Solutions for modelling moving heat sources in a semi-infinite medium and applications to laser material processing

Abstract

This study describes the analytical and numerical solution of the heat conduction equation for a localised moving heat source of any type for use in laser material processing, as welding, layered manufacturing and laser alloying. In this paper, the analytical solution for a uniform heat source is derived from the solution of an instantaneous point heat source. The result is evaluated numerically and is compared to existing solutions for the moving point source and a semi-ellipsoidal source. Next, the result is used to demonstrate how such model can be used to study the effect of the heat source geometry. Besides, this solution reveals that a melting efficiency higher than 0.37 (= 1/ e, a maximum value stated by Rykalin [N. Rykalin, A. Uglov, A. Kokora, O. Glebov, Laser Machining and Welding, Mir Publishers, Moscow, 1978]) can be obtained. To investigate the effect of the temperature dependence of the material parameters, in particular the latent heat of fusion, a finite difference model is implemented. It is shown that the enthalpy method is most suited to implement the latent heat of fusion. A numerical evaluation for Ti–6Al–4V, reveals that the effect of the latent heat is rather small, except when the conductivity is very low, e.g. when scanning in a loose powder bed. The results demonstrate that analytical and numerical solutions can be effectively used to calculate the temperature distribution in a semi-infinite medium for finite 3D heat sources. In this way, a tool to investigate the importance of different processing parameters in laser manufacturing is obtained.