A simple mathematical model for heat flow evaluation, used to design complex heat sources obtained from diffractive optical elements

Abstract

The use of diffractive optical elements (DOEs) to customise the spatial intensity profile of a laser beam enables the beam to be optimised to the process, not the process to the beam. This requires knowledge of the changes effected in the substrate by the laser beam in order to design the optimal heat source profile.This paper describes a mathematical model used to design optimised heat source profiles and shows the experimental correlation with the theory. The model simulates a moving complex heat source by a continuous distribution of point sources. The resultant temperature field is found by summing the individual point source contributions. Temperatures both outside and within the beam-substrate interaction zone can be calculated.For a complex shaped heat source the resultant temperature field is not analytically available but must be obtained using numerical integration. Modem mathematical software tools make complex numerical integration available to everyone but there are pitfalls that need to be avoided e.g. ensuring solution convergence, dealing with discontinuities and evaluating errors. Simple methods of removing discontinuities to allow solution convergence whilst retaining solution accuracy are given. Techniques are also described for dealing with finite substrate thicknesses and wedge shaped substrates.The model has been used to design heat sources for several processes e.g. conduction limited welding, surface alloying and cladding and transformation hardening. Substrates considered include steels, plastics and glass. Details of the transformation hardening and conduction limited welding processes are given, with supporting experimental results obtained using DOEs manufactured at Loughborough. The DOEs routinely have efficiencies of over 90% (spatially profiled power/incident power), ensuring beam profiles that match those used in the calculations. Correlation with experimental data is good.The use of diffractive optical elements (DOEs) to customise the spatial intensity profile of a laser beam enables the beam to be optimised to the process, not the process to the beam. This requires knowledge of the changes effected in the substrate by the laser beam in order to design the optimal heat source profile.This paper describes a mathematical model used to design optimised heat source profiles and shows the experimental correlation with the theory. The model simulates a moving complex heat source by a continuous distribution of point sources. The resultant temperature field is found by summing the individual point source contributions. Temperatures both outside and within the beam-substrate interaction zone can be calculated.For a complex shaped heat source the resultant temperature field is not analytically available but must be obtained using numerical integration. Modem mathematical software tools make complex numerical integration available to everyone but there are pitfalls that need to be a...