On thermal resistance in concentric residential geothermal heat exchangers

• Published in: Journal of engineering mathematics 2014-06, Vol.86 (1), p.103-124
• Main Authors: Frei, S., Lockwood, K., Stewart, G., Boyer, J., Tilley, B. S.
• Format: Article
• Language: English
• Subjects: Annular; Applications of Mathematics; Computational fluid dynamics; Computational Mathematics and Numerical Analysis; Fluid flow; Fluids; Geothermal; Heat transfer; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Soil (material); Theoretical and Applied Mechanics
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-013-9655-4

Residential geothermal ground-source heat pumps have been used for nearly 30 years as a low-cost, environmentally friendly alternative to traditional fossil-fuel systems. However, the limitation on a wider range of acceptance of the technology is the cost of the installation of a piping network through which the energy is transferred between the soil and the coolant. This cost is proportional to the piping length. We formulate a new mathematical modeling framework that calculates a characteristic streamwise length based on the geometry of the system, the operating conditions, and the material properties of the system materials and effective properties of the surrounding soil using a vertical concentric geothermal heat exchanger as an example. These concentric systems consist of a core flow (from the residence), which flows from the ground surface to the base of the well, and an annular return region in which the heat exchange between the fluid and the soil is expected to take place. Two modeling scenarios are considered: steady-state temperature profiles in the annular fluid region if the radial thermal resistance between the fluid and soil is fixed; a quasi-steady fluid temperature that captures the radial heat transfer from the fluid to the soil. For the first case, we find that the characteristic length is determined by the smallest eigenvalue of the separable thermal problem, where the velocity profile is laminar and there is no thermal transport between the core and the fluid. When this core-annular heat transfer is possible, the eigenvalue problem no longer satisfies the conditions for Sturm–Liouville theory, and through direct computation we find that energy transferred from the annular flow to the core reduces the temperature change. In the second case, we find that the temperature change is reduced over time, as the soil temperature near the exchanger responds to the energy transport. In both cases, the best thermal transport takes place when the annular gap is small. The impact of these results on system design considerations is discussed.