Abstract The rate of thaw beneath the center of a hot pipe buried in permafrost was calculated by the quasi-steady approximate method. The solution assumes a purely conductive heat transfer model. The heat loss at the surface was included for the case of a pipe buried at a finite depth. Functions are presented which allow the thaw Tate below the pipe and the heat transfer from the pipe to be calculated for any pipe radius, burial depth, temperature of the pipe surface, as well as initial temperature and thermal properties of the surrounding permafrost. The method allows a quick estimate to be made of the effect of burial depth, temperature and soil thermal properties on the phase change beneath a pipe. The same details can also be applied to the case of freezing around a buried pipe. Introduction THE PROBLEM OF THAWING beneath a hot pipe buried in permafrost, or in permafrost regions, has been of considerable interest. Unfortunately, the nonlinear nature of the problem precludes exact solutions when the geometry of the system is curvilinear. However, the exact solution for the semi-infinite region, known as the Neumann problem(1) is available and the details of this solution demonstrate the validity of the quasi-steady, or Stefan(2), approximate method. This idea relies on the fact that the rate of movement of the phase change interface is slow enough that the heat transfer for each region is essentially steady-state at any instant of time. This method was extended by Khakimov(3) to the case of a cylinder embedded in an infinite medium, where the concept was used of a finite volume around the cylinder in which the temperature is disturbed. Gupta'" extended the method to the analogous case of the sphere. There are numerous numerical solutions of the problem, such as those of Lachenbruch(5), but these methods are lengthy when applied to two-dimensional problems and are not convenient for quick estimates. Theory Consider the case of thawing of a medium initially frozen at temperature To. A hot cylinder with a surface temperature Ta is inserted into the medium at zero time. The thermal effect will extend, at any time, to a finite distance (a), i.e., the initial temperature of the medium will be unchanged at a location sufficiently far from the hot cylinder. This distance is denoted by the radius (a) and it can be assumed that the ratio α = a/R is a constant, using α = 4.5, as did Khakimov. Using the quasi-steady assumption and letting 0 = T ? To, the temperature for each region (see Fig. 1) will be the solution of the one dimensional conduction equation: (Equation Available In Full Paper) The energy required to thaw region 1 and increase the temperature of regions 1 and 2 is the sum of the latent plus the sensible heats and can be written as:
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