Miha Založnik

Effect of discretization of permeability term and mesh size on macro- and meso-segregation predictions

The effect of interpolation schemes used for discretization of permeability in numerical simulations on macrosegregation and channel segregation (mesosegregation) during solidification has been studied. The different ways to discretize the permeability term and its effect on the interdendritic velocity are illustrated by a simplified 1D model which solves the Darcy equation for a porous medium. The Darcy equation is solved numerically using the SIMPLE algorithm for the coupled velocity–pressure fields. For this simplified case, an analytical reference solution can also be obtained. In the numerical solution four different interpolation schemes for permeability discretization have been employed, and the results obtained are compared. For coarse mesh, different interpolation schemes produce large differences from the analytical reference solution. We thereafter, present simulation results for solidification of Sn–Pb alloy in a two-dimensional rectangular cavity using different discretization schemes. It is observed that solute-rich liquid flowing towards the bottom of the rectangular cavity in the mushy zone due to thermosolutal convection results in patches of thin structure known as channels. These channels are formed by perturbation by the interdendritic fluid flow in the mushy zone and in some cases by the localized remelting in some portions of the solid/mush interface. The role of discretization schemes and mesh size on the formation of channel segregates and macrosegregation is discussed.

Verification of a numerical model of macrosegregation in direct chill casting

Purpose – This paper aims to point out the critical problems in numerical verification of solidification simulation codes and the complexity of the verification and to propose and apply a procedure of generalized verification for macrosegregation simulation.Design/methodology/approach – A partial verification of a finite‐volume computational model of macrosegregation in direct chill (DC) casting of binary aluminum alloys, including the coupled transport phenomena of heat transfer, fluid flow and species transport, is performed. The verification procedure is conducted on numerical test problems, defined as subproblems with respect to the complexity of the physical model, geometry, and boundary conditions. The studied cases are thermal convection with solidification in DC casting, thermal natural convection of a low‐Prandtl‐number liquid metal in a rectangular cavity and 1D directional solidification of a binary Al‐Cu alloy. Grid‐convergence studies, code comparison with an alternative Chebyshev‐collocation...

Influence of Discretization of Permeability Term and Mesh Size on the Prediction of Channel Segregations

Macro- and meso-segregations correspond to compositional heterogeneities at the scale of a casting. They develop during the solidification process. One of the parameters that have an essential effect on these segregations is the mush permeability, which is highly nonlinear, and varies over a wide range of magnitudes. We present simulation results for solidification of a Sn-Pb alloy in a two-dimensional cavity, highlighting the role of (i) the numerical interpolation schemes used for the finite-volume discretization of the highly-nonlinear permeability term and (ii) of the mesh size on the prediction of mesosegregations and macrosegregation. We observe that solute-rich liquid flowing through the mushy zone due to thermo-solutal convection results in patches of thin channel structures, which develop into mesosegregations. We notice little sensitivity of the predicted macrosegregation to different discretization schemes for the permeability term. However, we found their influence on the prediction of channel segregates to be significant when using coarse computational grids, customary in the simulation of industrial castings. Mesh refinement is crucial for capturing the complex phenomena in the formation of channel segregates. With a very fine mesh channels have been captured with more than one grid point along their width, allowing the determination of their width.

A numerical simulation of columnar solidification: influence of inertia on channel segregation

We investigate the role of the inertia of the flow through the dendritic mushy zone in the numerical prediction of channel segregations during columnar solidification. The contribution of inertia is included in the momentum transport equation through the quadratic Forchheimer correction term. The study reveals a significant influence of the Forchheimer term in the vicinity of the liquidus front, i.e. at high liquid fractions. The natural convective flow field in this region is modified due to the additional inertial drag. This strongly influences the convective transport of solute and thereby incurs a modification of the dynamics of the advancement of the mushy zone. The most notable consequence is a significant decrease in the predicted channel segregation.

Modeling of the Coupling of Microstructure and Macrosegregation in a Direct Chill Cast Al-Cu Billet

The macroscopic multiphase flow and the growth of the solidification microstructures in the mushy zone of a direct chill (DC) casting are closely coupled. These couplings are the key to the understanding of the formation of the macrosegregation and of the non-uniform microstructure of the casting. In the present paper we use a multiphase and multiscale model to provide a fully coupled picture of the links between macrosegregation and microstructure in a DC cast billet. The model describes nucleation from inoculant particles and growth of dendritic and globular equiaxed crystal grains, fully coupled with macroscopic transport phenomena: fluid flow induced by natural convection and solidification shrinkage, heat, mass, and solute mass transport, motion of free-floating equiaxed grains, and of grain refiner particles. We compare our simulations to experiments on grain-refined and non-grain-refined industrial size billets from literature. We show that a transition between dendritic and globular grain morphology triggered by the grain refinement is the key to the explanation of the differences between the macrosegregation patterns in the two billets. We further show that the grain size and morphology are strongly affected by the macroscopic transport of free-floating equiaxed grains and of grain refiner particles.

