Methodology to Generate Accurate Solutions for Verification in Transient Three-Dimensional Heat Conduction

Abstract

This article describes the developmen t of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped . Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.

Conduction in rectangular plates with boundary temperatures specified

“…The series solutions available in some advanced heat conduction books for the temperature, and particularly the heat flux, converge slowly at the surface where a non-zero temperature is prescribed. Solutions with improved convergence are available [2,[4][5][6][7][8]. Even with these latter solutions, further improvement is possible based on a series identity given by Morse and Feshbach [9].…”

“…Exact transient heat conduction solutions have many uses including verification of numerical solutions [1][2][3][4], facilitating insight, and providing building blocks for the unsteady surface element method [5,Chapter 12]. These solutions are frequently the sum of steady-state and complementary transient components [2,3].…”

“…The time-partitioning method [2][3][4][5] is a powerful and general method, but numerical integration is used in [4] and large aspect ratios may cause difficulties. This paper incorporates some of the ideas in [6] and extends them by removing some slowly-convergent series for prescribed temperatures replacing them with two algebraic forms, which do not have convergence problems.…”

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

“…Examples of analytical solutions developed for verification or validation of approximate solutions obtained by numerical methods, such as algorithms based on finite differences, finite volumes or finite elements can be found in papers of Beck et al [3,4] or Macmasters et al [5] and others.…”

A heat-flux based “building block” approach for solving heat conduction problems

“…Notwithstanding this, the development of exact analytic solutions is relevant for verification purposes of large numerically based codes [8][9][10][11]. These solutions, in fact, not only can provide more accurate values of temperature and heat flux components within the domain of interest but their correctness can also be checked by means of intrinsic verification methods [2,12].…”