It is well known that for many heat transfer problems with phase change, their approximate solutions or numerical solutions can be derived. However, for multi-dimensional problems, even if using an approximate method, it is difficult to gain their approximate analytical solutions. For some, although they could be simplified, their eventual solutions can be obtained by using the numerical method. So the numerical method is a very important way for this type of problem. Be that as it may, we should not negate the approximate analytical method for it can give out the definite numeric relations among the physical quantities, the time variable and the spatial variables in the system. Moreover, the approximate analytical solutions can also be used to verify the reliability of the numeric solutions and the computer programs. In addition, when dealing with practical engineering, civil engineers would rather apply the simple analytical solution that well satisfies engineering precision than prepare and input the complicated data before running the computer programs.

Elmer (1932), Pekeris and Slichter (1939), Hwang (1965) and Slichter (1977) has made many researches on the freezing problem of the pipeline surface. Elmer (1932) obtained the pseudo-steady-state solution of ice formation on pipe surfaces. Pekeris and Slichter (1939) investigated a first-order correction to the pseudo-steady-state solutions, which were evaluated and found to be small. The obtained solution is applied to the problem of freezing soil around a long pipe. Hwang (1977) examined the validity of the two-dimensional solution for a buried pipe given by Carslaw and Jaeger, 1965H.S. Carslaw and J.C. Jaeger Conduction of Heat in Solids (2nd edn. ed.),, Clarendon Press, Oxford (1965).Carslaw and Jaeger (1965) with reference to numerical techniques. Until now, there are no analytical solutions for the freezing problem of the circular cross-sectional tunnels. In this paper, the approximate analytical solutions for the problem of the circular cross-sectional tunnels are obtained by using the perturbative technique. This type of solution could be used not only in engineering estimation, but could also be applied to verify the computer programs for numerical calculations.

According to the practical situation in permafrost regions, the governing differential equations of heat transfer in frozen zone and unfrozen zone are simplified. Then, using dimensionless and perturbative method, the approximate analytical solution for simplified heat transfer equation has been obtained. The research of the freezing process of circular tunnel with initial temperature near 0 °C has been made. Comparing the approximate analytical results with finite element results, it can be seen that the approximate analytical solution is of higher precision and this solution can meet the precision requirement of practical engineering. This solution can be used in both engineering computation and the examination of numerically calculated results.