1. ~0.5 m ~ 50  m solid ~5 mm Computational grid size ProcessREV representative ½ arm space sub-grid model g A Microsegregation Model – Vaughan Voller, University of Minnesota floor building column 1 of 11

2. ~0.5 m ~ 50  m solid ~5 mm Computational grid size ProcessREV representative ½ arm space sub-grid model g from computation Of these values need to extract Solidification Modeling -- 2 of 11

3. A C Primary Solidification Solver Transient mass balance equilibrium g ClCl T Iterative loop g model of micro-segregation (will need under-relaxation) 3 of 11

4. liquid concentration due to macro-segregation alone Micro-segregation Model new solid forms with lever rule on concentration transient mass balance gives liquid concentration q -– back-diffusion Need an easy to use approximation For back-diffusion Solute mass density before solidification Solute mass density of new solid (lever) (1/s) Solute mass density after solidification 4 of 11

5. The parameter Model --- Clyne and Kurz, solidification time parabolic growth Ohnaka Coarsening Voller and Beckermann suggest m = 2.33 5 of 11

6. The Profile ModelWang and Beckermann proposed modification steeper profile at low liquid fraction solidification time parabolic growth Coarsening Voller and Beckermann suggest Need to lag calculation one time step and ensure q >0 NOTE 6 of 11

7. Constant Cooling of Binary-Eutectic Alloy With Initial Concentration C 0 = 1 and Eutectic Concentration C eut = 5, No Macro segregation,  = 0.1 Use 200 time steps and equally increment 1 < C < 5 Calculating the transient value of g from Parameter or Profile Remaining Liquid when C =5 is Eutectic Fraction No Coarsening Coarsening 7 of 11

8. Parabolic solid growth – No Second Phase Use 10,000 time steps and set g =  1/2 at each step C 0 = 1,  = 0.13,  = 0.4 Use To calculate segregation ratio 8 of 11

9. Performance of Profile Model parabolic growth no second phase Prediction of segregation ratio in last liquid to solidify (fit exponential through last two time points)  =0.1  =0.4 9 of 11

10. A C Predict g predict C l predict T Calculate Transient solute balance in arm space Solidification Solver Two Models For Back Diffusion Profile Parameter Robust Easy to Use Possible Poor Performance at very low liquid fraction A little more difficult to use With this Ad-hoc correction Excellent performance at all ranges 10 of 11

11. .5m I Have a BIG Computer Why DO I need an REV and a sub grid model ~ 50  m solid ~5mm (about 10 6 nodes) Model Directly (about 10 18 nodes) Tip-interface scale 11 of 11 1000 2 0.6667 Year “Moore’s Law” current for REV of 5mm 2055 for tip Voller and Porte-Agel, JCP 179, 698-703 (2002