Simulation of macroscopic solidification with an incorporated one-dimensional microsegregation model coupled to thermody
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INTRODUCTION
NORMALLY, software packages calculating temperature fields for a casting during solidification use input files to obtain the necessary thermodynamic information of the cast alloy. Those files contain information about the liquidus and solidus temperature, the average heat capacity c p (T), the solid fraction f s (T), and the latent heat H f released during the solidification process. However, in reality, the amount of latent heat for a certain solidification interval and the development of fs depend on the local cooling rate. Thus, these values can neither reliably be pre-nor postcalculated in an adequate manner for the entire casting. Therefore, it seems to be obvious to couple both processes in a simulation. Apart from that, the parameters of microstructure such as the dendrite arm spacing and the distribution of phase amounts give a clue about the material properties. For example, weak sections of a casting can be avoided by specific variation of the local cooling conditions to avoid or minimize certain microsegregations such as ternary eutectics. Kraft [1] developed a segregation model based on the plate model of Brody and Flemmings[2] and the work of Roòsz et al. [3,4] Kraft’s model includes undercooling effects as well as dendrite arm coarsening for two-and three-component alloy systems. Chen and Chang[5] provide a geometrical description for the diffusion-controlled formation of solid phases along the monovariant eutectic of a three-component system. B. PUSTAL, A. LUDWIG, P.R. SAHM, and A. BÜHRIGPOLACZEK are with the Gießerei-Institut Aachen, D-52072 Aachen, Germany. Contact e-mail: [email protected] B. BÖTTGER is with ACCESS e.V., D-52072 Aachen, Germany. Manuscript submitted September 18, 2002. METALLURGICAL AND MATERIALS TRANSACTIONS B
They introduced the so-called “cross-sectional area” A for the one-dimensional plate model. As illustrated in Figure 1, A describes a local relative phase amount of the phases a and b divided by the length l of the control unit: Aa
fa 1 , fa fb l
Ab
fb fa fb
1 l
[1]
Here, f a and f b are phase fractions. The authors use a differential mass balance for the interface similar to the Gulliver[6] and Scheil[7] model, but extended by diffusion terms: fL dxkL (xLk xak ) dfa (xLk xbk ) dfb Dka Aa a
xbk xak b dt Dbk Ab a b dt s s
[2]
Here, xk means the mole fraction of the component k in the phase a, b, or liquid at the interface. The term D is the chemical diffusion coefficient for each component in each phase, t the time, and (x k /s) the concentration gradient in the direction of s. To calculate diffusion in the solid phases, the divergence of Fick’s first law extended by A is used. This leads to Eq. [3] in analogy to Fick’s second law but allowing a variable cross-sectional area A. Complete mixing is assumed in the liquid. xjk
xj aDjk Aj b, with j a, b Aj t s s k
[3]
Greven et al.[8] coupled Kraft’s model to a macroscopic FEM temperature calculation using the thermodynamic software package
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