Analytical heat transfer model for coaxial heat exchangers based on varied heat flux with borehole depth

In the classical analytical coaxial borehole heat exchanger (CBHE) models, the variations of the specific heat flux exchanged vertically along the depth of CBHE are ignored, which induces considerable prediction errors of the ground temperature. Based on the infinite line source model (ILSM), an improved varied heat flux model for the CBHE is developed using the superposition method, taking the vertical inhomogeneity of the specific heat flux into account. The mathematical expressions of the specific heat flux and the temperatures of the fluids and ground at different working conditions are obtained. The model is validated analytically and numerically. Finally, the influence of the mass flow rate, the outer-to-inner pipe radius ratio, and the thermal influence radius are investigated for optimal design. The results show that the specific heat flux is significantly related to the borehole depth, operation time, and fluid circulation direction. Compared with the finite line source method (FLSM), the relative errors of the ground temperature predicted by the proposed model can be reduced maximumly by 8.9% and 10.3% for heat extraction and injection, respectively. In addition, using higher mass flow rates and larger outer-to-inner pipe radius ratios can eliminate thermal interference to increase thermal efficiency. In general, the improved analytical model can provide an accurate and efficient method for this inhomogeneous problem and a valuable tool for designing and optimizing CBHEs for actual engineering applications.