Application of the heat equation to the study of underground temperature.

Modeling underground temperatures provides a practical application of the one-dimensional heat equation. In this work, the one-dimensional heat equation in surface soil is extended to include heat carried by the vertical flow of rainwater through the soil. Analytical solutions, with and without water flow, illustrate the influence of rainwater circulation on the sub-surface propagation of seasonal temperature variations, an important effect that is generally neglected in textbooks. The surface temperature variations are damped by the soil, and this effect was used by the Troglodytae in Egypt or the Petra in South Jordan to insulate against extreme temperatures. For a realistic case of horizontally layered geology, a finite volume Python code was developed for the same purpose. Subsurface temperatures were also measured over a full year at depths up to 1.8 m and used to estimate the thermal skin depth and thermal wavelength. This study provides students with a practical example of how a textbook physics problem can be modified to extract information of contemporary importance in geophysics and global warming. Editor's Note: Frequently, modeling the three-dimensional world is too challenging for a class exercise, and we instead teach students using unrealistic one-dimensional problems. So it is wonderful when a one-dimensional model turns out not only to be fairly accurate, but also to have real world implications. That is the case with heat flow through the soil, which can be treated with a one-dimensional model, and which is important to energy-efficient solutions such as underground housing and ground-source heat pumps. This paper finds the underground temperature as a function of depth through measurements, analytical solutions, and computational solutions. These solutions can form the basis for a very interesting class assignment. In addition to considering heat transport through conduction, the model also includes heat transport through the flow of rain water. [ABSTRACT FROM AUTHOR]