1. ~0.5 m Computational grid size Process ~5 mm REV Modeling Microsegregation In Metal Alloys – Vaughan Voller, University of Minnesota 1 of 14 Can we build a direct-simulation of a Casting Process that resolves to all scales? Difficult due to wide range of Time and Space Scales ~ 50  m solid representative ½ arm space sub-grid model g Scales in a “simple” solidification process model

2. chi ll A Casting The REV Nucleation Sites columnar equi-axed The Grain Envelope The Secondary Arm Space The Tip Radius   The Diffusive Interface ~ 0.1 m ~10 mm ~ mm ~100  m ~10  m ~1 nm 10 3 10 1 10 -1 10 -3 10 -5 10 -7 10 -9 10 -9 10 10 -3 10 -1 Length Scale (m) interface kinetics nucleation solute diffusion growth grain formation casting heat and mass tran. Time Scale (s) Scales in Solidification Processes To handle with current computational Technology require a “Micro-Macro” Model See Rappaz and co-workers Example a heat and Mass Transfer model Coupled with a Microsegregation Model 2 of 14 (after Dantzig)

3. ~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 -- 3 of 14 Micro segregation—segregation and solute diffusion in arm space

4. A C Primary Solidification Solver Transient mass balance equilibrium g ClCl T Iterative loop g model of micro-segregation (will need under-relaxation) 4 of 14 Give Liquid Concentrations

5. liquid concentration due to macro-segregation alone Micro-segregation Model In a small time step new solid forms with lever rule on concentration Q -– back-diffusion Need an easy to use approximation For back-diffusion transient mass balance gives liquid concentration Solute mass density before solidification Solute mass density of new solid (lever) Solute mass density after solidification 5 of 14 Solute Fourier No. ½ Arm space of length takes t f seconds to solidify

6. The parameter Model --- Clyne and Kurz, Ohnaka 6 of 14 For special case Of Parabolic Solid Growth And ad-hoc fit sets the factor and In Most other cases The Ohnaka approximation Works very well

7. The Profile Model Wang and Beckermann Need to lag calculation one time step and ensure Q >0 7 of 14 m is sometimes take as a constant ~ 2 BUT In the time step model a variable value can be use Due to steeper profile at low liquid fraction ----- Propose

8. Coarsening Arm-space will increase in dimension with time This will dilute the concentration in the liquid fraction—can model be enhancing the back diffusion  A model by Voller and Beckermann suggests If we assume that solid growth is close to parabolic In profile model m =2.33 in Parameter model 8 of 14

9. Remaining Liquid when C =5 is Eutectic Fraction 9 of 14 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 l < 5 Calculating the transient value of g from Parameter or Profile

10. Results are good across a range of conditions Note Wide variation In Eutectic 10 of 14

11. Parabolic solid growth – No Second Phase – No Coarsening Use 10,000 equal of  g C 0 = 1,  = 0.13,  = 0.4 Use To calculate segregation ratio 11 of 14

12. Performance of Models under parabolic growth no second phase Prediction of segregation ratio in last liquid to solidify (fit exponential through last two time points) 12 of 14

13. Parameter Robust Easy to Use Poor Performance at very low liquid fraction— can be corrected Profile A little more difficult to use With this Ad-hoc correction Excellent performance at all ranges Two Models For Back Diffusion A C Predict g predict C l predict T Calculate Transient solute balance in arm space Solidification Solver 13 of 14 Account for coarsening My Method of Choice

14. 1000 2 0.6667 Year “Moore’s Law” current for REV of 5mm Voller and Porte-Agel, JCP 179, 698-703 (2002).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) 14 of 14 2055 for tip Model Directly (about 10 18 nodes) Tip-interface scale