A mathematical model for the dissolution of particles in multi-component alloys

• Published in: Journal of computational and applied mathematics Vol. 126; no. 1; pp. 233 - 254
• Main Authors: Vermolen, F.J., Vuik, C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30-12-2000; Elsevier
• Subjects: Alloy homogenization; Exact sciences and technology; Finite differences; Fundamental areas of phenomenology (including applications); Heat transfer; Heat transfer in inhomogeneous media, in porous media, and through interfaces; Mathematical analysis; Mathematics; Newton–Raphson method; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Sciences and techniques of general use; Self-similar solution; Vector-valued Stefan problem; 80A22; 35A35; 65M06; Self-similar solution; Finite differences; Newton–Raphson method; 35R35; Alloy homogenization; Vector-valued Stefan problem; Stoichiometry; Stefan problem; Existence of solution; Alloys; Homogenization; Moving boundary; Partial differential equation; Newton Raphson method; Vector valued map; Selfsimilarity; Non linear equation; Parabolic equation; Boundary value problem; Discretization; Numerical solution; Asymptotic approximation; Alloy dissolution; Mathematical model; Analytical solution; Finite difference method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(99)00355-6

Dissolution of stoichiometric multi-component particles is an important process occurring during the heat treatment of as-cast aluminum alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. A necessary condition for existence of a solution of the moving boundary problem is proposed and investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an asymptotic approximation based on self-similarity is derived. The asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently, a numerical treatment of the vector valued Stefan problem is described. The numerical solution is compared with solutions obtained by the analytical methods. Finally, an example is shown.