Volumetric grain and nuclei density and envelope conservation:
The grains can grow from a chill wall, in form of columns, or from nuclei in the fully liquid region, in form of equiaxed grains. A nucleus is a precursor or seed for the nucleation of a grain. The nuclei can be of different types depending on their origin (fragments of the columnar region, inoculated particles or impurities) and size, thus, their nucleation undercoolings can be different. Bedel et al.  investigated the impact of the nuclei movement on the microstructure and on macrosegregation in direct chill casting of aluminum alloys. For that, they considered that the nuclei were advected by the liquid phase, solving a nuclei population balance. This balance is given by:@ @t Ni + r · h~vlil Ni nuc = − ˙Ni (1.34).
Where N is the nuclei density, which is defined as the number of nuclei per cubic meter and ˙N is the grain nucleation rate. The superscript i denotes each type of nuclei.
When nucleation occurs, equiaxed grains appear in the fully liquid region. In order to obtain geometrical information about the equiaxed grains, it is necessary to solve for the conservation of the volumetric grain density. For example, it is possible to relate the solid fraction to the averaged grain radius R by gs = 4 3NgR3 (in case of two-phase globular model), where Ng is the volumetric grain density. The volumetric grain density is advected by the solid phase and its conservation equation is given by a population balance: @ @t Ng + r · (h~vsis Ng) = mX i=1 ˙N i (1.35).
Coupling micro/macro modeling: Operator-splitting method for the solution of the coupled equations
The solidification model previously presented leads to a complex system of coupled partial differential equations. This system must be solved consistently to ensure precision and numerical stability on all computed fields. In literature, it is usually mentioned that iterative methods are used to couple the system of equations without further explanation. Založnik and Combeau [65, 79] proposed an operator-splitting scheme for the solution of the coupled system which basically consists in the scale separation (micro/macro) at the solution level.
They proposed a temporal scheme that, in a first stage, solves all the advective contributionsto the different conservation equations, neglecting all the interfacial exchanges due to grain growth. Next, in a second stage, all interfacial exchange terms are computed by means of a proper grain growth kinetics model. Finally, both contributions, advection and interfacial exchanges, are added as shown in Figures 1.8 and 4.11. For a quantity , the temporal splitting scheme is given by: @ @t = Atr + Bgr (1.37).
Solidification with a planar front in semiconductor processing (no mushy zone)
Directional solidification experiments under centrifugal conditions were conducted by Rodot et al.  using a PbTe and PbSnTe alloys which were doped with Ag atoms. Centrifugation was used to reduce the damaging effect of small lateral temperature gradients that occur due to imperfect thermal control of the crucible. In the study case, the axial temperature gradient (along the ampoule) was antiparallel to the centrifugation induced gravity (perpendicular to the rotation axis). Their results showed that increasing the level of centrifugation, the magnitude of Ag macrosegregation also increased due to liquid convection. However, they also reported that in cases with N = 5g; r = 18m and N = 2g; r = 5.5m (where N is the centrifugal acceleration and r the centrifuge arm size), the segregation profile was similar to the one observed in samples solidified in microgravity. Moreover, an improvement on the surface quality and on crystalline structure was noticed. In their work, they could not explain completely these results, but they indicate that a “convectionless regime” can be achieved combining properly centrifugation level (centrifuge rotation rate) and centrifuge arm size. In their cases, the “convectionless regime” was produced when N r 3. This study was the first one found that relates the segregation of alloying elements to centrifugation level.
Growth of Te-doped InSb cylindrically-shaped crystals were performed by Müller et al.  by using a Bridgman apparatus mounted in a centrifuge. The axial temperature gradient (along the cylinder centerline) was anti-parallel to the total apparent gravity ~gtot, sum of centrifugal and normal terrestrial gravity accelerations, in such a way, that “vertical” Bridgman growth, under variable gravity levels, could be performed. Temperature fluctuations in the themocouples and striations in the resulting crystal appeared when 1.65g < ~gtot < 2.5g. Increasing even more the total gravity, ~gtot > 2.7g, all temperature fluctuations and crystal striations disappeared. The authors pointed out that a flow regime transition took place in the melt convection when the total apparent gravity was increased. In order to have a better insight of the transition process, several numerical simulations of liquid convection were performed. In the simulations, the cylinder aspect ratio was maintained constant while the temperature gradient and centrifuge rotation speed were varied. All the analysis was given in function of the dimensionless Rayleigh number Ra = ~gtotTTh3 , where T is the thermal expansion coefficient, the thermal diffusivity, the viscosity and T the difference across the cylinder length h. The Rayleigh number is defined as the ratio between the time scale for thermal transport via diffusion and the time scale for thermal transport via convection. Two types of flow regime were found and characterized in the numerical study. The regime type I was characterized by a main circulation in the perpendicular plane to the rotation axis with small eddy currents in the corners. The circulation had a rotation sense opposite to the centrifuge. The authors conclude that this regime is quasi-stable since it had a small stability range when different parameters were varied. On the other hand, the regime type II was characterized by a main circulation in the perpendicular plane to the rotation axis and presented the same rotation sense to the centrifuge. This regime had a wide stability range and a much stronger convection compared to the regime type I. Both regimes could be achieved with the same set of Rayleigh, Prandtl, Taylor and aspect ratio. This work was the first one to describe the flow transition in cylinders mounted along centrifuge arms, in solidification cases. Although the importance of this work, the authors did not relate the regime transition with the dimensionless Rossby number, Ro = Uref 2!Lref (where Uref is the reference velocity, Lref the reference length scale and ! the centrifuge angular velocity), which quantifies the Coriolis effect. This study demonstrated that the Rayleigh number is insufficient to fully characterize the different types of flow regimes.
