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# A New Thermal Response Factor For Single U-Tube Borehole Heat Exchangers

The vertical borehole heat exchangers (BHEs), which often consist of multiple boreholes, have been widely used in ground heat pump system for developing shallow geothermal energy [1-2]. In studying BHEs’ thermal behavior analytically, the thermal response factor (TRF) is often been employed [1-7]. Several TRFs have been presented in literatures, such as infinite line source function, finite line source function, and g-function. Among these functions, the g-function proposed by Eskilson [7] is much more popular and has been widely used in practice. According to Eskilson’s work [7], g-function is defined as the dimensionless borehole wall temperature which is assumed to be uniform along borehole length.
To realize a high computation efficiency, Eskilson calculated many g-functions for bore fields with specified arrangements with finite difference method. Then pre-calculated g-functions will be stored for further calculations by interpolations. The g-function concept has been incorporated into some BHEs simulation software [8-9], such as EED and GLHEPRO, which also contain a large amount of g-functions for typical bore fields. Recently, Cimmino et al [10] proposed a semi-analytical method for calculating g-function, where boreholes are divided into small segments along borehole length. Temperature drop in soil can be calculated with the superposition principle while all segments can be considered as finite line sources (FLS). The semi-analytical method may facilitate g-function method, as it can be incorporated into software more flexibly and conveniently compared to the numerical method.
According to the original definition of multi-borehole g-function [7], all borehole wall temperatures are also assumed to be uniform, which is apparently unreasonable. In the multi-borehole field, thermal interferences exist among all boreholes. As a result, individual borehole wall temperature will depend on its relative position in the bore field.
In light of fluid flowing into each borehole wall from out from the same water distributor, a uniform inlet fluid temperature [11-12] can be assumed while fluid temperature variation along horizontal pipes is negligible. Subsequently, a new TRF of inlet fluid temperature can be developed under assumption of uniform inlet fluid temperature, as well as uniform borehole wall temperature. A semi-analytical method will be also built for the new thermal response function. Furthermore, In order to promote computation efficiency, the new TRF will be calculated by stages. In the early stage, the borehole length need not to be divided into small segments while finite line source can be replaced with the infinite line source function. Borehole wall temperature difference for a 3×3 bore filed will be also analyzed by the semi-analytical method.

REFERENCES
[1] Rees, S.: Advances in Ground-Source Heat Pump Systems, Woodhead Publishing Ltd (2016).
[2] Diao, N. R., and Fang Z. H.: Ground-coupled heat pump technology, Higher Education Press, Beijing, (2005) (In Chinese).
[3] Ingersoll, L. R., Zobel, O. J., and Ingersoll, A. C.: Heat conduction with engineering, geological, and other applications, Madison: The University of Wisconsin Press, (1954).
[4] Zanchini, E., and Lazzari, S.: New g-functions for the hourly simulation of double U-tube borehole heat exchanger fields, Energy, 70(6), (2014), 444-455.
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[6] Fossa, M. and Priarone, A.: Constant temperature response factors for fast calculation of sparse BHE field g-functions, Renewable Energy, (2018), 1236-1246.
[7] Eskilson, P., Thermal Analysis of Heat Extraction Boreholes, Thesis (PhD), University of Lund, Lund (1987).
[8] Hellstrom, G., and Sanner, B.: Earth energy designer: software for dimensioning of deep boreholes for heat extraction. Department of Mathematical Physics, Lund University, Sweden (1994).
[9] Spitler, J. D.: GLHEPRO—a design tool for commercial building ground loop heat exchangers, In: Proceedings of the fourth international heat pumps in cold climates conference, Aylmer, Quebec, August 17–18, (2000), 1-15.
[10] Cimmino, M., and Bernier, M.: A semi-analytical method to generate g-functions for geothermal bore fields, International Journal of Heat & Mass Transfer, 70, (2014), 641-650.
[11] Malayappan, V., and Spitler, J.: Limitations of using uniform heat flux assumptions in sizing vertical borehole heat exchanger fields, in Proceedings of Clima, Prague, Czech Republic, (2013), June 2013, 1-10.
[12] Luo, W., Tang, C., Yin, F., et al.: Thermal Performance Analyses of Multi-borehole Ground Heat Exchangers, Geofluids, 3, (2017), 1-11.