Transient Heat Conduction with No Steady State

2008 ◽  
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Transient Heat Conduction

EVALUATION OF COMBINED DELAUNAY TRIANGULATION AND REMESHING FOR FINITE ELEMENT ANALYSIS OF CONDUCTIVE HEAT TRANSFER

2004 ◽  
Vol 27 (4) ◽  
pp. 319-339 ◽  
Author(s):  
Sutthisak Phongthanapanich ◽  
Pramote Dechaumphai
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Heat Conduction ◽  
Steady State ◽  
Delaunay Triangulation ◽  
Transient Heat Conduction ◽  
Conductive Heat ◽  
Element Analysis ◽  
Adaptive Remeshing ◽  
Solution Accuracy

A finite element method is combined with the Delaunay triangulation and an adaptive remeshing technique to solve for solutions of both steady-state and transient heat conduction problems. The Delaunay triangulation and the adaptive remeshing technique are explained in detail. The solution accuracy and the effectiveness of the combined procedure are evaluated by heat transfer problems that have exact solutions. These problems include steady-state heat conduction in a square plate subjected to a highly localized surface heating, and a transient heat conduction in a long plate subjected to a moving heat source. The examples demonstrate that the adaptive remeshing technique with the Delaunay triangulation significantly reduce the number of the finite elements required for the problems and, at the same time, increase the analysis solution accuracy as compared to the results produced using uniform finite element meshes.

Higher Order Finite Elements for the Accurate Prediction of Temperature Gradients in Heat Conduction Problems

2014 ◽  
Author(s):  
Donovan A. Aguirre-Rivas ◽  
Karim H. Muci-Küchler
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Degrees Of Freedom ◽  
Neumann Boundary ◽  
Transient Heat Conduction ◽  
Field Variable ◽  
Higher Order ◽  
Temperature Gradients ◽  
Functional Representation ◽  
Higher Order Finite Elements

When the Finite Element Method (FEM) is used to solve heat conduction problems in solids, the domain is typically discretized using elements that only include the nodal values of the temperature as Degrees of Freedom (DOFs). If the values of the spatial temperature gradients are needed, they are typically computed by differentiating the functional representation for the temperature inside the elements. Unfortunately, this differentiation process usually leads to less accurate results for the temperature gradients as compared to the temperature values. For elliptic problems, like steady state heat conduction, with Neumann Boundary Conditions (BCs), recent research related to Adini’s element suggests that higher order elements that include spatial derivatives of the primary field variable as nodal DOFs are promising for obtaining accurate values for those quantities as well as providing a higher order of convergence than conventional elements. In this paper, steady state and transient heat conduction problems which involve Dirichlet BCs or both Dirichlet and Neumann BCs are studied and a new auxiliary BC is proposed to increase the accuracy of the FE solution when Dirichlet BCs are present. Examples are used to illustrate that Adini’s elements converge faster and are more computationally economical than the conventional Lagrange linear elements and Serendipity quadratic elements when auxiliary BCs are used.

Generalized Solution for Two-Dimensional Transient Heat Conduction Problems With Partial Heating Near a Corner

2019 ◽  
Vol 141 (7) ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Heat Conduction ◽  
Steady State ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Generalized Solution ◽  
Convective Heating ◽  
Two Dimensional ◽  
Partial Heating

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

scholarly journals Model reduction in computational homogenization for transient heat conduction

2019 ◽  
Vol 65 (1) ◽  
pp. 249-266 ◽  
Author(s):  
A. Waseem ◽  
T. Heuzé ◽  
L. Stainier ◽  
M. G. D. Geers ◽  
V. G. Kouznetsova
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Contrast Material ◽  
Transient Heat Conduction ◽  
Homogenization Method ◽  
Computational Homogenization ◽  
Computationally Efficient ◽  
Reduced Basis ◽  
Static Condensation ◽  
Separation Of Scales

Abstract This paper presents a computationally efficient homogenization method for transient heat conduction problems. The notion of relaxed separation of scales is introduced and the homogenization framework is derived. Under the assumptions of linearity and relaxed separation of scales, the microscopic solution is decomposed into a steady-state and a transient part. Static condensation is performed to obtain the global basis for the steady-state response and an eigenvalue problem is solved to obtain a global basis for the transient response. The macroscopic quantities are then extracted by averaging and expressed in terms of the coefficients of the reduced basis. Proof-of-principle simulations are conducted with materials exhibiting high contrast material properties. The proposed homogenization method is compared with the conventional steady-state homogenization and transient computational homogenization methods. Within its applicability limits, the proposed homogenization method is able to accurately capture the microscopic thermal inertial effects with significant computational efficiency.

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

2012 ◽  
Vol 4 (5) ◽  
pp. 519-542 ◽  
Author(s):  
Zhuo-Jia Fu ◽  
Wen Chen ◽  
Qing-Hua Qin
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Transient Heat Conduction ◽  
Meshless Methods ◽  
Boundary Integral ◽  
Functionally Graded ◽  
Local Boundary Integral Equation ◽  
Transient Heat Conduction Problem ◽  
Kirchhoff Transformation ◽  
Laplace Transform Technique

AbstractThis paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.

Analytical Solution of the Transient Heat Conduction in the Absorber Tube of a Parabolic Trough Solar Collector Under Quasi-Steady Conditions

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Roger Cundapí ◽  
Sara L. Moya ◽  
Octavio Cazarez
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Analytical Solution ◽  
Solar Collector ◽  
Transient Heat Conduction ◽  
Step Function ◽  
Operating Conditions ◽  
Fluid Temperature ◽  
Parabolic Trough ◽  
Parabolic Trough Solar Collector

Abstract Temperature fields and their transient behaviors are essential subjects to be considered for modeling and design of absorber tubes in concentrated solar power plants. Both subjects have been addressed by various authors. However, the first subject has been primarily solved in the steady state. While the second has been solved by considering transient variations in the environmental or operating conditions, but with a heat conduction model in steady state. To the best of our knowledge, there are no analytical transient two-dimensional (2D) (r, φ) solutions involving nonuniform heat flux distribution (NUHFD) on the absorber tube of a parabolic trough solar collector (PTC). This study aims to obtain an analytical solution for the transient heat conduction in 2D of the absorber tube. The analytical solution was obtained using the method of separation of variables and the superposition principle. Two NUHFD functions were analyzed: a step function and a local concentration ratio (LCR) function. To the first function, the effect of the inlet fluid temperature and efficiency were also studied. The results agree with experimental and numerical results from the literature. The maximum average root-mean-square was near 6.4% for the step function, while the maximum average error was 1% for LCR function. The theoretical energy balances corroborate the validity of the analytical solution. The analytical solution could be useful to compare other theoretical studies (e.g., to prove new numerical schemes), to simulate other parameters of design, and to calibrate experimental tests. Even this work could be extended for nonlinear boundary conditions.

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