Semi-Analytical Solution for Modelling Moving Heat Sources in a Semi-Infinite Medium with Radiative and Convective Boundary Conditions

The weld quality is highly related to the thermal history of the weld and there have been many trials to monitor the quality using an infrared (IR) sensor. To obtain the real temperature of a surface based on the brightness temperature values measured by an IR camera, the emissivity value must be derived. For an accurate assessment of the emissivity, one must be aware of the melting point isotherm. The temperature profiles only depend on three factors during laser processing, specified as constants the characteristics of the material: laser beam speed (v), laser beam diameter (d), and power (P). Predicting the width of the melted zone reached during the welding process as the parameters vary is a tool for helping a quality laser processing and for determination of true temperature in laser welding using IR camera. This study describes the semi-analytical (SA) solution of the heat conduction equation for a localized moving Gaussian heat source with constant parameters on a semi-infinite medium. The solution, simple and quick to obtain, provides information on the width of the melted zone with an average error < 5 %. The outcome is assessed numerically and contrasted with FEM solutions for a Gaussian source, the latter having undergone experimental validation. With two distinct defocus values, def0 and def-6, and by varying the speed and power settings, two separate types of experiments were run. Thus, the SA solution was obtained and compared after the FEM solution had been obtained with a good approximation (max err 4.3 %, average err 2.7 %). Only in regard to the 1AL test is an error more than 5 % detected; in the other case, the average error is 3.75 %. Two more tests at the defocus values of def-4 and def-8 were conducted to confirm the model's validity as the parameters varied. Overall, the average error between the semi-analytical and the FEM solution is 4.1%. The SA solution may be used to effectively estimate the isotherms related to the melting point of aluminum (770 K). This allows to obtain a tool which helps restoring the real temperature based on the brightness values measured by the IR camera during laser welding. At the same time, this effective tool allows to investigate the importance of different processing parameters in laser manufacturing.