Numerical Solution of Heat Conduction in a Heterogeneous Multiscale Material With Temperature-Dependent Properties

2016 ◽  
Vol 138 (6) ◽  
Author(s):  
James White
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Time Scales ◽  
Thermal Energy ◽  
Energy Equation ◽  
Three Dimensional ◽  
Homogeneous Material ◽  
Time Level ◽  
Temperature Dependent ◽  
Temperature Dependent Properties

Numerical solution of heat conduction in a heterogeneous material with small spatial and time scales can lead to excessive compute times due to the dense computational grids required. This problem is avoided by averaging the energy equation over the small-scales, which removes the appearance of the short spatial and time scales while retaining their effect on the average temperature. Averaging does, however, increase the complexity of the resulting thermal energy equation by introducing mixed spatial derivatives and six different averaged conductivity terms for three-dimensional analysis. There is a need for a numerical method that efficiently and accurately handles these complexities as well as the other details of the averaged thermal energy equation. That is the topic of this paper as it describes a numerical solution for the averaged thermal energy equation based on Fourier conduction reported recently in the literature. The solution, based on finite difference techniques that are second-order time-accurate and noniterative, is appropriate for three-dimensional time-dependent and steady-state analysis. Speed of solution is obtained by spatially factoring the scheme into an alternating direction sequence at each time level. Numerical stability is enhanced by implicit algorithms that make use of the properties of tightly banded matrices. While accurately accounting for the nonlinearity introduced into the energy equation by temperature-dependent properties, the numerical solution algorithm requires only the consideration of linear systems of algebraic equations in advancing the solution from one time level to the next. Computed examples are included and compared with those for a homogeneous material.

Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
James White
Keyword(s):  
Heat Conduction ◽  
Thermal Energy ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Scale Effects ◽  
Three Dimensional ◽  
Heterogeneous Material ◽  
Multiple Scale ◽  
Computational Effort ◽  
Short Scale

In order to better manage computational requirements in the study of thermal conduction with short-scale heterogeneous materials, one is motivated to arrange the thermal energy equation into an accurate and efficient form with averaged properties. This should then allow an averaged temperature solution to be determined with a moderate computational effort. That is the topic of this paper as it describes the development using multiple-scale analysis of an averaged thermal energy equation based on Fourier heat conduction for a heterogeneous material with isotropic properties. The averaged energy equation to be reported is appropriate for a stationary or moving solid and three-dimensional heat flow. Restrictions are that the solid must display its heterogeneous properties over short spatial and time scales that allow averages of its properties to be determined. One distinction of the approach taken is that all short-scale effects, both moving and stationary, are combined into a single function during the analytical development. The result is a self-contained form of the averaged energy equation. By eliminating the need for coupling the averaged energy equation with external local problem solutions, numerical solutions are simplified and made more efficient. Also, as a result of the approach taken, nine effective averaged thermal conductivity terms are identified for three-dimensional conduction (and four effective terms for two-dimensional conduction). These conductivity terms are defined with two types of averaging for the component material conductivities over the short-scales and in terms of the relative proportions of the short-scales. Numerical results are included and discussed.

Sensitivity Analysis for Nonlinear Heat Conduction

2000 ◽  
Vol 123 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Kevin J. Dowding ◽  
Bennie F. Blackwell
Keyword(s):  
Heat Conduction ◽  
General Procedure ◽  
Model Parameters ◽  
Nonlinear Heat Conduction ◽  
Temperature Dependent ◽  
Sensitivity Equations ◽  
Verification Problem ◽  
Temperature Variable ◽  
Numerical Solver ◽  
Temperature Dependent Properties

Parameters in the heat conduction equation are frequently modeled as temperature dependent. Thermal conductivity, volumetric heat capacity, convection coefficients, emissivity, and volumetric source terms are parameters that may depend on temperature. Many applications, such as parameter estimation, optimal experimental design, optimization, and uncertainty analysis, require sensitivity to the parameters describing temperature-dependent properties. A general procedure to compute the sensitivity of the temperature field to model parameters for nonlinear heat conduction is studied. Parameters are modeled as arbitrary functions of temperature. Sensitivity equations are implemented in an unstructured grid, element-based numerical solver. The objectives of this study are to describe the methodology to derive sensitivity equations for the temperature-dependent parameters and present demonstration calculations. In addition to a verification problem, the design of an experiment to estimate temperature variable thermal properties is discussed.

