Reduction of plane-radial freezing - thawing problems to plane-parallel ones

Abstract

Construction in northern conditions is complicated by the processes of seasonal freezing and thawing of soils. This factor is of particular importance in the area of permafrost distribution. The paper deals with the problems that arise when laying communication networks. Any engineering solution must be justified by a heat engineering calculation. In one case, it is necessary to exclude the thawing of water or sewer networks, in the other case – to prevent the destructive effect of the heating main on the permafrost soil. The most common mathematical model of the heat transfer process in the presence of phase transitions is the Stefan problem. When calculating the processes of freezing-thawing near the pipes of communications, it is rational to set the problem in cylindrical coordinates, using the property of plane-radial symmetry. But the Stefan problem has an exact solution only in Cartesian coordinates, for the case of plane-parallel symmetry. For this case, many approximate dependencies are also obtained. The paper presents a method that allows using the results of solving the plane-parallel Stefan problem to solve the plane-radial problem with the same values of the input parameters. Approximate values of the coordinate of the phase transition front as a function of time for the plane-parallel and plane-radial areas are obtained using the method of sequential change of stationary states. A one-to-one relationship is established between the obtained values. If there is a solution obtained by any method for a plane-parallel area, and then using this dependence, it can be applied to a plane-radial area. This technique is suitable not only for the Stefan problem, but also for many nonlinear heat conduction problems. The results given in the paper are of practical importance: using the above method, almost any solution obtained for one type of symmetry directly extends to other types of symmetry.