Explicit multipole formulas and thermal network models for calculating thermal resistances of double U-pipe borehole heat exchangers

Abstract

Double U-pipe boreholes are the second most used type of ground heat exchangers after single U-pipe boreholes. The borehole thermal resistance is a key design and performance parameter for the double U-pipe boreholes. Another parameter that is particularly important for double U-pipe boreholes is the internal thermal resistance between the U-pipes. There is, however, a general lack of methods that can be used to calculate the thermal resistances of the double U-pipe boreholes. This article presents explicit multipole formulas for calculating thermal resistances of double U-pipe boreholes with symmetrically positioned pipes. The presented formulas include zeroth-and first-order expressions for the borehole thermal resistance and the internal thermal resistance. The internal thermal resistance formulas cover various possible flow configurations of the heat carrier fluid in the double U-pipes. The accuracy of the presented zeroth-and first-order formulas is established by comparing them to the original 10th-order multipole method. The article also presents thermal network models for computing effective borehole thermal resistance of double U-pipe boreholes from the multipole resistances. Formulas for calculating effective borehole thermal resistance of double U-pipe boreholes are presented for two idealized cases of uniform heat flux and uniform average wall temperature along the borehole.

Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation

“…However, the first-and second-order multipole formulas provide an accuracy of better than 3% and 1%, respectively, compared to the original multipole method under all practical conditions [20]. Similar trends have also been observed for double U-pipe ground heat exchangers by Claesson and Javed [21]. The multipole formulas for single and double U-pipe ground heat exchangers are also applicable to energy piles with a corresponding number of U-pipes.…”

“…Claesson and Javed [20] presented zeroth-, first-, second-, and third-order multipole formulas for single U-pipe ground heat exchangers with symmetrically positioned pipes. Later, the authors also presented zeroth-and first-order multipole formulas for double U-pipe ground heat exchangers with symmetrically positioned pipes [21]. The accuracy of the presented formulas depends upon the order of their multipoles.…”

Coupling a Borehole Thermal Model andMT3DMSto Simulate Dynamic Ground Source Heat Pump Efficiency

“…If two branches of the U‐pipe are symmetrical, then = ; is the equivalent thermal conductivity between two branches. Given specific borehole design properties and operational conditions, the equivalent thermal conductivity in Equation 1 can be calculated based on thermal resistance factors (Hellström 1991; Claesson and Hellström 2011). The detailed mathematical descriptions are given in Appendix S1.…”

Energy-saving potential of ground source multiple chillers in simple and hybrid configurations for Mediterranean climates

“…Some were based on the finite element method (FEM) [43], while others relied on multi-pole methods [40]. These approaches were subsequentially extended to the case of double U-tube BHEs [39,44]. In this paper, the resistance R g was calculated using the FEM to solve the steady-state heat conservation equation in the space domain described by the data in Table 1.…”