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地埋管换热器动态热负荷下地层温度场的解析解

李嘉舒 戴传山 雷海燕 马非

李嘉舒,戴传山,雷海燕,等. 地埋管换热器动态热负荷下地层温度场的解析解[J]. 水文地质工程地质,2023,50(0): 1-9 doi:  10.16030/j.cnki.issn.1000-3665.202205040
引用本文: 李嘉舒,戴传山,雷海燕,等. 地埋管换热器动态热负荷下地层温度场的解析解[J]. 水文地质工程地质,2023,50(0): 1-9 doi:  10.16030/j.cnki.issn.1000-3665.202205040
LI Jiashu, DAI Chuanshan, LEI Haiyan, et al. Analytical solution of formation temperature distribution under dynamic heat load of borehole heat exchangers[J]. Hydrogeology & Engineering Geology, 2023, 50(0): 1-9 doi:  10.16030/j.cnki.issn.1000-3665.202205040
Citation: LI Jiashu, DAI Chuanshan, LEI Haiyan, et al. Analytical solution of formation temperature distribution under dynamic heat load of borehole heat exchangers[J]. Hydrogeology & Engineering Geology, 2023, 50(0): 1-9 doi:  10.16030/j.cnki.issn.1000-3665.202205040

地埋管换热器动态热负荷下地层温度场的解析解

doi: 10.16030/j.cnki.issn.1000-3665.202205040
基金项目: 国家重点研发计划(2019YFB1504205)
详细信息
    作者简介:

    李嘉舒(1994-),女,博士研究生,地热能开发利用。E-mail:li_jiashu@ tju.edu.cn

    通讯作者:

    雷海燕(1974-),女,博士,副教授,地热能开发利用。E-mail:leihy@tju.edu.cn

  • 中图分类号: TK521

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

  • 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。本研究提出了一种考虑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展开近似,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。本文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,便于工程实际的推广应用。
  • 图  1  无限长柱源导热模型示意图

    注:Rw为井筒半径,r为地层半径,R为地层远端半径,T为地层远端温度。

    Figure  1.  Schematic diagram showing the heat transfer model of BHE

    图  2  天津市气温与地埋管延米功率的变化

    Figure  2.  Ambient air temperature (a) and heat flux at the borehole wall (b) in Tianjin

    图  3  不同径向距离下地层温度解析解与数值解的对比

    Figure  3.  Comparison of analytical and numerical solutions of (a) temperature changes and (b) temperature deviations

    图  4  不同半径处的地层温度随时间的变化

    Figure  4.  Temperature changes at different radii of BHEs with time

    图  6  同年周期负荷下的地层温度随时间的变化

    Figure  6.  Comparison of temperature changes at different radii of BHEs in different area

    图  5  哈尔滨市年气温与地埋管延米功率变化

    Figure  5.  Air temperature (a) and (b) heat flux at the borehole wall in Harbin

    图  7  不同地区地埋管不同半径处地层温度总周期延米功率的解析解

    Figure  7.  Temperature changes at different radii of BHEs in different parts of analytical solution in (a) Harbin and (b) Tianjin

    表  1  地埋管几何与地层物性参数

    Table  1.   Geometrical and physical parameters given in the simulation

    参数名称参数名称
    Rw /mm55T0/°C15
    λs/(W·m−1·K−12.2(ρCp)s/(J·m−3·K−12.16×106
    运行周期/月
    (2×L)
    12运行时间/a10
    nm/a20nh/d500
    下载: 导出CSV

    表  2  不同地区的傅里叶展开级数

    Table  2.   Fourier expansion series in different areas

    地区$ {a_0} $$ {a_{_n}} (n=1,\cdots,n)$$ {b_{_n}} (n=1,\cdots,n)$
    天津$\dfrac{1}{6}\left(\dfrac{8}{\text{π} }{A}_{1}-\dfrac{8}{\text{π} }{A}_{2}\right)$$\dfrac{ {6{A_1}\left[ {\cos \left(\dfrac{ {2n{\text{π } } } }{3}\right) + 1} \right] - 6{A_2}\left[ \cos \left(\dfrac{ {n{\text{π } } } }{3}\right) + \cos (n{\text{π } }) \right]} }{ {(3 + 2n)(3 - 2n){\text{π } } } }$$\dfrac{ {6{A_1}\sin \left(\dfrac{ {2n{\text{π } } } }{3}\right) + 6{A_2}\left[ {\sin \left(\dfrac{ {n{\text{π } } } }{3}\right) + \sin (n{\text{π } })} \right]} }{ {(3 + 2n)(3 - 2n){\text{π } } } }$
    哈尔滨$\dfrac{1}{6}\left(\dfrac{4}{\text{π} }{A}_{1}-\dfrac{12}{\text{π} }{A}_{2}\right)$$\begin{array}{l}\dfrac{12{A}_{1}\left[\mathrm{cos}\left(\dfrac{n\text{π} }{3}\right)+1\right]}{(6+2n)(6-2n)\text{π} }(n=3,0)+\\ \dfrac{4{A}_{2}\left[\mathrm{cos}(\dfrac{5n\text{π} }{3})+\mathrm{cos}\left(\dfrac{2n\text{π} }{3}\right)\right]}{(2+2n)(2-2n)\text{π} }(n=1,-\dfrac{\sqrt{3} }{4})\end{array}$$\begin{array}{l}\dfrac{12{A}_{1}\left[\mathrm{sin}\left(\dfrac{n\text{π} }{3}\right)\right]}{(6+2n)(6-2n)\text{π} }(n=3,\dfrac{1}{6})+\\ \dfrac{4{A}_{2}\left[\mathrm{sin}(\dfrac{5n\text{π} }{3})+\mathrm{sin}\left(\dfrac{2n\text{π} }{3}\right)\right]}{(2+2n)(2-2n)\text{π} }(n=1,-\dfrac{1}{4})\end{array}$
    天津年周期负荷:${q}_{\text{TJ} }(t)={A}_{1}\mathrm{sin}\left(\dfrac{\text{π} }{4}{t}_{\text{m} }\right)\left(0\leqslant {t}_{\text{m} }\leqslant 4\;\text{m}\right)+{A}_{2}\mathrm{sin}\left(\dfrac{\text{π} }{\text{4} }{t}_{\text{m} }-\dfrac{\text{π} }{2}\right)(6\leqslant {t}_{\text{m} }\leqslant 10\;\text{m})$
    哈尔滨年周期负荷:${q}_{\text{HB} }(t)={A}_{1}\mathrm{sin}\left(\dfrac{\text{π} }{2}{t}_{\text{m} }\right)(0\leqslant {t}_{\text{m} }\leqslant 2\text{m})+{A}_{2}\mathrm{sin}\left(\dfrac{\text{π} }{6}{t}_{\text{m} }+\dfrac{\text{π} }{\text{3} }\right)(4\leqslant {t}_{\text{m} }\leqslant 10\text{m})$
    傅里叶展开公式:$q\left( t \right) \approx \displaystyle\sum\limits_{i = 0}^n { {q_i}(t) = \;} {a_0} + \displaystyle\sum\limits_{n = 1}^n {\left( { {a_n}\cos \dfrac{ {n{\text{π } } } }{L}t + {b_n}\sin \dfrac{ {n{\text{π } } } }{L}t} \right)}$
    下载: 导出CSV
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  • 收稿日期:  2022-01-05
  • 修回日期:  2022-06-03

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