In this paper, we state the mathematical modeling of heat transfer in the furnaces in the three-dimensional case. This modeling consists of the continuity equation, the Navier-Stokes equations, and the heat equation with the free boundary Stefan condition. We derive the variational formulation of the equations, and we invoke the finite element method technique to get the numerical solution of the system.

heat transfer, Navier-Stokes, finite element method, Stefan condition.