Numerical solution of solidification in a square prism using an algebraic grid generation technique Z. Dursunkaya, G. Odabas ¸ ı Abstract The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simulta- neous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the compu- tational grid structure is regenerated at every time step. Keywords Two dimensional solidification, Moving boundary problem, Coordinate transformation, Front fixing List of symbols C Specific heat D Side length of square prism J Jacobian J Jacobian matrix k Thermal conductivity L Latent heat of fusion m Number of nodes in n direction n Number of nodes in k direction St Stefan number, St = C(T f – T ref )/L T Dimensionless temperature t time V Interface velocity x Dimensionless Cartesian coordinate y Dimensionless Cartesian coordinate Greek letters a Thermal diffusivity d Interface position k Dimensionless transformed coordinate s Dimensionless transformed time n Dimensionless transformed coordinate Subscript f Fusion ref Reference value s Solid sur Surface x Differentiation with respect to x y Differentiation with respect to y k Differentiation with respect to k n Differentiation with respect to n Superscript * Dimensional quantity 1 Introduction The solutions of phase change problems involve the cal- culation of the advance of a phase change interface. The motion of the solidification/melting front is coupled to Received: 6 February 2002 Published online: 19 November 2002 Ó Springer-Verlag 2002 Z. Dursunkaya (&) Associate Professor of Mechanical Engineering, Middle East Technical University, Inonu Bulvari 06531, Ankara, Turkey E-mail: refaz@metu.edu.tr G. Odabas ¸ ı Senior Engineer, Roketsan A.S., Elmadag ˘ 06780, Ankara, Turkey Heat and Mass Transfer 40 (2003) 91–97 DOI 10.1007/s00231-002-0390-z 91