Conjugate direct resistance heating of metallic plates. multiplicities and stability

Abstract

A numerical bifurcation analysis is presented for an industrial application where direct resistance heating through a DC is applied to a flat metallic plate, which is cooled by a turbulent boundary layer and radiation. The process is modeled with a conjugate heat transfer between the plate and the cooling air steam. The convective part of the heat transfer mechanism is formulated in the framework of an integral approach, considering a turbulent core based on power law velocity and temperature profiles and a thin laminar sublayer thermally coupled with axial conduction along the plate. The analysis reveals that the problem admits two solutions: one stable and one unstable, separated by a limit point. The existence of multiple solutions is a consequence of the nonlinear electric resistivity–temperature relationship, allowing thermal equilibrium between heat generation and heat dissipation in multiple points. The application of realistic boundary conditions at the wall–fluid interface shows that the thermal to the hydrodynamic boundary layer thicknesses ratio along the plate is no longer close to the value of 1.25, as it is the case with a constant wall temperature. Instead, significant deviations occur due to the thermal coupling between the wall and the cooling fluid. The multiplicity structure and, consequently, the limit points depend on the plate Reynolds number and on the conduction–convection parameter. The locus of the limit points defines an instability threshold beyond which any excess applied current will trigger a thermal runaway phenomenon. This is also an equivalent of the maximum current carrying capacity of the plate.