Dimensionless fluid-to-ground thermal response of single-line bore fields with isothermal fluid

The design of a borehole-heat-exchanger (BHE) field is usually performed by means of dimensionless functions, called g-functions, that yield the time evolution of the mean temperature of the external surface of the BHEs, Tsm, produced by a time constant heat flux, which is usually considered as uniform. The mean temperature of the fluid, Tfm, is then evaluated by adding to Tsm the product of the heat flux per unit length and the BHE thermal resistance. This method overestimates the difference between Tfm and Tsm in the short term, and overestimates the g-function of the field in the long term. Methods to obtain more accurate results have been proposed, but require difficult and time-consuming numerical computations. In this paper dimensionless fluid-to-ground (ftg) functions that yield directly the time evolution of Tfm, in a time scale from a few minutes to hundreds of years, are provided for any single-line bore field subjected to a time constant heat flux, composed of up to four BHEs fed in parallel with the same inlet temperature. The ftg-functions are obtained by finite-element simulations implemented in COMSOL Multiphysics, and are reported in two Excel files that, after entering the dimensionless parameters of the BHE field under examination, instantly yield a short-term and a long-term ftg-function perfectly joined at the separation instant. The main novelties of this work are the characterization of each BHE field by a few dimensionless parameters, the improvement of the BHE model presented in Naldi and Zanchini 2020, the accuracy, speed and simplicity of use of the final results. The validations of the simulation codes for a single BHE and for fields of 3 and 4 BHEs, by comparison with analytical solutions, yielded root-mean-square deviations equal to 0.023%, 0.43%, and 0.49% of the mean value, respectively. The validation of the simulation code for two BHEs, performed with an extremely high distance between the BHEs, yielded a root-mean-square deviation equal to 0.054% of the mean value, with respect to the long-term ftg-function obtained for a single BHE.