The assessment of the consequences of hypothetical accidents in liquid-metal-cooled fast reactors often requires interaction between analysis and in-pile experiments, where experiments must provide geometry, boundary conditions, and thermal profiles that are prototypical of the accident scenario. Neutronic heating of test samples initially produces atypical thermal profiles, and a time period is required to elapse for thermal inversion. An analytic transient heat conduction analysis using multiregion eigenfunctions is provided to determine the space-time temperature profiles. With an assumed weak temporal dependence for eigenfunctions greater than the first, a determination of the motion of the position of maximum temperature is made, leading to a simple expression for the time to thermally invert completely, which requires knowledge of only the first eigenvalue and the expansion coefficient of the source for the fundamental mode, with similar analysis providing an estimate of the time to reach melting. A functional relationship is established between the operating reactor power, the thermal properties of the materials, and the boundary conditions to ensure satisfaction of both criteria of rapid thermal inversion and maximum temperatures above prescribed levels, such as melting. The analysis is then applied to a proposed in-pile experiment for studying pool boilup in internally heated fuel-steel pools with nuclear heated walls. It is shown that for a variety of external boundary conditions, a reactor power level may be chosen to ensure integrity of the insulating wall while simulating the pool boilup phenomena without the necessity of enrichment grading to enhance thermal inversion.