Mathematical models of heating and melting of particles of fine-dispersed powder

As it develops, mathematical modeling finds more and more new areas of application, remaining an effective tool, including engineering. Mathematical models go the way of evolutionary development, increasing the adequacy in accordance with real physical processes. One of the relevant areas of mathematical modeling is associated with the developing technologies of additive prototyping. For example, in the manufacture of products from metal powders by the methods of additive technologies, in particular, selective laser melting, one of the practical issues is the selection of optimal parameters for the 3D printer. The solution to the optimization problem x of the 3D printer operation parameters should be based on a mathematical model of the process of heating and melting of metal particles. An approach based on the formation and solution of the heat equation with boundary conditions that take into account the spherical shape of the particle, the energy distribution in the cross section of the laser beam, and the relative spatial position of the particle and the laser beam is used as the basic concept of modeling. It is noted that to assess the structure of the formed parts, this approach is redundant, and the algorithm for integrating the partial differential equation has high computational complexity. To simplify the analysis task, the initial micromodel is transformed into heating and melting macromodels in which the temperature distribution over the volume of the particle is considered constant, and the external effect on the particle is reduced to heat transfer through the surface of the ball, from the upper side from the laser beam to the particle, and from the lower side — from particle to environment. For the macromodel, we obtained time diagrams of the temperature increase and the accumulated internal particle energy in time. It is concluded that it is possible to divide the space around the particle into zones: complete and incomplete melting, as well as a heating zone insufficient for melting. It is shown that the presence of such zones leads to the friability of the structure of the parts formed on the 3D printer.Keywords: additive technology, laser heating, heat conduction equation, micro-model, macro-model, heating-melting timing charts.