Finite element modeling of borehole heat exchanger systems: Part 2. Numerical simulation
Introduction
The computation of borehole heat exchanger (BHE) systems is of high practical importance in modern geothermal heat extraction technologies. In the first part of the paper (Diersch et al., in press) the basic theory of BHE finite element modeling has been developed. For the local BHE problem both analytical (Eskilson and Claesson, 1988) and numerical (Al-Khoury et al., 2005, Al-Khoury and Bonnier, 2006) solution strategies have been preferred. Improved relationships for thermal resistances of BHE have been introduced. Diersch et al. (in press) have derived a direct, widely non-sequential (essentially non-iterative) coupling strategy for the BHE and porous medium discretization.
The second part of the paper focusses on meshing aspects for BHE discretizations, the verification of BHE solutions including mesh convergence studies, programming developments for coupling the simulation code FEFLOW (DHI-WASY, 2010) with TRNSYS (TRNSYS, 2004, Bradley and Kummert, 2005) and the numerical simulation of the real-site borehole thermal energy store system Crailsheim, Germany. Thermal energy storage may refer to a number of technologies that store energy in a thermal reservoir for later reuse. Four different types of seasonal thermal energy stores have been developed for the use in solar-assisted district heating systems with seasonal heat storage and realized several times in Germany: hot-water thermal energy store, gravel-water thermal energy store, borehole thermal energy store and aquifer thermal energy store (Bauer et al., 2010). Due to their simplicity, borehole thermal energy stores can be a reasonable technical and economical alternative—depending on the local geological and hydrogeological situation—to other techniques of seasonal heat storage (e.g. Dincer and Rosen, 2002).
Section snippets
Meshing BHE nodes
In using the numerical (Al-Khoury et al., 2005, Al-Khoury and Bonnier, 2006) or the analytical (Eskilson and Claesson, 1988) solution strategies a BHE is reduced to an internal boundary condition occupied at a single node in a horizontal view on the 3D finite element mesh of the global problem. It appears similar to a well node, where a pumping well with a rate Qb in the borehole is modeled at a singular node via a well function applied to the sink/source term Q of (see Diersch et al., in press
Numerical versus analytical solutions of BHE for steady-state conditions and given temperature at borehole wall
We directly compare the numerical and analytical solution strategies by Al-Khoury et al. (2005), Al-Khoury and Bonnier (2006) and Eskilson and Claesson (1988) for local BHE problems under steady-state conditions. The analytical BHE solutions are compared to the numerical BHE results for CXA, CXC, 1U and 2U-type BHE configurations with the parameters as listed in Table 1. Since Ts is here specified as a boundary condition the solid properties become irrelevant for the present comparison
Application to borehole thermal energy stores
Borehole thermal energy stores (BTES) consist of a large number of borehole heat exchangers typically installed with spacing in the range of 2–5 m as the thermal interaction of the individual borehole heat exchangers is essential for an efficient storage process. BTES may be very sensitive to groundwater flow. Both for permit procedures required by the authorities and for plant-engineering issues, simulations are needed which are capable of predicting the 3D temperature profile in the
Summary and conclusions
Using BHE in regional discretizations optimal conditions of mesh spacing around singular BHE nodes are recommended to attain numerical accuracy. The direct estimation of the nodal distance can be sufficient under practical conditions. Such optimal meshes have shown superior to such discretizations which are either too fine or too coarse. Commonly, over-refined meshes around BHE nodes require the assignment of high contrast of thermal conductivity for elements within the physical BHE radius.
Acknowledgements
The authors acknowledge Rafid Al-Khoury (Delft University of Technology, The Netherlands) for his interest, suggestions and useful discussions of the work. This work was supported by the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety under Grant no. AZ 02E2-41V5034. The authors are grateful for this support.
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