Fluid and borehole wall temperature profiles in vertical geothermal boreholes with multiple U-tubes

An analytical model for the calculation of fluid temperature profiles in geothermal boreholes with multiple U-tubes is presented. A linear system of first order differential equations is built based on a steady-state thermal resistances approach to the heat transfer between the U-tube pipes and the borehole wall. Analytical expressions for fluid temperature profiles are provided for boreholes with arbitrary borehole wall temperature profiles as well as piece-wise uniform and uniform borehole wall temperature for different U-tube configurations: independent, parallel and series. The analytical model is coupled to a finite line source model of the heat transfer between the borehole and the ground. Fluid and borehole wall temperature profiles are solved simultaneously using numerical Laplace transforms to consider the time variation of the temperatures and heat extraction rates. Results are verified against a finite difference model of the borehole. Differences between the fluid temperatures calculated with the numerical and analytical models are smaller than 0.003 °C.