Numerical simulation of solidification process using the Scheil model

2007 
In the paper the macro model of volumetric solidification is discussed. The mathematical description of the process bases on a one domain approach, this means in order to take into account the evolution of latent heat, the substitute thermal capacity is introduced to the energy equation. This parameter is determined using the well known Scheil model of macrosegregation. In the first part of paper the theoretical considerations are presented, while in the second one the example of simulation are presented (solidification of Al-Si alloy is considered). The numerical algorithm bases on the finite differences method. 1. Mathematical description of the problems At first the mass balance of alloy component of solidifying material Ω will be presented. Considering two successive time levels t and t + ∆t one obtains the following form of balance discussed ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t z t t m t t z t t m t z t m t z t m L L S S L L S S ∆ + ∆ + + ∆ + ∆ + = + (1) where zS , zL are the concentrations of alloy component in solid and liquid phases, mS is a mass of solidified part of domain considered, mL is a mass of molten metal. Assuming that the mass densities ρS = ρL one can transform the balance (1) to the form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t z t t V t t z t t V t z t V t z t V L L S S L L S S ∆ + ∆ + + ∆ + ∆ + = + (2) where VS, VL are the volumes of liquid and solid in domain Ω. The change of VS (t+∆t) − VS (t) (volumetric solidification) is conventionally shown in Figure 1. The values of VS, VL and zS, zL at the moment t+∆t can be found using the Taylor formula ( ) ( ) ( ) t t t V t V t t V S S S ∆ + ≈ ∆ + d d (3) Please cite this article as: Jaroslaw Siedlecki, Romuald Szopa, Wioletta Wojciechowska, Numerical simulation of solidification process using the Scheil model, Scientific Research of the Institute of Mathematics and Computer Science, 2007, Volume 6, Issue 1, pages 253-260. The website: http://www.amcm.pcz.pl/ R. Szopa, J. Siedlecki, W. Wojciechowska 254 ( ) ( ) ( ) t t t V t V t t V L L L ∆ + ≈ ∆ + d d (4) ( ) ( ) ( ) t t t z t z t t z L L L ∆ + ≈ ∆ + d d (5) ( ) ( ) ( ) t t t z t z t t z S S S ∆ + ≈ ∆ + d d (6)
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