Problem 5 – Hint 1

There are two transport processes taking place simultaneously: (a) radially, forming a warm “sausage” around the pipe, and (b) along the pipe longitudinally. The former is slower, purely diffusive process. For such diffusive process with a quasi-stationary boundary condition (the temperature at the pipe wall equals the temperature of the hot water), the soil around the pipe can be divided into three regions. First, in the region close to the pipe, a quasi-stationary temperature profile will be established: in that region, heat flux only “flows through”, i.e. the accumulation of heat by soil can be neglected (the soil is already somewhat warm, and the incoming heat flux is much bigger than what would be needed to slow heating). The region with a quasi-stationary heat profile is slowly expanding, at the diffusion rate, and so the second region is the region where the still-cold soil is heating up. Thus, the radius of the first region grows diffusively, at the rate defined by the heat conductivity, specific heat, and density of the soil. Finally, at even larger distances, the soil is still cold: there, the temperature increase due to the hot pipe is exponentially small.

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