Analysis of the non-linearity of the heat transfer equation in case of a time-dependent heat source: application to the $$3\omega $$ method

Abstract

The 3w method may be used to estimate the thermal conductivity of an electric conducting wire. In this method, an alternating voltage with an angular frequency w is applied to the wire. The resulting low tension 3 of angular frequency 3w that appears in the total tension is extracted by a lock-in amplifier. The amplitude of 3w is directly linked to the thermal conductivity of the wire and enables its estimation. All authors using the 3w method for the determination of the thermal conductivity of an electric conducting wire considered that the heat flux produced by Joule effect in the wire is constant. This hypothesis leads to a linear form of the heat transfer equation. In this paper, an analytical model taking into account the dependence of the heat flux on the temperature is developed, it leads to a non-linear form of the heat transfer equation. The importance of the non-linearity in certain cases is demonstrated and the analytical solution is used to define a unique criterion that must be verified to ensure the validity of the linear solution.