Analytical solutions of steady-state temperature field for large-section freezing with rectangular layout of single-ring holes

The temperature field is the basis for assessing the mechanical state and water-sealing performance of the frozen wall, which is an important research direction of the artificial freezing theory. For the freezing pipes in the form of a closed circumferential arrangement, there are only analytical solutions under regular annular conditions, including single-circle and double-circle models. However, the rectangular arrangement of freezing pipes is also very common in practical projects, especially for the subway station projects that use frozen concealed excavation, and the temperature field has not yet been answered. According to the geometric consistency of rectangular and annular layouts, based on the four-pipe model, a method of "replacing squares with circles" is firstly proposed for the rectangular problem. Furthermore, considering the boundary separable properties of the steady-state heat conduction control equation and the superposition principle of potential functions, the analytical solutions of the temperature field for rectangular arrangement with eight pipes and the generalized rectangular arrangement with multiple pipes are solved. By comparing with the transient numerical results the model test ones, the correctness and the applicability of the analytical solutions are verified. The results show that the temperature field exhibits a highly rectangular distribution characteristic near the pipe layout line, and the isotherm gradually transforms to a circular shape as it moves away from the freezing pipes. The inner side of the rectangular freezing wall develops faster than the outer side, and the temperature field inside and outside the 0℃ line is significantly affected. The influences of the freezing pipe arrangement on the geometric characteristics of the freezing wall should be reasonably considered in the design of freezing scheme.