Analysis of a numerical benchmark for columnar solidification of binary alloys

Abstract

During the solidification of metal alloys, chemical heterogeneities at the scale of the product develop. It is referred to as "macrosegregation". Numerical simulation tools exist in the industry. However, their predictive capabilities are not validated and are still limited. A 2D numerical benchmark is presented, based on the solidification of metallic Pb-Sn alloys. Concerning the numerical benchmark, a "minimal" common model of solidification is assumed, including columnar growth without undercooling, fixed solid, isotropic permeability of the mushy region, local thermodynamic equilibrium, lever-rule assumption for the local average composition. We focus our attention on the numerical method used to solve the average conservation equations: Finite Volume, Finite Element, Velocity-Pressure coupling treatment, scheme for convective terms, etc. At this stage of the work, we cannot exhibit a reference solution. However we draw some conclusions on the effects of the grid dependency, in particular on the location and sizes of the segregate channels. The development of both thermally and solutal driven convections in the first stage of the process (cf. low Prandtl and high Lewis numbers) and the relative independency of the convective scheme are also discussed. This presentation also have the goal to call other contributors to join this benchmark [1] in order to enrich the exercise and to reach a reference solution for this important problem in metallurgy.

Simple metrics for verification and validation of macrosegregation model predictions

“…In practice there are many factors that can control the fluid flow in the mushy region, in particular coarsening of the arm spaces (leading to a transient permeability) and the movement of solid grains (non-fixed microstructure). As a result, although it may be obvious to state, we believe that it is in the modeling treatment of the mushy region flow that drives the observed divergence between macrosegregation code predictions observed in benchmark studies [2]. Therefore, in moving towards simulations that can consistently and faithfully match those seen in experiment or plant, it is key to focus on the development of robust physical models of the flow in the mushy region.…”

“…Previously introduced as the macrosegregation number [2] it provides a basic measure of the level of macrosegregation. The third moment is the sample skewness, which based on the SKEW function found in spreadsheet software, can be defined as:…”

“…The standard model to suppress fluid flow, as full solidification is reached, is to add a Darcy like sink term, scaling as ~ (inverse permeability x velocity), to the momentum equation. In the base case, the permeability is modeled as K=κ 0 f 3 /(1-f) 2 , where the constant κ 0 =1.0x10 -10 m 2 . In this way as we approach full solidification f1 the inverse permeability K -1 , effectively suppressing the flow velocity in the mush.…”

Simulation of a macrosegregation benchmark in a cylindrical coordinate system with a meshless method

“…The benchmark resembles the macrosegregation benchmark presented in [5]. A more complex macrosegregation benchmark, where the channel segregates are present, has also been solved and presented [6,7,8]. Yet, a related benchmark in axisymmetry, which requires a different implementation of the mathematical operators, has not been proposed and solved yet.…”

Parallel solution of multiphase transport problem

“…The main complexities in numerical treatment of the solidification models are moving boundaries, high gradients in governing fields, strong couplings between the transport equations, coupling between different flow regimes, unstable flow of metallic fluids and completely advective transport. Regardless the substantial effort and resources invested to study the behavior of different numerical methods in the prediction of segregation [4][5][6], the commonly agreed solution is still not available. Although, a macrosegregation maps, i.e.…”