1 Copyright © 2003 by ASME Proceedings of NHTC’03 ASME Summer Heat Transfer Conference Las Vegas, July 21-23, 2003 HT2003-47308 REDUCING CONVECTION EFFECTS IN SOLIDIFICATION BY APPLYING MAGNETIC FIELDS HAVING OPTIMIZED INTENSITY DISTRIBUTION Marcelo J. Colaço 1 Department of Mechanical Engineering, EE/COPPE Federal University of Rio de Janeiro, UFRJ Cid Universitaria, Cx. Postal 68503 Rio de Janeiro, RJ, 21945-970 BRAZIL colaco@pobox.com George S. Dulikravich 2 Department of Mechanical and Aerospace Engineering Multidisciplinary Analysis, Inverse Design & Optimization (MAIDO) Institute, UTA Box 19018 The University of Texas at Arlington Arlington, TX 76019, U.S.A. dulikra@mae.uta.edu Thomas J. Martin 3 Pratt & Whitney Engine Company Turbine Discipline Engineering & Optimization Group, M/S 169-20 400 Main Street, East Hartford, CT 06108, U.S.A. thomas.martin@pw.utc.com ABSTRACT This paper presents a numerical procedure for achieving desired features of a melt undergoing solidification by applying an external magnetic field whose intensity and spatial distribution are obtained by the use of a hybrid optimization algorithm. The intensities of the magnets along the boundaries of the container are described as B-splines. The inverse problem is then formulated as to find the magnetic boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the gravity direction are minimized. For this task, a hybrid optimization code was used that incorporates several of the most popular optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). Transient Navier-Stokes and Maxwell equations were discretized using finite volume method in a generalized curvilinear non-orthogonal coordinate system. For the phase change problems, an enthalpy formulation was used. The code was validated against analytical and numerical benchmark results with very good agreements in both cases. 1 Postdoctoral Fellow. Lecturer. 2 Professor and Director of MAIDO. Fellow of ASME. 3 Systems Engineer. Member of ASME. NOMENCLATURE C P specific heat at constant pressure B x magnetic flux component in x-direction B y magnetic flux component in y-direction g acceleration of the gravity Gr Grashoff number f solid fraction k thermal conductivity L latent heat of solidification/melting h enthalpy Ht Hartmann number n partition coefficient in Scheil’s equation (5) P pressure Pr Prandtl number Ra Rayleigh number t time T temperature u velocity component in x-direction v velocity component in y-direction x, y Cartesian coordinates Greek letters α thermal diffusivity β thermal expansion coefficient μ fluid viscosity μ m magnetic permeability