Effect of Magnetic Field and Wall Conductivity on Conjugate Natural Convection in a Square Enclosure

Abstract

Steady, laminar, conjugate natural convection flow in the presence of a magnetic field in a square enclosure is considered. The left vertical wall of the enclosure is thick with a finite thermal conductivity, while the other three walls are taken to be of zero thickness. The enclosure is filed with liquid gallium and subjected to horizontal temperature gradient: the vertical boundaries are isothermal at different temperatures, whereas the remaining walls are adiabatic. Finite volume method is used to solve the dimensionless governing equations. The physical problem depends on seven parameters: Rayleigh number (500 ≤ Ra ≤ 106), the Prandtl number (Pr = 0.020), the wall to fluid thermal conductivity ratio (0.1 ≤ Kr ≤ 10), wall thickness (D = 0.2), the Hartmann number (50 ≤ Ha ≤ 200), the inclination angle of the magnetic field (ϕ = 0°, 45° or 90°) and the wall to fluid thermal diffusivity ratio (α* = 1). The main focus of the study is on examining the effect of both inclination angle and Hartmann numbers on fluid flow and heat transfer. The effect of Rayleigh number and conduction in the left wall is also considered. The obtained results, in the absence of a magnetic field, show that natural convection can be strengthened by the increase of both Rayleigh number and conductivity ratio, because of the increase of the effective temperature difference driving the flow. For poor conducting wall (Kr = 0.1), where the solid part is an insulated material and the thermal resistance is more important, the average Nusselt number is approximately constant and having low values comparing with equal (Kr = 1) and high (Kr = 10) conducting wall, indicating that most of heat transfer is by heat conduction. The interface temperature is found to be quite non-uniform, especially for high Ra and Kr. This non-uniformity tends to make the flow pattern in the enclosure asymmetric. In the presence of a magnetic field, the results show that for a given Ra, as the value of Hartmann number increases, convection is suppressed progressively and the rate of heat transfer is reduced in the enclosure. Convection mechanism is also affected by the direction of the magnetic field. It is found that the rate of convection heat transfer is more reduced with the x direction of the magnetic field (inclination angle ϕ = 0°). Also the results show that for poor conducting wall (Kr = 0.1), where the convection is dominated by heat conduction, the presence of a magnetic field is not important and its effect in this case can be neglected.