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 phi 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 phi 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 phi in quasi-steady working conditions can be used for a simple calculation of the effective borehole thermal resistance.