Microsegregation Models and their Role In Macroscale Calculations

1. Microsegregation Models and their Role In Macroscale Calculations Vaughan R. Voller University of Minnesota

2. What is Macrosegregation Partitioned solute at solid-liquid interface After Flemings (Solidification Processing) and Beckermann (Ency. Mat) Cliquid Csolid 1m Redistributed by Fluid and solid motion shrinkage grain motion convection

3. Macro (Process) Scale Equations Equations of Motion (Flows) mm REV Heat: Solute Concentrations: Assumptions for shown Eq.s: -- No solid motion --U is inter-dendritic volume flow To advance to the next time step we need find REV values for • T temperature • Cl liquid concentration • gs solid fraction • Cs(x) distribution of solid concentration

4. Need four relationships which can be obtained from a micro-scale model • Under the assumptions of: • Equilibrium at solid-liquid interface • Perfect solute mixing in the liquid • Identification of a solid-liquid interface length scale • (e.g., a ½ secondary arm space) Possible Relationships are Definitions of mixture terms in arm space 4. Account of local scale redistribution of solute during solidification of the arm space 1. 2. 1 Thermodynamics 3. T=G(Cl1,Cl2…….) Primary + Secondary Xs(t) ~ 10’s mm Clk <--> Csk (interface) Xl(t)~t1/3 gs= Xs/Xl Clk=F(Cl1,Cl2…….) coarsening

5. The Micro-Scale Model Macro Inputs ~ 1 mm Xs(t) solid Liquid Xl(t)~t1/3 Define: Average Solute Concentration in Xl-Xsold During time step Dt . Treat like initial sate for a new problem back coarsening macro thermo new solid Iteration: Guess of T in [H]  Xs –Xsold  Assume Lever on C*  Cl--, T

6. Requires a Micro-Segregation Modelthat to estimate “back diffusion” of solute into the solid at the solid-liquid interface ( Three Approaches) Xs(t) ~ 1 mm solid Liquid Xl(t)~t1/3 1. Numerical Solution in solid 3***. At each time step approx. solid solute profile as 2. Approximate with “average” parameter Choose to satisfy Mass Balance Function of a can be corrected for coarsening

7. Testing: Binary-Eutectic Alloy. Cooling at a constant rate Predictions of Eutectic Fraction at end of solidification solid Numerical back diff model coarsening Approx profile model

8. A uni-directional solidification of a Of binary alloy cooled from a fixed chill. Effect of Microsegregation On Macrosegregation g Zliq Zeut Microsegregation (back diffusion into solid) modeled in terms off rate of change of solute in liquid z Solute concentration in mushy region Coarsening lever a, b = 1 a = 0.2 Gulliver -Scheil No Coarsening a=0, b = 0

9. Key features -- Simple Equilibrium Thermodynamics -- External variables consistent with macro-scale conservation statements -- Accurate approximate accounting of BD and coarsening at each step based on current conditions Summary From macro variables 1m Find REV variables Accounting for solid ~ 1 mm ~ 1 mm Microsegregation And Themodynamics T ,g, Cs and Cl at solid-liquid interface

10. I Have a BIG Computer Why DO I need an REV and a sub grid model solid ~ 1 mm ~mm 1m (about 109) Tip-interface scale Model directly (about 1018)

11. Well As it happened not currently Possible 1000 20.6667 Year “Moore’s Law” 2055 for tip 2010 for REV of 1mm Voller and Porte-Agel, JCP 179, 698-703 (2002) Plotted The three largest MacWasp Grids (number of nodes) in each volume

12. Modeling the fluid flow requires a Two Phase model That may need to account for: Both Solid and Liquid Velocities at low solid fractions A switch-off of the solid velocity in a columnar region A switch-off of velocity as solid fraction g  o. An EXAMPLE 2-D form of the momentum equations in terms of the interdentrtic fluid flow U, are: Extra Terms turbulence Buoyancy Friction between solid and liquid Accounts for mushy region morphology Requires a solid-momentum equation