for the calculation of the three-dimensional borehole thermal resistance with the multipole method (Claesson and Hellstrom 2011).
A comparison of the three-dimensional borehole thermal resistance calculated using the multipole method (Claesson and Hellstrom 2011) and the Hellstrom (1991) extension, whose flexibility allowed the above considerations to be taken into account, suggested that only TRT-1 and TRT-3 had thermal performances below expectations.
The flexibility associated with the multipole method and its validation with a comparison of borehole thermal resistances determined with numerical simulations (Lamarche et al.
In contrast, the effective resistances inferred in the field for all the GHEs were in the range of three-dimensional resistances that can be calculated with the multipole method (Claesson and Hellstrom 2011) and the Hellstrom (1991) extension, taking into account a [+ or -] 10% variation of the pipe spacing and the grout thermal conductivity.
Multipole method to compute the conductive heat transfer to and between pipes in a composite cylinder.
Multipole method to calculate borehole thermal resistances in a borehole heat exchanger.
The solution of Equation 26 for the multipoles has a similar structure.
n,j] of the multipoles are determined from the boundary conditions in Equations 12 and 13 at the N pipes.
Here, the resistances in the first sum give the solution without multipoles (J = 0):
Equations to determine the strength of multipoles up to order J
The Fourier components from multipoles and line sources up to order J are given by the sum in k from 1 to J in Equation 31.