Theoretical and Practical Stefan Problems

This provides a brief description of the history of the Stefan problem, followed by the formulation of the governing equations and boundary conditions (including mushy regions) for the classical case. After a discussion of appropriate scales, the system is written in nondimensional form. Parameter values for typical phase change materials are presented, permitting the reader to better understand the physical problem. This sets the scene for subsequent chapters, where the nondimensional system is analysed in a variety of settings and, in later chapters, the governing equations are extended beyond the classical form. This chapter, and all subsequent chapters, ends with a set of exercises (solution outlines are presented at the end of the book).

There exist very few exact solutions to practical Stefan problems. The key solutions are presented in the first section for both one- and two-phase systems. Approximate solution techniques are then presented, including the use of boundary fixing transformations, perturbation methods, small and large time solutions, and Heat Balance Integral Methods. The use of these methods is illustrated through the study of laser-induced sublimation. Although the book does not focus on numerical techniques, a topic that could occupy a whole book, a brief description of the main methods (with references) is presented in the final section.

When dealing with thin layer flow, a key technique is the well-known lubrication theory. It is shown how the lubrication approximation is achieved through a systematic reduction of the Navier-Stokes equations. Subsequently, the equations are applied to channel and free surface flows. The channel flow version is employed to model solidification in a microchannel with a flowing fluid (in the context of phase change microvalves) and also contact melting. Aircraft ice accretion is an infamous problem in the aviation industry. The mathematical model is analogous to ice growth on any structure, such as wind turbines, power cables, structures, and ships. The final sections deal with solidification in the presence of a moving thin fluid layer with a free surface (in the context of in-flight aircraft ice accretion) as well as a recent modification for ice crystal icing, which involves the enthalpy formulation.

Up to this point the analysis has dealt with situations that fall into the category of standard Stefan problems, from now on we diverge from this by permitting changes in properties, such as the specific heat, density in each phase as well as the phase change temperature. The density jump between phases is often neglected but plays an important role – density change in confined spaces can cause damage while, in terms of the problem formulation, it induces motion in the fluid, which then introduces kinetic energy into the system. Consequently, when properties change, a new form of Stefan condition applies. Starting from conservation laws for mass, momentum, mechanical energy and total energy, a more general form for the governing equations is derived. It is shown how these may be reduced to apply to one-dimensional Cartesian and spherical problems. When the phase change temperature varies the simple one-phase Stefan problem is known to lose energy. This issue is explained and the energy conserving form presented.

Following from the new formulations of Chap. 4, here the focus is on Stefan problems where the phase change temperature is a variable. In the first section, physical reasons for this change are discussed and quantified. Models for the solidification of supercooled fluids, where the phase change temperature varies with the front velocity, are analysed for both linear and nonlinear kinetic undercooling cases. At the nanoscale, the phase change temperature may vary due to curvature-induced stress, and this effect is investigated for the melting of spherical nanoparticles and nanowires. The variable temperature effect can help explain the experimentally observed sudden disappearance of nanoparticles. An analogous problem in mass transfer, concerning the growth of nanocrystals from a monomer solution, is also presented. Here the solubility (equivalent to the phase change temperature) varies with crystal size. By introducing multiple crystals, the effect of Ostwald ripening, where smaller particles dissolve and are then consumed by larger particles, is explained.

As electronic devices decrease in size, heat management at the nanoscale becomes a crucial issue. Nanoscale heat flow may be significantly different to that at the macroscale: At the macroscale, due to the large number of phonons and consequent frequent collisions, the process may be viewed as diffusive; at the nanoscale, thermal energy transport may be viewed as a ballistic process driven by infrequent, random collisions. The breakdown of Fourier’s law, at both small length and time scales, has been predicted theoretically, demonstrated via molecular dynamics and observed experimentally. In this chapter, we begin by analysing heat flow in a nanowire through the Guyer-Krumhansl (GK) formulation, rather than Fourier’s law. Once the GK equation has been established as a reliable descriptor of heat flow at the nanoscale, we extend the equations to deal with a one-dimensional phase change problem. Results from GK, Maxwell-Cattaneo, and Fourier models are compared for a solidifying silicon material, demonstrating a similarity in the phase change rate but significant differences in the heat flow behaviour.

Substitute for the solid and liquid values: solid gold \(\tau \approx 78.54\) s, liquid gold \(\tau \approx 266\) s, ice \(\tau \approx 8572\) s, and liquid water \(\tau \approx 111,075\) s.