Effect of Temperature Dependent Properties on the Applicability of the Heat Conduction Equations for Rapid Heat Deposition Applications

2016 ◽  
Author(s):  
John G. Michopoulos ◽  
Andrew Birnbaum ◽  
Athanasios P. Iliopoulos
Keyword(s):  
Differential Equation ◽  
Thermal Properties ◽  
Heat Conduction ◽  
Effect Of Temperature ◽  
Temperature Dependent ◽  
Proper Form ◽  
Heat Deposition ◽  
Hyperbolic Form ◽  
Parabolic Form ◽  
Temperature Dependent Properties

Despite significant efforts examining the suitability of the proper form of the heat transfer partial differential equation (PDE) as a function of the time scale of interest (e.g. seconds, picoseconds, femtoseconds, etc.), very little work has been done to investigate the millisecond-microsecond regime. This paper examines the differences between the parabolic and one of the hyber-bolic forms of the heat conduction PDE that govern the thermal energy conservation on these intermediate timescales. Emphasis is given to the types of problems where relatively fast heat flux deposition is realized. Specifically, the classical parabolic form is contrasted against the lesser known Cattaneo-Vernotte hyperbolic form. A comparative study of the behavior of these forms over various pulsed conditions are applied at the center of a rectangular plate. Further emphasis is given to the variability of the solutions subject to constant or temperature-dependent thermal properties. Additionally, two materials, Al-6061 and refractory Nb1Zr, with widely varying thermal properties, were investigated.

Thermal and Roughness Effects in a Slider Bearing Considering Conduction Through Both the Pad and the Slider

2008 ◽  
Author(s):  
Prawal Sinha ◽  
Getachew Adamu
Keyword(s):  
Heat Conduction ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Momentum Equation ◽  
Regular Domain ◽  
Slider Bearing ◽  
Irregular Domain ◽  
Temperature Dependent ◽  
Roughness Effects ◽  
Different Characteristics

This paper analyses the thermal and roughness effects on different characteristics of an infinitely long tilted pad slider bearing considering heat conduction through both the pad and slider. The roughness is assumed to be stochastic, Gaussian randomly distributed. Density and viscosity are assumed to be temperature dependent. The irregular domain of the fluid due to roughness is mapped to a regular domain so that the numerical method can be easily applied. The modified Reynolds equation, momentum equation, continuity equation, energy equation and the heat conduction equations on the pad and slider are coupled and solved using finite difference method to yield various bearing characteristics. The solutions with respect to different pad and slider boundary conditions are elaborated through tables and figures.

scholarly journals Numerical study of transient three-dimensional heat conduction problem with a moving heat source

2011 ◽  
Vol 15 (1) ◽  
pp. 257-266 ◽  
Author(s):  
Ivana Ivanovic ◽  
Aleksandar Sedmak ◽  
Marko Milos ◽  
Aleksandar Zivkovic ◽  
Mirjana Lazic
Keyword(s):  
Heat Conduction ◽  
Heat Source ◽  
Numerical Solution ◽  
Numerical Study ◽  
Heat Conduction Problem ◽  
Gaussian Function ◽  
Three Dimensional ◽  
Moving Heat Source ◽  
Input Parameters ◽  
Flux Boundary Conditions

A numerical study of transient three-dimensional heat conduction problem with a moving source is presented. For numerical solution Douglas-Gunn alternating direction implicit method is applied and for the moving heat source flux distribution Gaussian function is used. An influence on numerical solution of input parameters figuring in flux boundary conditions is examined. This include parameters appearing in Gaussian function and heat transfer coefficient from free convection boundaries. Sensitivity of cooling time from 800 to 500?C with respect to input parameters is also tested.

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