Finite Element Method for Transport Phenomena in the Three-Dimension Case

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Аннотация

The mathematical modeling of heat transfer in the Garnissage furnace for glass melting in a three dimension domain is carried out. Since the furnace has a cylindrical form, all calculations are in the cylindrical coordinate system. We applied the fundamental physical conservation laws, that is the mass, the momentum, and the energy conservation laws. Using these laws we obtain the equations of continuity, the Navier-Stokes, and the heat transfer. On the stationary part of the boundary we use the classical boundary conditions. The free boundary which formed between the solid and liquid phases is modeled using three dimensional version of Stefan condition.

When we derived the mathematical modeling of the transport phenomena, we started to rearrange the system under consideration to the weak formulation by handling the sufficient test functions. In this part we divided the domain by cubes where every cubic part had 24 tetrahedrons. Then we defined the test functions on the tetrahedrons and we obtained the coefficients for the continuity system, and the same approach was used to get the other coefficients of the system (Navier-Stokes and energy conservation system). Then, after solving all the integrals on the mentioned tetrahedrons, we will complete the finite element techniques. All calculations are done using the software Mathematica.