3D Fast Multipole Boundary Element Method Analysis of Heat Exchange Performance of Buried Pipe Groups
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摘要: 基于三节点三角形线性单元，为克服单元跨叶子积分难题，将三维位势问题快速多极边界元法与几乎奇异积分的半解析算法相结合，实现了三维边界元法中几乎奇异积分的准确计算，该方法适用于U型地埋管薄体结构的换热分析. 在制冷、制热两种工况下研究了U型地埋管壁厚对换热量的影响，并进一步分析了管群间的热相互作用. 计算结果显示，当管壁导热系数一定时，管壁越厚，对管内流体和土壤之间的换热影响越大. 当钻孔间距一定时，管群中埋管数量越多，热干扰现象越强烈，提高管群换热量的主要措施是降低管群间热干扰. 因准确计算了几乎奇异积分，三维快速多极边界元法可以有效计算薄体和厚体耦合的三维热传导问题. 该文方法和分析结果可为地埋管换热器系统的工程应用提供参考.Abstract: Based on the 3-node triangular linear element and to overcome the element cross-leaf integration problem, a new 3D fast method was formulated for 3D potential problems through combination of the fast multipole boundary element method (FMBEM) with the semi-analytical algorithm of nearly singular integral, to realize the accurate calculation of the nearly singular integral in the 3D boundary element method (BEM). This method is applicable to the heat exchange of thin-wall structures of U-type buried pipe groups. In the cooling and heating conditions, the effects of the wall thickness of the U-type buried pipe group were analyzed by means of the new FMBEM, and the thermal interaction between multiple buried pipes was discussed. The calculation results show that, for a constant thermal conductivity of the pipe wall, the thicker the pipe wall is, the greater the effect on the heat exchange between the pipe fluid and the soil will be. For a constant borehole spacing, the bigger the number of buried pipes in a group is, the stronger the thermal interference between the pipes will be. The main strategy to increase the heat exchange of the pipe group is to reduce the thermal interference between the heat exchange pipes. Due to the accurate calculation of the nearly singular integral, the proposed 3D FMBEM can effectively solve the 3D heat exchange problems of thin-thick coupled bodies. This method and the results for the provide references for the engineering application of buried pipe heat exchangers.
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图 4 FMBEM在Z=10 mm处沿AB路径计算所得温度TFMBEM及相对误差Δ
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 4. Temperature TFMBEM and relative error Δ calculated with the FMBEM along the AB path at Z=10 mm
图 6 不同管壁厚度时FMBEM计算温度分布图
Figure 6. FMBEM calculated temperature distribution diagrams for different tube wall thicknesses
图 8 4×4管群FMBEM计算温度分布图
Figure 8. The 4×4 pipe system's FMBEM calculated temperature distribution diagrams
图 9 4×4地埋管群单位井深换热量对比
Figure 9. Comparison of heat transfer fluxes of 4×4 buried pipe groups
表 1 地埋管换热器设计参数
Table 1. The U-tube buried pipe design parameters
number parameter name value unit 1 borehole depth 50 m 2 borehole radius 75 mm 3 U-tube pipe outer radius 16 mm 4 U-tube pipe inner radius 13 mm 5 shank spacing 100 mm 6 soil radius 1.5 m 7 U-tube (PE pipe) thermal conductivity 0.4 W/(m·℃) 8 fill material thermal conductivity 2.4 W/(m·℃) 9 ground thermal conductivity 2.0 W/(m·℃) 10 inlet water temperature in summer 35 ℃ 11 outlet water temperature in summer 32 ℃ 12 inlet water temperature in winter 7 ℃ 13 outlet water temperature in winter 10 ℃ 14 undisturbed ground temperature 18 ℃ 
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表 2 不同管壁厚度时地埋管单位井深换热量(单位: W/m)
Table 2. Heat transfer fluxes of buried pipes with different wall thicknesses (unit: W/m)
wall thickness cooling heating Qi Qo Q=i+Qo Qi Qo Q=Qi+Qo 0 -29.466 -17.230 -46.696 20.433 8.205 28.638 3 -28.081 -13.300 -41.381 20.061 5.300 25.361 6 -21.922 -15.910 -37.832 14.640 8.548 23.188 8 -18.118 -14.471 -32.589 11.826 8.147 19.973 10 -15.927 -13.017 -28.944 10.332 7.407 17.739
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表 3 壁厚3 mm时4×4管群单位井深换热量(单位: W/m)
Table 3. Heat transfer fluxes of 4×4 pipe groups for a wall thickness of 3 mm (unti: W/m)
heat exchange cooling heating tube of №.① tube of №.② tube of №.③ tube of №.④ tube of №.① tube of №.② tube of №.③ tube of №.④ Qi -7.901 -10.999 -11.640 -14.856 6.055 8.218 8.390 10.352 Qo -1.530 -5.077 -5.637 -8.999 0.379 1.822 2.216 4.275 Q=Qi+Qo -9.431 -16.076 -17.277 -23.855 6.434 10.040 10.606 14.627
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表 4 不考虑壁厚时4×4管群单位井深换热量(单位: W/m)
Table 4. Heat transfer fluxes of 4×4 pipe groups regardless of the wall thickness (unit: W/m)
heat exchange cooling heating tube of №.① tube of №.② tube of №.③ tube of №.④ tube of №.① tube of №.② tube of №.③ tube of №.④ Qi -9.768 -14.009 -13.776 -17.944 7.966 10.579 10.447 13.002 Qo -0.519 -3.829 -4.128 -8.281 2.320 0.337 0.535 3.079 Q=Qi+Qo -10.287 -17.838 -17.904 -26.225 10.286 10.916 10.982 16.081
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