽  
Rectangular Domain ◽  
Temperature Fields ◽  
Heat Sources ◽  
Point Heat Source ◽  
Analytical Formulation ◽  
Analytical Technique

An analytical technique is presented for treating heat conduction problems involving a body experiencing oscillating heat flux on its boundary. The boundary heat flux is treated as a combination of many point heat sources, each of which emits heat intermittently based on the motion of the flux. The working function of the intermittent heat source with respect to time is evaluated by using the Fourier series and temperature profile of each point heat source is derived by using the Duhamel’s theorem. Finally, by superposition of the temperature fields over all the point heat sources, the temperature profile due to the original moving heat flux is determined. Prediction results and verification using finite element method are presented for an oscillatory heat flux in a rectangular domain.

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