A Three-Dimensional Analysis of Homogeneous and Functionally Graded Thermoelectric Cylinders

This paper analyzes a homogeneous thermoelectric (TE) circular cylinder as well as an exponentially graded TE cylinder under three-dimensional (3D) conditions. Analytical solutions of the temperature fields are obtained using an eigenfunction expansion method. The energy conversion efficiencies are determined and the material property gradation and leg geometry effects are examined. For the homogeneous TE material, it is found that the efficiency of the TE cylinder is almost the same as that of the cuboid TE leg with the side 2 W equal to the diameter 2R of the cylindrical leg. The efficiency of the cylindrical leg, however, is slightly higher than that of the cuboid leg if the two legs have the same length and cross-sectional area. In this case, the difference in energy efficiency between the two leg geometries depends on the ratio of the lateral dimension to the leg length and vanishes when the lateral dimension goes to infinity. For the exponentially graded TE cylinder, the numerical examples show that the efficiency is increased by appropriate material property gradations but is decreased by the 3D effects. Finally, the analytical solution indicates that besides the figure of merit, the two dimensionless parameters R/L and hL/k, where L is the leg length, h is the surface heat transfer coefficient, and k is the thermal conductivity, are also important factors that influence the energy efficiency of TE cylinders in 3D.