Semi-Analytical Solution for a Single-Phase Conduction Problem of a Finite-Slab With a Growing or Receding Boundary

Abstract

Thermal conduction considerations of a solid media with moving boundaries are of great interest in many research areas. Unfortunately, it is very difficult to find analytical or semi-analytical solutions for the single-phase heat equation in real time with a growing or receding boundary. While non-numerical solutions for infinite and semi-infinite domains are available, these can not accurately model many common situations. In order to overcome this shortcoming, a semi-analytical solution for the heat equation for a single phase, homogeneous, and finite-slab with a growing or receding boundary under unit loading was derived using the Laplace transform method and Zakian's series representation of the inverse Laplace transform. Predictions were compared to finite element solutions with good agreement obtained for low to moderate growth or recession rates with improvements seen by using a heuristic approach. Applications of this work could include the direct or inverse prediction of temperatures during machining, wear, corrosion, and/or additive manufacturing via cold-spray.