Analytical solution of steady-state temperature field of arc-shaped three-pipe liquid nitrogen freezing under adiabatic boundary condition

Abstract

Sudden seepage and leakage in underground engineering retaining structures are often mitigated using artificial liquid nitrogen freezing. However, the development of frozen soil curtains during liquid nitrogen freezing is frequently influenced by adjacent pile foundations and other structures. Currently, no theoretical solution exists for the temperature field distribution under these conditions. To investigate the temperature field distribution under adiabatic boundaries, such as adjacent pile foundations, and to understand the development of the freezing curtain under local adiabatic boundary constraints, a steady-state temperature field analytical solution of arc-shaped three-pipe liquid nitrogen freezing was derived using thermal potential superposition theory and the mirror method. This analytical solution for different characteristic sections was then contrasted with numerical simulations and model test results to verify its accuracy and applicability. Results indicate that the analytical solution aligns with the steady-state numerical solution, and the freezing model transitions from an unsteady to a quasi-steady state. The error compared to the transient numerical solution decreases over time, from 15.3 °C on the 10th day to 0.3 °C by the 50th day. The applicability and accuracy of the analytical solution are further validated using auxiliary interfaces. Comparing the analytical solution with model test results reveals that isotherms are perpendicular to the adiabatic boundary, with heat flow parallel to the boundary and no normal heat flow. The adiabatic boundary notably enhances the temperature field distribution and freezing efficiency. Finally, the accuracy of the analytical solution of the liquid nitrogen freezing model meets the requirements of practical engineering applications.