Stefan problem for a finite liquid phase and its application to laser or electron beam welding

Abstract

An exact solution of a heat conduction problem with the effect of latent heat of solidification (Stefan problem) is derived. The solution of the one dimensional Stefan problem for a finite liquid phase initially existing in a semi-infinite body is applied to evaluate temperature fields produced by laser or electron beam welding. The solution of the model has not been available before, as Carslaw and Jaeger [Conduction of Heat in Solids, 2nd ed. (Oxford University Press, New York, 1959)] pointed out. The heat conduction calculations are performed using thermal properties of carbon steel, and the comparison of the Stefan problem with a simplified linear heat conduction model reveals that the solidification rate and cooling curve over 1273 K significantly depend on which model (Stefan or linear heat conduction problem) is applied, and that the type of the thermal model applied has little meaning for cooling curve below 1273 K. Since the heat conduction problems with a phase change arise in many important industrial fields, the solution derived in this study is ready to be used not only for welding but also for other industrial applications.