scholarly journals SEMI-ANALYTICAL SOLUTION OF THE HEAT CONDUCTION IN A PLATE WITH HEAT GENERATION

2017 ◽  
Vol 16 (1) ◽  
pp. 58
Author(s):  
L. C. da Silva ◽  
D. J. N. M. Chalhub ◽  
A. L. O. Calil ◽  
R. S. de Moura
Keyword(s):  
Heat Conduction ◽  
Heat Generation ◽  
Truncation Error ◽  
Integral Transform ◽  
Solution Process ◽  
High Concentration ◽  
Classical Integral ◽  
Steady State Heat ◽  
Central Finite Difference ◽  
Integral Transform Technique

In the present work, a formulation for the solution of the two-dimensional steady state heat conduction with heat generation is presented. The classical integral transform technique (CITT) is used to solve the problem in a semi- analytical manner. CITT deals with expansions of the sought solution in terms of infinite orthogonal basis of eigenfunctions, keeping the solution process always within a continuous domain. For the particular problem, the resulting system is composed of a set of uncoupled differential equations which can be solved analytically. However, a truncation error is involved since the infinite series must be truncated to obtain numerical results. For comparison and validation purposes, the second order central finite difference method (FDM) is also implemented. The convergence analysis showed that CITT has a greater performance having no difficulties to obtain accurate results with very few terms in the solution summation. The FDM had convergence troubles specially for the positions near the center and for high concentration of heat generation in the center of the plate.

Heat Conduction in Three Dimensional Stacked Packages

2005 ◽  
Author(s):  
Anand Desai ◽  
James Geer ◽  
Bahgat Sammakia
Keyword(s):  
Heat Conduction ◽  
Thermal Management ◽  
Heat Generation ◽  
Analytical Study ◽  
Three Dimensional ◽  
The Other ◽  
Heat Generation Rate ◽  
Potential Applications ◽  
Steady State Heat ◽  
Spatially Varying

This paper presents the results of an analytical study of steady state heat conduction in multiple rectangular domains. Any finite number of such domains may be considered in the current study. The thermal conductivity and thickness of these domains may be different. The entire geometry composed of these connected domains is considered as adiabatic on the lateral surfaces and can be subjected to uniform convective cooling at one end. The other end of the geometry may be adiabatic and a specified, spatially varying heat generation rate can be applied in each of the domains. The solutions are found to be in agreement with known solutions for simpler geometries. The analytical solution presented here is very general in that it takes into account the interface resistances between the layers. One application of this analytical study relates to the thermal management of a 3-D stack of devices and interconnect layers. Another possible application is to the study of hotspots in a chip stack with non uniform heat generation. Many other potential applications may also be simulated.

Nonhomogeneous Dual-Phase-Lag Heat Conduction Problem: Analytical Solution and Select Case Studies

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Simon Julius ◽  
Boris Leizeronok ◽  
Beni Cukurel
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Generation ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Relaxation Times ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Dual Phase

Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the wavelike heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the dual-phase solution depicts wavelike temperature propagation—the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the dual-phase model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

Steady-State Temperatures in an Infinite Medium Split by a Pair of Coplanar Cracks

1988 ◽  
Vol 110 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Shangchow Chang
Keyword(s):  
Steady State ◽  
Integral Transform ◽  
Continuation Method ◽  
Infinite Medium ◽  
Analytical Continuation ◽  
Crack Plane ◽  
Solution Methods ◽  
Steady State Heat ◽  
Integral Transform Technique ◽  
Coplanar Cracks

This article presents a study on the steady-state heat conduction in an infinite medium containing two coplanar cracks. Using an integral transform technique, formal temperature solutions have first been worked out for both the fundamental symmetric and antisymmetric cases. The explicit and exact expressions for temperatures are then developed via both the conventional inversion transform approach and an analytical continuation method proposed in this paper. Numerical results prepared from analytic and numerical methods are presented in graphic form for temperatures on the horizontal crack plane and on a plane slant to the cracks. The relative merit of various possible solution methods is also discussed.

scholarly journals Heat conduction in rectangular solids with internal heat generation

2020 ◽  
pp. 235-235
Author(s):  
Zhipeng Duan ◽  
Hao Ma
Keyword(s):  
Heat Conduction ◽  
Heat Generation ◽  
Shape Factor ◽  
Heat Conduction Problem ◽  
Rectangular Cross Section ◽  
Internal Heat ◽  
Engineering Practice ◽  
Constant Wall Temperature ◽  
Temperature Boundary ◽  
Steady State Heat

A representative steady-state heat conduction problem in rectangular solids with uniformly distributed heat generation has been investigated analytically. An analytical solution is provided by solving a nonhomogeneous partial differential equation. A simple and accurate model is proposed to predict the dimensionless shape factor parameter for the first time. The dimensionless shape factor is obtained in the light of the solution of Poisson equation with constant wall temperature boundary conditions. The area-mean temperature is found by integration on the rectangular cross-section. The model is very concise and nice for quick real world approximations, and it provides acceptable accuracy for engineering practice.

