, Thermophysical properties: Pb-18wt, A.1.1

, Thermophysical properties: water/glycerol mixture for the rotating annulus case

, Thermophysical properties: Sedimentation column and Hebditch-Hunt benchmark cases (Sn-5wt.%Pb)

, Thermophysical properties: cases of solidification of an unitary control volume (Al-5wt.%Si)

, Scalar conservation equations for solidification modeling written in terms of the rotating reference frame

, Thermophysical properties used in the simulations

, Total apparent gravity (g is the normal terrestrial gravity), centrifuge angular velocity, ?, reference velocity, U ref and Rossby number

, Thermophysical properties of the water/glycerol mixture used in the simulation case of Sect. 3.1

A. ;. , Thermophysical properties of Sn-5wt.%Pb alloy used in the sedimentation column, p.156

). .. , 157 RE-1Schéma du four et de l'échantillon de TiAl dans la centrifugeuse. L'échantillon est représenté en gris et les éléments chauffants en orange et jaune. L'angle d'inclinaison du four est indiqué par l'angle ?, la vitesse de vol du four par V f light (en bleu), la gravité apparente totale par g tot (en orange) et le gradient de température principal de ?T (en rouge)

×. Le, étude en fin de solidification pour différents niveaux de centrifugation. Contours à C = C 0 (ligne noire)

]. .. ,

. .. , Effect of the grain movement on macrosegregation in a case of Pb-Sn. A stainless steel mesh was placed at the half-height of the mold (de Groh [48]), p.16

, Each case has a solid (black), mush (gray) and liquid (white) regions. No solutal buoyancy effects considered

, Red circle shows the main sense of the liquid flow

, Representative elementary volume (REV) for the representation of columnar and equiaxed grains

, Schematic of solute diffusion at solid-liquid interface in the two-phase approach for microscopic modeling

, Schematic of the intradendritic liquid solute balance in the three-phase approach where g e denotes the envelope fraction, g li the intradendritic liquid and g le the extradendritic liquid. The intradendritic liquid is considered to be perfectly mixed, p.24

, Complete algorithm proposed by Zalo?nik and Combeau [65] for solidification modeling

, Note the "seagull wing" shape of the radial columnar growth. g cent denotes the direction of the centrifugal acceleration. Vector g denotes the direction of the normal terrestrial gravity

, Fixed reference frame (black) and rotating reference frame (red) which is dependent in time, vol.36

. .. , Fixed reference frame (black) and a moving reference frame (blue) at constant velocity V f . The axes of both reference frames are aligned, p.43

. .. , Schematic representation of how the same velocity vector is observed from a fixed reference frame u f x f , t and a rotating reference frame u ( x, t), p.44

. Bellet, 60 3.2 Comparison between "castFoam" and macroS3D: Instantaneous intrinsic liquid velocity magnitude and liquid fraction at t = 120 s along the H 2 line, Schematic and instantaneous result of the Pb-18wt%Sn solidification case, p.61

. .. , Comparison between "castFoam" and macroS3D: Averaged solute concentration of tin at the end of solidification in H 1 and H 2 lines (See Fig. 3.1a), p.62

, Tin segregation map at the end of solidification. Comparison between macroS3D and "castFoam" [4] (re-plotted)

, Schematic of the rotating annulus case along with the three-dimensional mesh used to perform the numerical simulations

, Comparison between results obtained with macroS3D and experimental measurements. The inner side (blue line) is 4 ? C colder than the outer side (red line)

, Comparison between macroS3D numerical results, experimental measurements and numerical results from, Mean azimuthal velocity vs radial position, vol.91, p.66

, H3") and orange ("H1" and "H2") circles. At t = 0, the heaters temperatures are T H1 = 1623 ? C , T H2 = 1623 ? C and T H3 = 1543 ?

, 68 4.7 Solid fraction on the vertical certerline over time for different numerical configurations on macroS3D. Solid line: macroS3D (FVM), Temperature measured by the thermocouples in the three stages of the furnace protocol

, Comparison between SOLID R and macroS3D for the Case A. "glob" denotes results obtained using a two-phase globular model

, Comparison between SOLID R and macroS3D for the Case B. "glob" denotes results obtained using a two-phase globular model

, Comparison between SOLID R and macroS3D for the Case C. "glob" denotes results obtained using a two-phase globular model

, The overall operator splitting algorithm for solidification implemented in macroS3D. The variable ? denotes an arbitrary field. The index m denotes the iteration "m" within a time-step. The index t denotes the old time-step (previous macrotimestep). The index n denotes the iterations for the energy equation, p.119

, Transport stage in macroS3D. The index m denotes the iteration "m" within a time-step. The index t denotes the old time-step (previous macro-timestep), p.120

, Nucleation stage in macroS3D. The index m denotes the iteration "m" within a time-step. The index t denotes the old time-step (previous macro-timestep), p.121

, The variable ? denotes an arbitrary field. The index t denotes the old time-step (previous macro-timestep). The index p denotes the internal iterations of the grain growth algorithm, Growth stage in macroS3D. The index m denotes the iteration "m" within a time-step

, 16 Comparison between the solutions given by SOLID R and macroS3D (upwind, mesh 80x60) at various instants for the average solute concentration (color map), liquid intrinsic velocity (vectors) and formation of the bed of packed grains (black contours for solid fraction equals to 0.1, 0.2 and 0.3), Schematic of the solidification domain in the Hebditch and Hunt case, vol.123

, Comparison between the solutions given by macroS3D for the macrosegregation pattern (color map) at the end of solidification using different mesh sizes and different numerical schemes

. .. , Schematic of the columnar and equiaxed regions considered in the simulations. The predefined CET was adjusted to z CET = 35 mm. The TiAl sample is shown in gray. The heaters are shown in purple and orange, p.130

?. C. , Liquid velocity and liquid fraction in the main plane of study inside the columnar region for the case G1P3N9. Violet contours delimit the mushy zone. Temperature contours are in green for 1490, 1500, 1503, 1508, 1510, p.133

?. C. , Liquid velocity and liquid fraction in the main plane of study inside the columnar region for the case G5P3N9. Violet contours delimit the mushy zone. Temperature contours are in green for 1490, 1500, 1503, 1508, 1510, p.133

, Evolution of the liquid velocity pattern in the main plane of study in the equiaxed region for the case G1P3N9. Temperature contours in black: 1400,1450,1500 and 1550 ? C. Violet dashed line delimits the columnar region. The green contours delimit the slurry region

. .. , Liquid intrinsic velocity and different fields in the packed equiaxed region for the case G1P3N9 at t = 450 s. Lines in black limit the packed equiaxed region. From left to right: liquid fraction, averaged solute concentration in liquid and buoyancy liquid density ( C ref ), p.136

, Evolution of the liquid velocity pattern in the main plane of study in the equiaxed region for the case G5P3N9. Temperature contours in black: 1400,1450,1500 and 1550 ? C. Violet dashed is the predefined CET position. The green contours delimit the slurry region

, a) Case under normal terrestrial gravity. (b) Case with centrifugation

. Drag-coefficient, Green region corresponds to Stokes flow. Blue and black-dashed regions correspond to Ni and Beckermann [60] for g s = 10 ?4 and g s = 0.05, respectively

, Ratio between the characteristic time of drag response and the characteristic time of rotation vs the grain diameter for different levels of apparent gravity (centrifuge arm is assumed 4 m)

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