Mathematical Model of the Processes of Heat and Mass Transfer and Diffusion of the Magnetic Field in an Induction Furnace

Abstract

We propose a mathematical model describing the motion of a metal melt in a variable inhomogeneous magnetic fi eld of a short solenoid. In formulating the problem, we made estimates and showed the possibility of splitting the complete magnetohydrodynamical problem into two subproblems: a magnetic fi eld diffusion problem where the distributions of the external and induced magnetic fi elds and currents are determined, and a heat and mass transfer problem with known distributions of volume sources of heat and forces. The dimensionless form of the heat and mass transfer equation was obtained with the use of averaging and multiscale methods, which permitted writing and solving separately the equations for averaged fl ows and temperature fi elds and their oscillations. For the heat and mass transfer problem, the boundary conditions for a real technological facility are discussed. The dimensionless form of the magnetic fi eld diffusion equation is presented, and the experimental computational procedure and results of the numerical simulation of the magnetic fi eld structure in the melt for various magnetic Reynolds numbers are described. The extreme dependence of heat release on the magnetic Reynolds number has been interpreted.Keywords: induction melting, superalloy, variable magnetic fi eld, magnetic fi eld diffusion, inductive current, heat and mass transfer, convection, multiscale method, averaging method.Introduction. Induction heating is widely used in various fi elds of metallurgy and in precision production of semiconductor dielectric elements: induction melting, zone melting, strengthening of the surface of metal products, crystal growing, and production of optical fi bers. The main idea of induction heating is the use of the Joule heat of currents induced in a conducting material by a variable external magnetic fi eld. The power released by the currents leads to local heating of the material and, consequently, to changes in the density and equilibrium concentration of elements. The description of the heat and mass transfer in such systems requires, as a rule, solving a system of electrodynamical, thermal energy transfer, and mass transfer equations.If the pulse rate of the fi eld is fairly high, then the resulting magnetic fi eld composed of the external fi eld of the inductor and the fi elds of inductive currents attenuates into the depth of the conductor and becomes practically zero at some distance ζ from its surface [1]. Within the limits of the boundary layer ζ, inductive currents (heat sources) and bulk electromagnetic forces acting on the moving melt are localized. Such an approach is convenient if the intensity near the conductor surface is known.For the case where the fi elds cannot be assumed to be homogeneous [2], a popular approach is describing the fi eld is terms of the vector potential A whose rotor is equal to the magnetic fi eld intensity H = 1

Experimental Study of Liquid Metal Flow for the Development of a Contact-Less Control Technique

“…The induction mechanism does not have such a disadvantage, but instead, it can generate a vortex stirring flow, the influence of which needs to be reduced. The behavior of the liquid metal flow in the above configuration is studied in detail in the context of induction melting problems [15][16][17][18], where the height of the cell is less than the height of the solenoid. Paper [19] numerically studies natural convection in an inclined cylinder placed in an alternating magnetic field.…”