Analytical and semi-analytical models of multi-applicability for medium-deep borehole heat exchanger

Medium-deep coaxial borehole heat exchanger models are essential for design and analysis in geothermal heating. However, current models suffer from high complexity and computational demands. Hence, a novel model was developed based on borehole fluid energy balance and infinite line-source model coupling at the borehole wall, deriving analytical, layered semi-analytical, and time–space discrete semi-analytical solutions. Model's universality was enhanced through verification using data from four representative regions, various borehole configurations and operating conditions. Model demonstrates exceptional accuracy, with short-term prediction errors not exceeding 3.75 % for heat extraction and ± 0.90 °C for temperatures, and no-load circulation prediction errors within ± 0.6°C. Long-term predictions over 3600 h maintain an absolute error of ± 1.65 °C. Comparative analysis confirms the model's leading-edge precision and adaptability to diverse scenarios. Analytical solution facilitates convenient heat transfer calculations with known inlet temperatures, layered semi-analytical form incorporates layered geological variations, and the time–space discrete semi-analytical solution enables dynamic load coupling, better reflecting actual operation. Achieving equivalent heat extraction in “inner → outer” flow requires higher flow rates and pump consumption compared to “outer → inner” mode. Excessive load fluctuation frequency and amplitude can make the inlet–outlet temperature drop faster.