无限大区域内少量冻结管稳态温度场解析解

在人工地层冻结法施工中，为满足一些特殊的工况需求，常采用布置两管或三管等少量冻结管的形式，虽然现在已有2根冻结管排列与3根冻结管以等间距直线排列时的稳态温度场解析解，但是遇到三管以其它形式排列的情况时，这些解已经不能满足需求。采用势函数叠加理论推导了3根冻结管以任意形式排列时的稳态温度场解析解。在此基础上得出了3根冻结管以等间距和不等间距直线、等腰三角形及直角三角形等排列形式下的温度场特定解。采用热学数值模拟方法对解析解进行了验证。结果表明，解析解计算结果与数值模拟结果较吻合，解析解正确。同时验证了采用势函数叠加法推导出的两管排列与三管等间距直线排列时的稳态温度场解析解与已有解完全相同。

Analytical solution of steady state temperature field of a few freezing pipes in infinite region

A few pipes freezing method is often applied in artificial ground freezing projects for special construction condition requirement，such as two freezing pipes or three freezing pipes.Although there exist analytical solutions of steady-state temperature field for two freezing pipes and for three freezing pipes arranged in a straight row with equal spacing，the solutions cannot meet the requirements of other arrangements of three freezing pipes.Based on the potential superposition method，authors gave analytical solutions of steady state frozen temperature for three freezing pipes arranged in form of arbitrary triangle.Particular solutions were derived for some particular arrangements，such as three freezing pipes in linear arrangement with equal or unequal spacing，right triangle arrangement and isosceles triangle arrangement.Comparison of the analytical solutions with the numerical thermal analysis shows that the analytical solutions are precise enough.Furthermore，analytical solutions of steady state temperature field for two freezing pipes and for three freezing pipes arranged in a straight row with equal spacing，derived by potential function superposition，are verified to be the same as the existing solutions.