3D numerical modeling of vertical geothermal heat exchangers. Ozudogru, T., Olgun, C., & Senol, A. Geothermics, 51:312-324, 2014.
3D numerical modeling of vertical geothermal heat exchangers [pdf]Paper  3D numerical modeling of vertical geothermal heat exchangers [link]Website  abstract   bibtex   
This paper presents the development and validation of a 3D numerical model for simulating vertical U-tube geothermal heat exchangers (GHEs). For minimizing the computational effort, the proposed numerical model uses 1D linear elements for simulating the flow and heat transfer inside the pipes. These linear elements are coupled with the 3D domain using the temperature field along the exterior surface of the pipe and an optimized finite element mesh for reducing the number of elements. The discretization of geometry, finite element mesh generation and the specifics of the system physics and boundary condition assignments are explained in detail. The model is used to simulate two generic cases, a borehole with a single U-tube and an energy pile with double U-tubes. In each case, a constant heating followed by a recovery period (i.e., no heating) is simulated. A review of the theory of finite line source model is also presented, along with modifications to account for variable heat rate. Moreover, a method to estimate the steady state thermal resistances in the borehole/energy pile is presented in order to calculate the fluid temperatures analytically. The validation of the model is carried out by comparing the numerical results with the results obtained from the analytical model.

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