Computational grid size ~0.5 m Process ~5 mm REV Maco-Micro Modeling— Simple methods for incorporating small scale effects into large scale solidification models – Vaughan Voller, University of Minnesota 1 of 19 Can we build a direct-simulation of a Casting Process that resolves to all scales? Scales in a “simple” solidification process model ~ 50  m solid representative ½ arm space sub-grid model g Enthalpy based Dendrite growth model

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 Length Scale (m) interface kinetics nucleation solute diffusion growth grain formation casting heat and mass tran. Time Scale (s) Scales in Solidification Processes 2 of 19 (after Dantzig) Can we build a direct-simulation of a Casting Process that resolves to all scales?

Well As it happened not currently Possible Year “Moore’s Law” Voller and Porte-Agel, JCP 179, (2002) Plotted The three largest MacWasp Grids (number of nodes) in each volume 2055 for tip 6 decades 1 meter 1 micron 3 of 19

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 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 4 of 19 (after Dantzig)

~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 -- 5 of 19 Micro segregation—segregation and solute diffusion in arm space

A C Primary Solidification Solver Transient mass balance equilibrium g ClCl T Iterative loop g model of micro-segregation (will need under-relaxation) 6 of 19 Give Liquid Concentrations

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 7 of 19 Solute Fourier No. ½ Arm space of length takes t f seconds to solidify

The parameter Model --- Clyne and Kurz, Ohnaka 8 of 19 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

The Profile Model Wang and Beckermann Need to lag calculation one time step and ensure Q >0 9 of 19 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

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 10 of 19

Remaining Liquid when C =5 is Eutectic Fraction 11 of 19 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

Results are good across a range of conditions Note Wide variation In Eutectic 12 of 19

Predictions of Eutectic Fraction With constant cooling C o = 4.9 C eut = 33.2 k = 0.16 Comparison with Experiments Sarreal Abbaschian Met Trans of 19

Parabolic solid growth – No Second Phase – No Coarsening Use 10,000 equal of  g C 0 = 1,  = 0.13,  = 0.4 Use To calculate evolving segregation ratio 14 of 19

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) 15 of 19

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 16 of 19 Account for coarsening My Method of Choice

Year “Moore’s Law” current for REV of 5mm Voller and Porte-Agel, JCP 179, (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) 17 of for tip Model Directly (about nodes) Tip-interface scale

chill solid mushy liquid riser y Application – Inverse Segregation in a binary alloy 100 mm Shrinkage sucks solute rich fluid toward chill – results in a region of +ve segregation at chill Fixed temp chill results in a similarity solution Parameter Current estimate empirical 18 of 19

Ferreira et al Met Trans 2004 Comparison with Experiments 19 of 19