Solidification or Melting Initiation over a Limited Portion of an Edge in a Plate

This paper proposes a method for obtaining short-time analytical solutions to problems in pure heat conduction, and heat conduction with phase change. The method employs the notion of fictitious initial temperatures in some fictitious extensions of the original phase region. Analytical results obtained in the case of a heat conduction problem in a rectangular plate are presented first and compared with numerical solutions. This analytical solution is also required later in the determination of liquid temperature in the phase change problem. The method is then extended to a two-phase solidification problem in which solidification starts over a limited portion of one of the vertical edges of the rectangular plate. The freezing front in this case consists of spread along the vertical edge and growth towards the interior. The spread along the edge can have asymptotic behaviour not commonly found. The method is applicable to other geometries, e.g. inside and outside of a long cylinder, a three-dimensional slab, etc.