Prediction of Macrosegregation in Steel Ingots: Influence of the Motion and the Morphology of Equiaxed Grains

Abstract

Although a significant amount of work has already been devoted to the prediction of macrosegregation in steel ingots, most models considered the solid phase as fixed. As a result, it was not possible to correctly predict the macrosegregation in the center of the product. It is generally suspected that the motion of the equiaxed grains is responsible for this macrosegregation. A multiphase and multiscale model that describes the evolution of the morphology of the equiaxed crystals and their motion is presented. The model was used to simulate the solidification of a 3.3-ton steel ingot. Computations that take into account the motion of dendritic and globular grains and computations with a fixed solid phase were performed, and the solidification and macrosegregation formation due to the grain motion and flow of interdendritic liquid were analyzed. The predicted macrosegregation patterns are compared to the experimental results. Most important, it is demonstrated that it is essential to consider the grain morphology, in order to properly model the influence of grain motion on macrosegregation. Further, due to increased computing power, the presented computations could be performed using finer computational grids than was possible in previous studies; this made possible the prediction of mesosegregations, notably A segregates.

Appendix

Adding the average solute mass balance in the solid and liquid phases (Eqs. [6] and [7]) and accounting for relation [21], one gets

$$ \frac{{\partial (\rho _m C_m )}} {{\partial t}} + \nabla (\rho _s g_s C_s \vec{v}_s ) + \nabla (\rho _l g_l C_l \vec{v}_l ) = 0 $$

(A1)

with $\rho _m C_m = \rho _s g_s C_s + \rho _l g_l C_l$.

By applying mass conservation (Eq. [3]) and assuming equal and constant densities of the solid and liquid phases, Eq. [A1] becomes

$$ \frac{{\partial (C_m )}} {{\partial t}} = - g_s \vec{v}_s \cdot \nabla (C_s ) - g_l \vec{v}_l \cdot \nabla (C_l ) - \left( {C_s - C_l } \right) \cdot \nabla (g_s \vec{v}_s ) $$

(A2)

Assuming in addition, the local thermodynamic equilibrium in the entire liquid and solid phases (lever rule), the partial time derivative of the average local solute mass fraction can then be expressed as

$$ \frac{{\partial (C_m )}} {{\partial t}} = - \underbrace {\frac{{\left( {g_s k_P \vec{v}_s + g_l \vec{v}_l } \right)}} {{m_L }} \cdot \nabla (T)}_{{\text{motion coupled with phase change}}} - \underbrace {\left( {\frac{{\partial (g_s )}} {{\partial t}} - \frac{1} {{\rho _s }}\left( {\Gamma _s - \Phi _s } \right)} \right)\left( {1 - k_P } \right)C_l }_{{\text{grain transport}}} $$

(A3)

For a negative liquidus slope m _L , the first term of Eq. [A3] means that any motion of the solid and liquid phases, such that the velocity component parallel to the thermal gradient is oriented in the same direction as the thermal gradient, will induce an increase in the average solute mass fraction. At the reverse, if the parallel velocity component is in the direction opposite to the thermal gradient, a decrease in the average solute mass fraction will be observed. The second (grain transport) term quantifies the contribution of the transport of solid grains (purely passive transport, i.e., without phase change) on the average solid mass fraction. Note that this equation allows a direct interpretation of the circulation of the solid and liquid phases on the variation in the local solute content; in the more general case, however, in which shrinkage is accounted for, it is possible to derive only a relation between the relative motion of the phases and the partial derivative of the solute mass fraction of the liquid vs the volume fraction of the liquid.[2]