An exact transient analytical solution of the temperature fields and heat fluxes for the modified transient plane source (MTPS) method is presented. The MTPS sensor is made of a thin disc with a heater/sensor and a heated guard ring on its surface, providing a one-dimensional heat flow to a semi-infinite sample in contact with its surface. The MTPS sensor measures the thermal effusivity of the sample. Two different solutions are derived, one is in the form of an infinite series with no thermal resistance, the other is in integral form and includes thermal resistance between the MTPS sensor surface and the measured sample. The theoretical model is supplemented by experimental results.
The problem of a three-dimensional (3D) heat flow from a circular heat source (CHS) embedded inside a composite solid of two isotropic but different semi-infinite media is solved for the first time in this paper. This CHS asymmetrical measurement setup is useful when two identical samples are not available for measurement. Two different time-dependent temperature fields are derived for the composite semi-infinite media, as well as their corresponding heat fluxes. The derivation of the 3D solution uses first principles with basic assumptions and employs the Hankel and Laplace transforms. The Laplace inversion theorem is used to find the inverse Laplace transform of the temperature functions, since no tabulated inverse transform functions are available for this case. The solution is exact with no approximations and is given in an integral form, which can easily be evaluated numerically. This solution is a generic one and can be applied to more complex asymmetrical setups, such as the case involving thermal contact resistances.
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