摘要
The multi-piped freezing method is usually applied in artificial ground freezing （AGF） projects to fulfill special construction requirements, such as two-, three-, or four-piped freezing. Based on potential superposition theory, this paper gives analytical solutions to steady-state frozen temperature for two, three, and four freezing pipes with different temperatures and arranged at random. Specific solutions are derived for some particular arrangements, such as three freezing pipes in a linear arrangement with equal or unequal spacing, right and isosceles triangle arrangements, four freezing pipes in a linear arrangement with equal spacing, and rhombus and rectangle arrangements. A comparison between the analytical solutions and numerical thermal analysis shows that the analytical solutions are sufficiently precise. As a part of the theory of AGF, the analytical solutions of temperature fields for multi-piped freezing with arbitrary layouts and different temperatures of freezing pipes are approached for the first time.
题目：少量管冻结稳态温度场数学模型目的：少量任意布置冻结管冻结的稳态温度场无解析解。建立任意布置少量管冻结稳态温度场模型，获得解析解，解决人工冻结温度场理论问题。创新点：1．基于势函数叠加原理，确立人工地层冻结中少量管冻结稳态温度场的通用求解方法；2．建立任意布置的3根和4根非等温冻结管下冻结稳态温度场数学模型，获得其解析解通解及特解。方法：1．通过理论分析，将冻结管简化为热汇点源，确定人工地层冻结热势的拉普拉斯方程表述；2．应用势函数叠加原理建立少量管冻结稳态温度场的通用求解方法；3．建立少量管冻结稳态温度场的数学模型，通过理论推导获得温度场解析解；4．通过数值模拟，验证所提方法、数学模型和解析解的正确性和准确性。结论：1．将冻结管简化为点源（热汇），其冻结形成的热势场服从拉普拉斯方程，其解即为热势函数；2．多根冻结管冻结时，将单根冻结管的热势函数叠加，由冻结管的位置决定每根冻结管的热流，再根据边界条件定解。这一方法（即势函数叠加法）可以用于任意布置冻结管冻结稳态温度场解析解的求解：3．将冻结管简化为点源导致获得的解析解存在一定的误差，但误差仅发生在冻结管附近极小的范围内，并且误差微小，完全满足工程上的精度要求。
基金
Project supported by the National Natural Science Foundation of China （Nos. 51178336 and 51478340）, the Natural Science Foundation of Zhejiang Province, China （No. LZ13E080002）, and the China Ministry of Communications Construction Science ＆ Technology Projects （No. 2013318R11300）
