Investigation on Three-Dimensional Morphology of Channel-Type Macrosegregation in DC Cast Al-Mg Billets Through Numerical Simulation

Abstract

Channel-type segregation is one of the defects in aluminum casting technology, particularly in the direct chill (DC) process, having a significant influence on the cast quality. Nevertheless, the formation mechanism of three-dimensional (3D) channel-type segregations remains poorly understood. In order to clarify the formation mechanism of defects of this type, we conducted a 3D numerical simulation of the DC casting process of an Al-Mg alloy billet considering the melt flow, heat and mass transfer, solidification, and the motion of the solidified ingot, coupled with the alloy phase diagram. The simulation results showed that the channel-type segregations have strip-patterns and formed easily at high casting speeds, being localized at a distance of a half-radius from the billet centerline. We also compared two-dimensional axisymmetric and 3D simulations, and the results indicated that a two-dimensional axisymmetric simulation is incapable of properly predicting the behavior of channel-type segregations.

Appendices

Appendix A

Numerical model for thermodynamic relationship

The thermodynamic relationship was coupled with the governing equations through the numerical model proposed by Prakash and Voller.³³ Through a linearized phase diagram, the mass fraction of liquid, f_l, was calculated as:

$${f}_{l}=1-\frac{1}{1-{k}_{p}}\frac{T-{T}_{liq}}{T-{T}_{m}}$$

(A1)

where k_p is the partition coefficient, T_m the melting temperature of pure aluminum, and T_liq the liquidus temperature, which is described as:

$${T}_{\mathrm{liq}}={T}_{m}+\left({T}_{e}-{T}_{m}\right)\frac{C}{{C}_{e}}$$

(A2)

where T_e is the eutectic temperature, and C_e the eutectic concentration. By rearranging Eq. A1, the following equation can be obtained:

$$T=F\left({f}_{l},C\right)=\frac{{T}_{liq}-\left(1-{f}_{l}\right)\left(1-{k}_{p}\right){T}_{m}}{1-(1-{f}_{l})(1-{k}_{p})}$$

(A3)

A dependence of enthalpy on the liquid fraction can be given as:

$$h={c}_{p}T+{f}_{l}L$$

(A4)

where L is the latent heat. The discretized governing equations and thermodynamic relationship were coupled by using Eq. A3 and (A4) through outer iterations. By substituting Eq. A3 into Eq. A4, the following equation can be obtained at the n-th outer step:

$${c}_{p}F\left({f}_{l}^{n+1},{C}^{n}\right)+{f}_{l}^{n+1}L ={c}_{p}{T}^{n}+{f}_{l}^{n}L$$

(A5)

Finally, Eq. A5 can be transformed into the following:

$$a{\left({f}_{l}^{n+1}\right)}^{2}+b\left({f}_{l}^{n+1}\right)+d=0$$

(A6)

where a, b, and d are coefficients of the quadratic equation, which are described as:

$$a=1-{k}_{p}$$

(A7)

$$b={k}_{p}-{f}_{l}^{n}\left(1-{k}_{p}\right)+\frac{{c}_{p}}{L}\left(1-{k}_{p}\right)({T}_{m}-{T}^{n})$$

(A8)

$$d=-\left({f}_{l}^{n}+\frac{{c}_{p}}{L}{T}^{n}\right){k}_{p}+\frac{{c}_{p}}{L}\left[{T}_{liq}^{n}-\left(1-{k}_{p}\right){T}_{m}\right]$$

(A9)

Finally, the mass fraction at the n-th outer step was calculated as:

$${f}_{l}^{n+1}=\text{min}\left(\text{max}\left(0, \frac{-b+\sqrt{{b}^{2}-4ad}}{2a}\right),1\right)$$

(A10)

During the outer iterations, liquidus and solidus concentration and the physical properties were updated. By the outer iterations, the mass fraction was converged to a certain value. In this simulation, the convergence criterion of mass fraction was set to 1 × 10⁻⁶. The average number of outer iterations was approximately 7.