Analytical solution for surrounding rock temperature of cold-region tunnel considering phase change

The temperature of the surrounding rock in cold-region tunnels is crucial for antifreeze design, and the water-ice phase transition is essential to addressing the temperature field. This paper proposes a refined method that equates the latent heat of the ice-water phase transition to heat capacity and establishes a one-dimensional radial heat transfer model considering phase change. By defining an average thermal diffusivity coefficient through the concept of equal accumulated temperature, this method overcomes the limitations of classical heat transfer theory in directly solving the temperature field of three zone (unfrozen zone, freezing zone and frozen zone). Additionally, by employing the variable separation method and Fourier integral transformation method, the analytical formula for the transient temperature field considering phase change is derived. Then, the analytical solution was verified based on the field data. The results calculated using this method exhibit greater consistency with field temperature data and outperform the modified Stephan formula in determining the maximum frozen depth of the surrounding rock. Finally, the simplified form of the established analytical solution was further discussed. The research results can provide a theoretical basis for the analysis of the temperature field of the surrounding rock of the tunnel in cold regions and its antifreeze design.