The zero-phase Stefan problem

What is it about?

The mathematical description of melting and solidification of a medium is called a Stefan problem. When convection (energy transport through medium motion) is disregarded, the mathematical problem becomes much more simple — the resulting heat diffusion equation on either phase is linear and can be solved by many methods — the presence of phase change; however, introduces a geometrical difficulty, since each phase domain (or phase extent) becomes part of the problem. Therefore, despite the underlying mathematical simplicity of the energy diffusion process, only few Stefan problems have known exact solutions, and they usually involve unrealistic geometries — such as infinite (probably larger than the entire universe!) or semi-infinite portions delimited by an infinitely long planar face — and unrealistic conditions, such as imposed temperatures, as well as absence of any resemblance with its practical applications of heat charge and discharge cycles. The analytical simplifications that existed before this work did not solve these problems. The classical Stefan problems are named based on the number of phases in which the energy diffusion process have to be solved — hence the classical "one-phase" and "two-phase" Stefan problems. This work presents a yet simpler analytical approximation to the Stefan problem — called "zero-phase modeling" — that corresponds to the limit of infinite heat conductivity of the phase change material with respect to the heat interactions at its boundaries (that is technically expressed as the limit of vanishing Biot number, mentioned in the Abstract). In this regime, interesting things happen: (i) the phase change domain becomes isothermal at the phase change (melting) temperature — a uniform temperature condition, (ii) the heat transport equation within each phase becomes a trivial one and need not to be solved — hence the "zero-phase" name, and (iii) the only thing that is left to be solved is the equation for the solid-liquid interface position, or, the so-called Stefan condition, which becomes a very simple integral equation. This work formally postulates the "zero-phase Stefan problem" by applying zero-phase modeling to the classical one-phase Stefan problem. Under such dramatic simplifications, the problem finally becomes mathematically tractable, and more complex geometrical configurations are shown to yield solutions.

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Why is it important?

The "zero-phase Stefan problem" is extremely important in the sense that it is the only Stefan problem to admit exact solutions in (i) finite geometries, subject to (ii) heat charge and discharge events, in one Euclidean dimension in (iii) Cartesian, (iv) cylindrical, and (v) spherical geometries. Moreover, zero-phase modeling can be used to approximate situations of small (but finite) Biot numbers. Furthermore, owing to the isothermal domains (and isothermal boundaries), the zero-phase Stefan problem (i) decouple otherwise coupled heat transfer problems, and also (ii) represent an internally reversible domain with respect to heat interactions, and (iii) the resulting models can also be thought of as the lumped capacitance equivalent for phase change problems.