Two-Energy Modeling of Heat Conduction in a Heterogeneous Spherical Fuel Element

2009 ◽  
Author(s):  
Carlos V. Pessoa ◽  
Jian Su ◽  
Claudio L. de Oliveira
Keyword(s):  
Heat Conduction ◽  
Fuel Element ◽  
Integral Transform ◽  
Equation Model ◽  
Fast Evaluation ◽  
Graphite Matrix ◽  
Hybrid Solution ◽  
The Matrix ◽  
Steady State Heat ◽  
Fuel Particles

In this work, heat conduction in a typical spherical fuel element (pebble) of a pebble-bed high temperature reactor was studied. The fuel element is composed by a particulate region with spherical inclusions, the UO2 fuel particles (TRISO), dispersed in a graphite matrix. The two energy equation model was applied to the particulate region, generating two macroscopic temperatures, respectively, of the fuel and of the matrix. Analytical solutions are obtained for steady-state heat conduction. Transient analysis was carried out by using the generalized integral transform technique (GITT), which requires low computational efforts and allows a fast evaluation of the two macroscopic transient temperatures of the particulate region. The solution by GITT leads to a system of ordinary differential equations with the unknown transformed potentials, which is solved numerically to obtain the hybrid solution of the original partial differential equation. Numerical results for several testing cases are presented.

scholarly journals SOLUTION OF MULTIPHASE HEAT CONDUCTION PROBLEMS VIA THE GENERALIZED INTEGRAL TRANSFORM TECHNIQUE WITH DOMAIN CHARACTERIZATION THROUGH THE INDICATOR FUNCTION

2016 ◽  
Vol 15 (2) ◽  
pp. 53
Author(s):  
H. A. Machado ◽  
N. G. C. Leite ◽  
E. Nogueira ◽  
H. Korzenowisk
Keyword(s):  
Heat Conduction ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Transport Equations ◽  
Indicator Function ◽  
Vast Number ◽  
Convection Diffusion ◽  
The Common ◽  
Integral Transform Technique ◽  
Diffusion Problems

The Generalized Integral Transform Technique (GITT) has appeared in the literature as an alternative to conventional discrete numerical methods for partial differential equations in heat transfer and fluid flow. This method permits the automatic control of the error and is easy to program, since there is no need for a discretization. The method has being constantly improved, but there still a vast number of practical problems that has not being solved satisfactory. In several brands of engineering, the transport equations have to be solved for a combination of different phases or materials or inside irregular domains. In this case, the mathematical resource of the Indicator Function can be employed. This function is a representation of the phases or parts of the domain with the numbers 0 and 1 for each phase. According to the method, the Indicator Function is defined by Poisson’s equation, which is added to the system of the transport equations. An integral is done along the curve that defines the interface that will generate the source term in Poisson’ equation used to calculate the Indicator Function distribution. The solution of the system of equations is done using the common GITT approach. Then, an analytical expression for each transformed potential of the indicator function and the other variables are available. Once the transformed potentials are known, the Indicator Function can be analytically operated, and the interface can be represented by an analytical continuous function. In this work, the use of the GITT in conjunction with the Indicator Function is proposed. The methodology is described and some previous results are presented. GITT is applied to a two-dimensional heat conduction problem in a multiphase domain with an irregular geometry, inside a square domain. The methodology presented here can be extended to all brands of convection-diffusion problems already solved via GITT.

scholarly journals UMA CONSTRUÇÃO HISTÓRICA DAS TÉCNICAS DA TRANSFORMADA INTEGRAL CLÁSSICA (CITT) E GENERALIZADA (GITT)

2020 ◽  
Vol 7 (20) ◽  
pp. 80-92
Author(s):  
Reynaldo D'Alessandro Neto
Keyword(s):  
Integral Transform ◽  
De Se ◽  
Classical Integral ◽  
Integral Transform Technique

Estes resultados parciais de uma pesquisa de doutorado do Programa de Pós-Graduação em Educação Matemática da UNESP – Rio Claro, insere-se na linha de pesquisa Relações entre História e Educação Matemática e tem como objetivo descrever a evolução histórica que culmina na concepção da Técnica da Transformada Integral Clássica, e as motivações que levaram a sistematização do seu modelo generalizado. As técnicas têm como foco resolver Equações Diferenciais Parciais (EDP) a princípio não tratáveis pelas teorias clássicas, como o conhecido método da separação de variáveis. Pretendemos fazer uma construção histórica, considerando o contexto do seu surgimento e desenvolvimento, passando pelas diversas modificações ao longo dos estudos e necessidades de se tornar uma técnica mais competitiva para a evolução do mundo tecnológico. Para atingir esse objetivo, faremos uma abordagem historiográfica que começa ao descrevermos algumas motivações históricas dos desenvolvimentos da Transformada Integral, e as principais ideias da Transformada Integral Finita por N.S. Koshlyakov. Além dos estudos detalhados realizados por G.A. Grinberg (1948), que generaliza os métodos de Koshlyakov, para o caso de mudança das propriedades do meio na direção da coordenada ao longo da qual a transformação é executada. E a aplicação de M.D. Mikhailov (1972), que propõe um núcleo de núcleo de processamento geral que unificou as várias transformações desenvolvidas até então, obtendo a solução para a equação da difusão linear em regiões finitas. Para assim, podermos entender esses movimentos que são precursores da proposta da Técnica da Transformada Integral Clássica (CITT – Classical Integral Transform Technique), de Özisik e Murray (1974). E, por fim, dos conceitos que surgiram com o formalismo da Técnica Transformada Integral Generalizada (GITT - Generalized Integral Transform Technique), proposta por Özisik e Mikhailov (1984). Nesses resultados parciais de pesquisa, apresentamos os passos descritos acima até a contribuição de Mikhailov (1972), que serão finalizados com a análise dos escritos que fundamentam a CITT e GITT.

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