Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media

Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation. 1. Introduction Heat diffusion during the change of state, for example, melting and solidification, has important applications in science and technology including casting of nuclear waste materials, vivo freezing of biological tissues, and solar collectors. Problems involving heat conduction in materials in which a phase change occurs are known as moving boundary problems. First time Stefan [1] discussed this problem to study the melting of polar ice, so this is referred as Stefan problem. Solidification is the phase change problem in which phase transformation takes place by descending the energy of the system, that is, heat extraction from the liquid region. The problem related to the heat conduction involving solidification has drawn the attention of many researchers. Solidification process can be grouped into two major categories, that is, solidification of pure substances and alloy. Solidification of pure substances is usually characterized by a sharp solid or liquid interface. On the other hand, a mixed phase region characterizes solidification of an alloy. The mixed phase regions are commonly termed as the mushy region; this is a combination of liquid solute and solid crystals [2]. Fractals and fractional calculus have been used to improve the modelling accuracy of many phenomena in natural sciences. The most important advantage of using fractional differential equations is their nonlocal property. This means that the next state of a system depends not only upon its current state but also depends upon all of its historical states. These are more realistic and one of reasons to make the fractional calculus more and more popular [3, 4]. The fractional model is appropriate for modelling complex dynamics as the ions are undergoing anomalous diffusion. Fractional order models are more general and useful than the previously used integer order model. Fractional derivative enables the description of the memory and hereditary inherent properties in various processes governed by