Similarity Criterion Based on the Analytical Solution of Heat Conduction Equation in Frozen Soil

[Objective] The evolution pattern of temperature field in frozen walls serves as a key factor in optimizing design schemes during artificial ground freezing. At present, laboratory model tests are a critical approach for investigating the development of temperature fields. This study aims to propose a method for deriving similarity criteria for temperature field model tests, offering essential theoretical guidance for the design of laboratory experiments. [Methods] First, analytical solutions to one-dimensional heat conduction differential equations, under both constant and nonlinear thermal parameters, were obtained using the method of separation of variables with different boundary conditions. Based on these solutions, similarity transformation techniques were employed to derive similarity criteria for frozen soil model tests, accounting for both heat exchange and non-heat-exchange conditions. Finally, the finite element software ABAQUS was utilized to conduct numerical simulations of the temperature fields for both prototype and model soils under constant and nonlinear thermal conductivity conditions, verifying the accuracy of the derived criteria. [Results] Results indicated that under non-heat-exchange conditions, when the first-type (Dirichlet) and second-type (Neumann) boundary conditions were combined, the time similarity coefficient equaled the square of the geometric similarity coefficient, enabling rapid determination of model test durations once the geometric scaling ratios were predefined. Similarly, under the combination of second-type and third-type (Robin) boundary conditions, the time similarity constant coefficient remained the square of the geometric similarity constant coefficient. This consistency held regardless of whether thermal parameters were constant or nonlinear, meaning that the time similarity coefficient was the square of the heat conduction geometric similarity coefficient. ensuring uniform criteria between prototype and model cases. When heat exchange was considered, the temperature similarity coefficient was no longer constant. In such cases, the test soil must be replaced and the similarity coefficients of thermal properties such as thermal conductivity, specific heat capacity, and density must satisfy specific quantitative relationships with those of the prototype soil. [Conclusion] The simulation results showed that under the corresponding time and geometric scaling conditions derived in this study, the temperature fields of the prototype and model closely matched, further validating the accuracy of the proposed similarity criteria. The similarity criteria derived from analytical solutions to heat conduction equations fully incorporate the effects of heat exchange boundary conditions and provide a fast and accurate method for determining scaling relationships when similarity coefficients for relevant thermophysical parameters are known. These findings are expected to offer a theoretical basis for solving nonlinear heat conduction problems and for guiding the design and execution of frozen soil model tests.