Analytical solution of formation temperature distribution under dynamic heat load of borehole heat exchangers
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摘要:
垂直地埋管换热器（borehole heat exchanger，BHE）是利用浅层地热能的主要换热装置，如何确定合理的地埋管间距对地源热泵系统（ground source heat pump system，GSHPs）的传热性能与经济性影响很大。以往工程应用中未考虑地埋管热负荷的动态变化，常采用最大延米热/冷负荷（即最不利情况下）的影响半径作为设计依据，使设计参数趋于保守，很难实现地源热泵系统的技术和经济优化，而考虑负荷变化的数值模拟方法耗时复杂，不便于工程应用。文章提出了一种在地埋管实际热冷负荷动态变化条件下，计算地埋管换热器影响半径的简单数学方法。该法首先推导了地埋管换热器在周期性热流边界条件下，井筒周围地层温度场的解析解，在此基础上将地面建筑物全年周期下的实际波动热冷负荷进行傅里叶级数近似展开，最后通过线性叠加每个周期函数对应的解析解，得到建筑物实际动态热冷负荷下的地层温度动态分布。提出的解析解实时耦合了地面建筑动态热冷负荷，计算结果接近实际应用，具有计算精度高、简单方便快捷的优点，便于在工程实际中推广应用。
Abstract: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.
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表 1 地埋管几何与地层物性参数
Table 1. Geometrical and physical parameters given in the simulation
参数名称 值 参数名称 值 Rw /mm 55 T0/°C 15 λs/（W·m−1·K−1） 2.2 (ρCp)s/（J·m−3·K−1） 2.16×106 tm
（2L）/月12 t/a 10 nm 20 nh 500 注：nm、nh为年、天周期傅里叶展开级数。 
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表 2 不同地区的傅里叶展开级数
Table 2. Fourier expansion series in different areas
地区 
天津 
哈尔滨 
天津年周期负荷： 
哈尔滨年周期负荷：
傅里叶展开公式：
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