A homeowner, whose water pipes have frozen during a period of cold weather, decides to melt the ice by passing an electric current I through the pipe wall. The inner and outer radii of the wall are designated as r1 and r2, and its electrical resistance per unit length is designated as R’e (Ω/m). The pipe is well insulated on the outside, and during melting the ice (and water) in the pipe remain at a constant temperature Tm associated with the melting process.

(a) Assuming that steady-state conditions are reached shortly after application of the current, determine the form of the steady-state temperature distribution T(r) in the pipe wall during the melting process.

(b) Develop an expression for the time tm required to completely melt the ice. Calculate this time for I = 100 A, Re = 0.30 Ω/m, and r1 = 50 mm.