Relation between mean fluid temperature and outlet temperature for single U-tube boreholes

Ground-coupled heat pump (GCHP) systems usually utilize buried vertical heat exchangers, named borehole heat exchangers (BHEs). The accurate design or simulation of a GCHP system requires the calculation of the time-dependent outlet temperature from the BHEs, T out. However, the most widely employed BHE simulation models yield the time evolution either of the mean temperature of the BHE-ground surface, T sm, or of that of the fluid, T fm. In transient regime, it is not easy to relate T out to either T sm or T fm. In this paper we determine, through 3D finite element simulations, simple expressions of a dimensionless coefficient φ allowing the calculation of T out by means of a simulation model that yields T fm. These expressions hold for single U-tube BHEs, both in quasi-steady and in unsteady working conditions. We validate our 3D simulation code by comparison with an analytical BHE model. Then, we present applications of our expressions of φ to calculate the time-dependent values of T out through a BHE model that yields those of T fm. Finally, we show that the values of φ in quasi-steady working conditions can be used for a simple calculation of the effective borehole thermal resistance.