Eulerian derivation of Navier-Stokes equations in a rotating reference frame
Kageyama and Hyodo  used an Eulerian approach to show the derivation of the Coriolis force in a rotating reference frame. Contrarily to a Lagrangian approach, in an Eulerian approach, the velocity is described in a point which is fixed in space instead of following a fluid parcel.
According to the authors, there are three advantages of using an Eulerian approach: (1) it is general, since it can be used in the transformation of any vector field (e.g. Maxwell’s equations in rotating reference frame), (2) the physical meaning of the Eulerian derivation is clear and (3) it is mathematically rigorous. Although the authors clearly presented the methodology, it was only used in the transformation of incompressible Navier-Stokes equations, not reviewing its application in cases of scalar conservation equation or more complex models, such as, multiphase flows models. In this section, the methodology used in  will be presented, detailing all the intermediate steps that were omitted in the original publication. In the following sections the methodology will be applied to derive averaged scalar conservation equations and averaged momentum conservation equations for macroscopic modeling of solidification.
Consider an arbitrary velocity vector ~uf and a reference frame in constant rectilinear movement with a ~V f velocity. The moving reference frame will be denoted as L0, as shown in Fig. 2.1. Note that vector ~uf , observed from L0 is ~u0 = ~uf − ~V f since both reference frames are in relative movement. A Galilean transformation for velocity can be defined in order to relate how the same vector is observed from the different reference frames: ~u0 ~x0, t = G ~V n ~uf ~xf , t o = ~uf ~xf , t − ~V f (2.1).
Prediction of macrosegregation
macroS3D was validated for solidification cases using the Pb-18wt%Sn benchmark solidification case proposed by Bellet et al. . This benchmark case was well documented by Combeau et al. , who reported results obtained with various numerical implementations. The case consists of the solidification of a two-dimensional cavity (50mm wide and 60mm high), which is cooled from the right side, whereas the left side is a symmetry plane and the upper and lower sides are thermally insulated. Figure 3.1a presents a schematic of the solidification domain and its boundary conditions.
In order to perform the numerical simulation, a quadrilateral mesh containing 200 cells along the width and 240 along the height was created using the OpenFOAM standard tool blockMesh. The first order upwind interpolation scheme was used for all advective terms and a linear interpolation scheme for all diffusion terms. The alloy thermophysical properties used to simulate this case are reported in Appendix A.1.1.
Table of contents :
Chapter 1 Literature review
1.1 Solidification and its macroscopic modeling
1.1.2 Volume averaging method
1.1.3 Microscopic modeling
1.1.4 Macroscopic modeling
1.1.5 Coupling micro/macro modeling: Operator-splitting method for the solution of the coupled equations
1.2 Rotating fluids and centrifugal casting
1.2.1 Solidification in centrifugal conditions
1.2.2 Rotating reference frame: Apparition of centrifugal and Coriolis
Chapter 2 Solidification model for centrifugal casting
2.1 Eulerian derivation of Navier-Stokes equations in a rotating reference frame
2.2 Derivation of scalar conservation equation
2.3 Derivation of momentum conservation equations
Chapter 3 Thermosolutal buoyancy convection and macrosegregation during solidification in a centrifugated system
3.1 Simplified model for solidification modeling
3.2 Simulation of TiAl samples solidified in the “Large Diameter Centrifuge” (LDC)
3.2.1 Furnace thermal protocol
3.2.2 Solidification path and alloy phase diagram
3.2.3 Numerical setup
3.3 Results and discussion
3.3.1 Furnace thermal protocol: Temperature field and heat transfer
3.3.2 Liquid flow
3.3.3 Aluminum macrosegregation
3.3.4 Comparison with aluminum measurements
Chapter 4 Equiaxed grain motion and grain growth kinetics
4.1 Macroscopic conservation equations
4.1.1 Mass balance of phase k
4.1.2 On the coupling of enthalpy, temperature and solid fraction
4.1.3 Momentum conservation and mass conservation equations
4.1.4 Other macroscopic conservation equations
4.1.5 Validation of the transport model in macroS3D
4.2 Microscopic modeling in macroS3D
4.2.1 Grain growth kinetics
4.2.2 Grain nucleation and nuclei re-injection
4.2.3 Validation of the microscopic modeling in macroS3D
4.3 Coupling of macroscopic conservation equations with microscopic modeling
4.3.1 Validation of the multiscale modelling in macroS3D: The Hebditch and Hunt case
4.4 Simulations of the GRADECET experiments
4.4.1 Results and discussion
General conclusions and perspectives
Appendix A Thermophysical properties
A.1.1 Thermophysical properties: Pb-18wt.%Sn alloy
A.1.2 Thermophysical properties: water/glycerol mixture for the rotating annulus case
A.1.3 Thermophysical properties: Sedimentation column and Hebditch-Hunt benchmark cases (Sn-5wt.%Pb)
A.1.4 Thermophysical properties: cases of solidification of an unitary control volume (Al-5wt.%Si)
List of Tables
List of Figures