One-dimensional thermal model including the dependence of absorptivity on temperature using Hagen-Rubens equation

In laser surface treatment of materials it is a matter of practical importance to determine relations combining the various processing parameters. For a particular laser setup and for a certain practical requirement, such a relation is quite useful for the purpose of process optimisation. For instance, in pulsed laser irradiation when the requirement is that a definite value of temperature is attained at the surface (e.g. the melting temperature), a relation between power density and pulse duration should be looked for. If heat conduction is one-dimensional and the material properties are considered as temperature independent, a first estimate can be done by using a simple analytical model, which gives a temperature rise at the surface proportional to tn. Considering the effect of the various properties on the thermal field, the absorptivity plays a special role, because it controls directly the heat input to the sample, while being itself a function of surface temperature. In order to take this dependence into account, the one-dimensional heat conduction equation was solved for a semi-infinite geometry using variable absorptivity given by Hagen-Rubens equation and assuming constant thermal conductivity and thermal diffusivity. It is shown that the surface temperature satisfies an Abel integral equation with an approximate analytical solution that gives a temperature rise at the surface linear with t. Comparison is made between analytical approximate expressions, numerical solutions, previous models and experimental results.