Comparison between conduction models and models including the energy balance along the flow, for the simulation of U-tube borehole heat exchangers

Most ground-coupled heat pumps exchange heat with the ground by means of a borehole-heat-exchanger (BHE) field. The time evolution of the fluid temperature at the outlet of the BHE field, employed for the design of the heat pump, is often determined starting from that of the mean temperature of the surface between the BHEs and the ground, evaluated by means of a dimensionless function called g-function. In order to obtain an accurate thermal response to peak loads, this method must be coupled with a short-term simulation tool. Simulation models that yield accurately the time evolution of either the outlet fluid temperature or the mean fluid temperature both in the short and in the long term are also available. The simplest of these models, called here conduction models, represent the fluid by a solid that receives or generates heat. Models that include the energy balance for the flow along the pipes, called here complete models, have also been proposed. Complete models are the most accurate, but can hardly be applied for long-term simulations. In this paper, the results of finite-element simulations of U-tube BHES performed by conduction models are compared with those obtained by complete models, implemented through the COMSOL Pipe Flow Module. It is shown that there is an excellent agreement between the models in the short term, while complete models yield slightly higher values of the mean fluid temperature in the medium and long term. It is also shown that one can obtain by a conduction model the same time evolution of the mean fluid temperature that would be yielded by a complete model, replacing the real 2D BHE thermal resistance, R_b2D, with a virtual one, equal to the asymptotic value of the 3D thermal resistance yielded by the complete model, R_bPF. For BHEs with length 100 m, diameter 152 mm, grout thermal conductivity 1.6 W/(m K), ground thermal conductivity 1.8 W/(m K) and flow rate 14 L per minute, the ratio R_bPF/R_b2D is, for instance, 1.032 for a single U-tube BHE with shank spacing s = 94 mm, and 1.090 for a double U-tube BHE with s = 85 mm. Dimensionless correlations yielding the value of R_bPF/R_b2D for any U-tube BHE will be provided in a future paper.