Mathematical models of heating and melting of particles of fine-dispersed powder
https://doi.org/10.22405/2226-8383-2019-20-2-488-498
Abstract
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, micromodel,
macro-model, heating-melting timing charts.
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, micromodel,
macro-model, heating-melting timing charts.
About the Authors
Eugene Vasilyevich LarkinRussian Federation
Alexander Nikolaevich Privalov
Russian Federation
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Review
For citations:
Larkin E.V., Privalov A.N. Mathematical models of heating and melting of particles of fine-dispersed powder. Chebyshevskii Sbornik. 2019;20(2):488-498. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-2-488-498
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