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

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