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• W9360671 abstract "Problems connected with the mathematical description of pure metals solidification (macro approach) are often called the Stefan ones. The second generation models (micro/macro approach) discussed in this paper base on a theory presented by Kolmogoroff (Mehl-Johnson-Avrami-Kolmogoroff models). Both macro and micro/macro problems can be analyzed using the numerical methods. The aim of investigations presented here was a comparison of numerical solutions obtained by use of macro and micro/macro approach. On a stage of numerical modelling the finite difference method has been applied. 1. Governing equations Solidification of pure metals or eutectic alloys proceeds at a constant temperature (solidification point T). The mathematical macroscopic model of the process discussed is called in literature 'the Stefan problem' [1-3]. The domain Ω being a sum of molten metal ) ( 1 t Ω and solid state ) ( 2 t Ω sub-domains is considered. The position of interface ) ( 12 t Γ is time-dependent. So the Stefan problem belongs to a group of moving boundary ones. The temperature field in domain of molten metal is described by the well known Fourier equation [ ] ) , ( ) ( ) , ( ) ( 1 1 1 1 t x T T t t x T T c ∇ λ ∇ = ∂ ∂ (1) where c1, λ1 are the volumetric specific heat and thermal conductivity of material, T, x, t denote the temperature, spatial co-ordinates and time. The similar equation determines the temperature field in a solidified part of metal [ ] ) , ( ) ( ) , ( ) ( 2 2 2 2 t x T T t t x T T c ∇ λ ∇ = ∂ ∂ (2) where c2, λ2 are the volumetric specific heat and thermal conductivity of solid body. It should be pointed out that only heat conduction in Ω is considered (it results from the form of equations (1) and (2)). On the interface ) ( 12 t Γ the following Please cite this article as: Romuald Szopa, Edyta Pawlak, Jaroslaw Siedlecki, Stefan and Kolmogoroff models of solidification. Comparison of numerical solutions, Scientific Research of the Institute of Mathematics and Computer Science, 2008, Volume 7, Issue 1, pages 185-192. The website: http://www.amcm.pcz.pl/ R. Szopa, E. Pawlak, J. Siedlecki 186 boundary condition is taken into account n v L n t x T n t x T + ∂ ∂ λ − = ∂ ∂ λ − ) , ( ) , ( 2 2 1 1 (3) where L is a volumetric latent heat, νn is a solidification rate in a normal direction, n ∂ ∂ denotes a normal derivative (Figure 1). Additionally the temperatures . ) , ( ) , ( * 2 1 T t x T t x T = = Fig. 1. Domain considered On the outer surface of the system the boundary condition in general form 0 ) , ( ), , ( 2 2 =       ∂ ∂ Φ n t x T t x T (4) is given. The initial temperature distribution and the initial position of interface are also known. In literature one can find the analytical solution of this problem. They concern the very simple geometrical and boundary conditions. In a practice the problem of pure metals solidification can be solved using the numerical methods. The other approach to the Stefan problem results from the considerations concerning the crystallization processes proceeding in a micro scale (micro/macro model of solidification). Then one considers the following energy equation [ ] t t x f L t x T T t t x T T c S ∂ ∂ + ∇ λ ∇ = ∂ ∂ ) , ( ) , ( ) ( ) , ( ) ( (5) where fS is a volumetric solid state fraction at the point x. The energy equation (5) is the typical Fourier equation with additional term (source function) controlling the evolution of latent heat L, but the capacity of internal heat sources results from the laws determining the nucleation and nuclei 1 Ω 2 Ω" @default.
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• W9360671 date "2008-01-01" @default.
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• W9360671 title "Stefan and Kolmogoroff models of solidification. Comparison of numerical solutions" @default.
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