Analytical solution of formation temperature distribution under dynamic heat load of borehole heat exchangers

The borehole heat exchanger (BHE) is a key component using shallow geothermal energy in ground source heat pump systems (GSHPS), and reasonable pipe spacing design has a great impact on the heat transfer performance and economy of the GSHPs. In most of real applications, the thermal disturbance radius of the maximum heat load per unit length (that is, the most unfavorable case) is often used as the design basis, and this makes it difficult to achieve the technical and economic optimization of the ground source heat pump system. This paper proposes a simple but more practical mathematical method to obtain the thermal disturbance radius of the borehole heat exchanger. The method first derives an analytical solution of the formation temperature distribution around the borehole under the boundary condition of periodic heat flow. On this basis, the actual dynamic building heating and cooling load is approximately expanded into a finite of sine and cosine periodic functions with the Fourier series. By superimposing the analytical solution corresponding to each periodic function obtained by Fourier series expansion of the original dynamic load, the variation of formation temperature distribution under the actual dynamic heating and cooling load conditions can be obtained.