Solving two-phase Stefan problems in materials with internal heat generation in cylindrical geometry

Abstract: We consider the evolution of the solid-liquid interface in a material with constant internal heat generation during melting and solidification in an infinite cylinder. The boundary of the cylinder is held either at a constant temperature or has a prescribed heat flux. The problem generalizes a classical Stefan problem by allowing heat generated internally, e.g. by radioactive decay. The transition between liquid and solid phases is modeled by a sharp interface approach. We assume that the interface moves slowly compared to the change of temperature in both phases. This weakly time-dependent assumption allows us to split the solution in each phase into quasi-transient and quasi steady-state solutions and solve the quasi-transient part by the method of separation of variables and obtain the solution for the quasi steady-state part by direct integration. We then derive the first-order differential equation for the interface, which involves infinite Fourier-Bessel series terms and can be solved numerically by standard ODE methods. For comparison, we also simulate the problem by using the front catching into a space grid node method as well as the enthalpy-porosity method. Even though the sharp-interface approach does not include a mushy zone assumption, it produces an overheated region in the solid phase during the melting process, that resembles a mushy zone. An application of this problem includes melting of nuclear fuel rods during a nuclear accident or meltdown scenario analysis.

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