The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and Benchmark Exercises

TL;DR

This paper analyzes and improves the heat-balance integral method with an unspecified parabolic profile, introducing additional physical constraints to enhance accuracy and robustness across various heat conduction problems.

Contribution

It introduces a new approach to determine the parabola exponent using physical assumptions, improving the classical heat-balance integral method's effectiveness.

Findings

Enhanced method accurately models thermal layer penetration.

Improved boundary condition handling at thermal layer front.

Validated through solutions of 4 different 1-D heat conduction problems.

Abstract

The heat-balance integral method of Goodman has been thoroughly analyzed in the case of a parabolic profile with unspecified exponent depending on the boundary condition imposed. That the classical Good man's boundary conditions defining the time-dependent coefficients of the prescribed temperature profile do not work efficiently at the front of the thermal layers if the specific parabolic profile at issue is employed. Additional constraints based on physical assumption enhance the heat-balance integral method and form a robust algorithm defining the parabola exponent . The method has been compared by results provided by the Veinik's method that is by far different from the Good man's idea but also assume forma tion of thermal layer penetrating the heat body. The method has been demonstrated through detailed solutions of 4 1-D heat-conduction problems in Cartesian co-ordinates including a spherical problem (through change of vari ables) and over-specified boundary condition at the face of the thermal layer.