End Effects in the Three-Omega Method to Measure Gas Thermal Conductivity

A two-dimensional analytical solution is derived for the three-omega method for measurement of thermal conductivity of materials with a fine wire. The analytical solution includes the wire heat capacity and the effect of heat losses from the ends of the wire. To derive the solution, finite Fourier transforms are applied in the direction parallel to the wire axis. The solution is compared with a one-dimensional solution and experimental data. It is found that heat losses from the wire ends have a significant effect on the 3ω components at low frequency and tend to be less important at high frequency. Moreover, it is shown that two-dimensional effects will be severe for nano-scale wires, even if the wire length-to-diameter ratio is very large.