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.

Simulation of Nonlinear Heat Conduction in Perforated Material by Improved Moving Particle Semi-Implicit Method

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
Jianqiang Wang ◽  
Xiaobing Zhang
Keyword(s):  
Heat Conduction ◽  
Laplacian Operator ◽  
Nonlinear Heat Conduction ◽  
Temperature Dependent ◽  
One Dimensional ◽  
Particle Distributions ◽  
Temperature Dependent Conductivity ◽  
Irregular Particle ◽  
Moving Particle ◽  
Temperature Dependent Thermal Conductivity

Abstract An improved moving particle semi-implicit (MPS) method is presented to simulate heat conduction with temperature-dependent thermal conductivity. Based on Taylor expansion, a modified Laplacian operator is proposed, and its accuracy in irregular particle distributions is verified. Two problems are considered: (1) heat conduction in a one-dimensional (1D) slab and (2) heat conduction in a perforated sector with different boundary conditions. Consistent results with a mesh-based method are obtained, and the feasibility of the proposed method for heat conduction simulation with temperature-dependent conductivity is demonstrated.

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.

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