Analytical, quasi-stationary Wilson-Rosenthal solution for moving heat sources

Abstract

This paper presents the quasi-stationary Wilson-Rosenthal solution for isothermal contours in a semi-infinite medium due to a moving heat source. There is a single dimensionless governing parameter which could correspond, for example, to a conduction-limited heat flow in a point-source driven, fusion process, such as that witnessed in modern metal additive processes. Variables involved in this heat transfer problem are the power and speed of the heat source as well as the thermal properties of the domain being heated. On one hand, the forwardmost location for a generic isothermal contour profile, for example the melting isotherm, is calculated by (i) a fixed-point numerical method, (ii) second-, third- and fourth-order Taylor series expansions, and (iii) a direct calculation of the location using the Laplace-Carson transform and an approximate anti-transform. On the other hand, the rearmost point of the isothermal contour is found to be invariant with the source speed parameter while the horizontal coordinate corresponding to the maximum depth of the contour is predicted by the semi-sum of the forwardmost and rearmost coordinates. Experimental results suggest that this analytical solution provides accurate physical understanding of the process, for example, when the melt pool contour geometry of a conduction-limited fusion process requires to be controlled during an specific heat driven process. In this latter situation, a single dimensionless governing parameter can be of benefit.