Analytical Solution to Steady-State Heat-Conduction Problems With Irregularly Shaped Boundaries

1971 ◽  
Vol 93 (4) ◽  
pp. 449-454 ◽  
Author(s):  
D. M. France
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Analytical Solution ◽  
Least Squares ◽  
Series Solution ◽  
Point Matching ◽  
State Heat ◽  
Temperature Heat ◽  
Steady State Heat ◽  
Least Squares Approach

A method of obtaining an analytical solution to two-dimensional steady-state heat-conduction problems with irregularly shaped boundaries is presented. The technique of obtaining the coefficients to the series solution via a direct least-squares approach is compared to the “point-matching” scheme. The two methods were applied to problems with known solutions involving the three heat-transfer boundary conditions, temperature, heat flux, and convection coefficient specified. Increased accuracy with substantially fewer terms in the series solution was obtained via the least-squares technique.

AN INTERPOLATING ELEMENT-FREE GALERKIN METHOD FOR STEADY-STATE HEAT CONDUCTION PROBLEMS

2014 ◽  
Vol 06 (03) ◽  
pp. 1450024 ◽  
Author(s):  
HONGPING REN ◽  
LIPING WANG ◽  
NA ZHAO
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Least Squares ◽  
Shape Function ◽  
Moving Least Squares ◽  
Computational Accuracy ◽  
State Heat ◽  
Element Free Galerkin ◽  
Steady State Heat ◽  
Element Free

In this paper, an interpolating element-free Galerkin (IEFG) method is presented for steady-state heat conduction problems with heat generation and spatially varying conductivity. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed, then combining the shape function constructed by the IMLS method and Galerkin weak form of the two-dimensional steady-state heat conduction problems, the IEFG method for heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless methods is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method not only has high computational accuracy, but also enhances computational efficiency greatly.

Steady State Heat Conduction Modeling by the Generalized Finite Element Method

2018 ◽  
Author(s):  
Humberto Alves da Silveira Monteiro ◽  
Guilherme Garcia Botelho ◽  
Roque Luiz da Silva Pitangueira ◽  
Rodrigo Peixoto ◽  
FELICIO BARROS
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Steady State ◽  
Generalized Finite Element Method ◽  
State Heat ◽  
Steady State Heat ◽  
Generalized Finite Element ◽  
Element Method

Study on Radial Temperature Distribution of Aluminum Dispersed Nuclear Fuels: U3O8-Al, U3Si2-Al, and UN-Al

2018 ◽  
Vol 4 (3) ◽  
Author(s):  
Jayangani I. Ranasinghe ◽  
Ericmoore Jossou ◽  
Linu Malakkal ◽  
Barbara Szpunar ◽  
Jerzy A. Szpunar
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Steady State ◽  
Temperature Gradient ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Temperature Profiles ◽  
Nuclear Fuels ◽  
State Heat ◽  
Steady State Heat

The understanding of the radial distribution of temperature in a fuel pellet, under normal operation and accident conditions, is important for a safe operation of a nuclear reactor. Therefore, in this study, we have solved the steady-state heat conduction equation, to analyze the temperature profiles of a 12 mm diameter cylindrical dispersed nuclear fuels of U3O8-Al, U3Si2-Al, and UN-Al operating at 597 °C. Moreover, we have also derived the thermal conductivity correlations as a function of temperature for U3Si2, uranium mononitride (UN), and Al. To evaluate the thermal conductivity correlations of U3Si2, UN, and Al, we have used density functional theory (DFT) as incorporated in the Quantum ESPRESSO (QE) along with other codes such as Phonopy, ShengBTE, EPW (electron-phonon coupling adopting Wannier functions), and BoltzTraP (Boltzmann transport properties). However, for U3O8, we utilized the thermal conductivity correlation proposed by Pillai et al. Furthermore, the effective thermal conductivity of dispersed fuels with 5, 10, 15, 30, and 50 vol %, respectively of dispersed fuel particle densities over the temperature range of 27–627 °C was evaluated by Bruggman model. Additionally, the temperature profiles and temperature gradient profiles of the dispersed fuels were evaluated by solving the steady-state heat conduction equation by using Maple code. This study not only predicts a reduction in the centerline temperature and temperature gradient in dispersed fuels but also reveals the maximum concentration of fissile material (U3O8, U3Si2, and UN) that can be incorporated in the Al matrix without the centerline melting. Furthermore, these predictions enable the experimental scientists in selecting an appropriate dispersion fuel with a lower risk of fuel melting and fuel cracking.

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