Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: Analytical solution and numerical modeling

A thermo-electric problem is solved analytically for an electrically conducting particle in a form of an ellipsoid of revolution immersed in the liquid metal and subjected to a temperature gradient. It is shown that the density of the thermoelectric current is constant inside the particle and its value depends on the eccentricity of the ellipse in the meridian plane of the ellipsoid, but does not depend on the size of the particle. Another parameter which affects the value of the thermoelectric current is the orientation of the ellipsoid with respect to the imposed temperature gradient. The vector of the thermoelectric current inside the particle and the vector of the imposed thermal gradient are co-planar, but a planar angle between these vectors exist and its value is also a function of the eccentricity of the ellipse and its orientation in a thermal field. Limiting minimal and maximal value of the thermoelectric current inside a very elongated particle are found and compared with values obtained in simulations for a dendrite grain. Numerical simulation performed with FEM software for two orientations of an elongated ellipsoid with respect to the imposed thermal gradient provided results similar to analytical solutions with the relative error less than 0.1%.