Frequency-dependent thermal resistance of vertical U-tube geothermal heat exchangers

Abstract

Geothermal ground-source heat pumps have been used for nearly 30 years as an environmentally friendly alternative to fossil fuel systems. The limitations on a wider range of acceptance of the technology depends on the cost of installation of a piping network through which energy is transferred between the soil and the heat-transfer fluid. The cost is proportional to the piping length. The most common of these exchangers is a U-tube system, which involves a single flexible tube that is put in a vertical borehole so that both ends remain at the ground surface. The tube radius is typically much smaller than the depth of the borehole, and this presents the opportunity to simplify the modeling through asymptotic techniques. We consider a simple Cartesian model that consists of two finite-length parallel channels carrying heat-transfer fluid embedded in a soil. One channel carries fluid from the surface to the bottom of the borehole, while the other carries the fluid from the bottom of the borehole to the surface. Heat transfer is driven in the fluid by advection and conduction, while only conduction is found in the soil regions, and we assume that the temperature in the fluid is quasi-steady on the soil conduction time scale. Applying asymptotic techniques, we find a separable boundary-value problem for the Laplace transforms of temperature difference in the channels and the average temperature of the fluid temperatures at a particular depth. By means of the basis functions found computationally from this boundary-value problem, we construct solutions for heat-transfer fluids entering the system at constant temperature and at time-harmonic entering temperature. From these results, we observe that optimal system performance depends on the frequency of the input temperature and the separation between the channels, with the exchanger performance depending on the thermal capacity of the internal soil layer.