Three-dimensional study of macro- and mesosegregation formation in a rectangular cavity cooled from one vertical side

In metal alloys the solidification process leads to heterogeneities in chemical composition, which result in macroscopic (on the scale of the product) and mesoscopic (on the scale of several grains) segregation structures in the casting. Mesosegregations are a severe form of segregation in which solute concentration changes abruptly with respect to the surrounding regions and can induce structural heterogeneities in the casting. In this paper we move on from [1,2], in which channel mesosegregations and their numerical prediction were studied on a two-dimensional configuration – a small (10 × 6 cm) ingot of a Sn-10wt%Pb binary alloy. We extend the study to a slender three-dimensional cavity (10 × 6 × 1 cm) and we investigate three-dimensional effects in the formation of channel mesosegregations in a columnar mushy zone. We investigate the effects of the three-dimensionality on the structure of channels in the mushy zone and of channel mesosegregations. We present the influence of the permeability of the mushy zone on the channel structures. We show that channels can form either as lamellar or as tubular structures: at higher permeability secondary instabilities across the cavity thickness promote a transition from lamellar to tubular channels.

Modelling of Columnar-to-Equiaxed and Equiaxed-to- Columnar Transitions in Ingots Using a Multiphase Model

We present a new method to handle a representative elementary volume (REV) with a mixture of columnar and equiaxed grains in ingot castings in the framework of an Eulerian volume averaged model. The multiscale model is based on a previously established fully equiaxed model. It consists of a three-phase (extra-granular liquid, intra-granular liquid and solid) grain-growth stage coupled with a two-phase (solid and liquid) macroscopic transport stage accounting for grain and nuclei movement. In this context, we take into account the formation of a columnar structure and its development using a simplified front-tracking method. Columnar solidification is coupled with the growth of equiaxed grains ahead of the columnar front. The particularity of the model is the treatment of concurrent growth of mixed columnar and equiaxed structures only in the volumes that contain the columnar front. Everywhere else, the structure is considered either fully columnar or fully equiaxed. This feature allows for reasonable computational times even in industrial size castings, while describing the solutal and mechanical blocking phenomena responsible for the Columnar-to-Equiaxed Transition. After a validation of the model, we discuss the numerical results for a 6.2-ton industrial steel ingot by comparison with experimental measurements. Final maps for macrosegregation and grain structures size and morphology are analysed. Furthermore, we quantify the impact of nuclei formation through fragmentation along the columnar front on the result. An attempt at predicting the occurrence of the Equiaxed-to-Columnar Transition in the later phases of the process is also made.

Multi-scale Unite element modelling of solidification structures by a splitting method taking into account the transport of equiaxed grains

In solidification processes of large industrial castings and ingots, the transport of solid in the liquid has an important effect on the final grain structure and macrosegregation. Modeling is still challenging as complex interactions between heat and mass transfers at microscopic and macroscopic scales are highly coupled. This paper first presents a multi-scale numerical solidification model coupling nucleation, grain growth and solute diffusion at microscopic scales with heat and mass transfer, including transport of liquid and solid phases at macroscopic scales. The resolution consists of a splitting method, which considers the evolution and interaction of quantities during the process with a transport stage and a growth stage. This splitting reduces the nonlinear complexity of the set of considered equations and provides an efficient numerical implementation. It is inspired by the work of Založnik et al. [1,2], which used a finite volume method (FVM). The present work develops the solution based on the finite element method (FEM). Numerical results obtained with this model are presented and simulations without and with grain transport are compared to study the impact of solid-phase transport on the solidification process and on the formation of macrosegregation.

Mesoscopic modeling of columnar solidification and comparisons with phase-field simulations

We use two complementary modeling approaches for the simulation of columnar growth in directional solidification of organic alloys: a phase field model and a mesoscopic envelope model of dendritic growth. While the phase-field method captures the details of the dendritic structure and of the growth dynamics, the mesoscopic model approximates the complex dendritic morphology by its envelope. The envelope growth is deduced from the velocities of the dendrite tips, calculated by an analytical LGK-type tip model that is matched to the temperature and concentration fields in the stagnant film around the envelope. The computational cost of the mesoscopic model is several orders of magnitude lower and can bridge the gap between phase-field and macroscopic models. We demonstrate the applicability of the mesoscopic model to columnar growth and we discuss in particular its capabilities to predict the primary dendrite arm spacing.

The effect of finite microscopic liquid solute diffusion on macrosegregation formation

We study the effect of solidification kinetics, driven by local limited diffusion in the liquid, on macrosegregation. If the diffusion in the liquid surrounding a growing grain is slow, the local average liquid concentration is lower than the thermodynamic equilibrium concentration at the interface. The redistribution of solute by the flow of intergranular liquid on the macroscopic scale is affected by the modified microsegregation in the liquid. We study this phenomenon using a two-phase model based on the volume-averaging method, describing macroscopic transport coupled to a microscopic grain growth model. The growth kinetics is resolved by accounting for finite diffusion in the liquid and solid phases, assuming an equiaxed globular morphology. To accurately model the diffusion field around the grain, we propose an improved approximation for the solutal boundary layer thickness accounting for the growth conditions and liquid convection. The effect of growth kinetics on macrosegregation is then investigated in the case of solidification of a binary alloy in a small cavity where the solid phase is fixed and fluid flow is driven by natural convection. We show that it is important to accurately model the diffusion field around the grain to capture correctly the effect of growth kinetics on the weakening of macrosegregation.