1. Session III: Computational Modelling of Solidification Processing Analytical Models of Solidification Phenomena V. Voller QUESTION: As models of solidification process and phenomena become more complex do analytical solutions of limit cases become less useful? Short Answer: Yes, limit cases that admit analytical solutions are often physically too far removed from the process/phenomena of interest to be useful. Counter Answer: With a bit of searching and a little innovation it is possible to build physically sophisticated limit solutions of process/phenomena that admit analytical solutions.

2. A Key Moment In History of the Computational Modelling of Solidification Processing 65 years ago Heating of a steel Ingot Followed the standard modeling paradigm of Validation ---comparing computations to measurement observed calculated A very early (first ?) paper using numerical modeling of heat transfer in metals processing. Pre-digital The Differential Analyser: An analog machine Build for ~ $30 in 1934. Crank, 1947

3. A Key Moment In History of the Computational Modelling of Solidification Processing At that time these were using state of the art computations First use of Enthalpy Method for Solidification Model Early application of Crank-Nicolson And they recognized the need to Verify their calculations via comparison with appropriate analytical solutions

4. But in today's world with solutions obtained with sate of the art digital technologies Differential Analyzer Allow us to solve much more complex systems solidmushliquid cool mold One-D solidification of an alloy controlled by heat conduction vs Crystal growth in an under cooled alloy vs. Distributed Graphics Processing Units (GPUS) Is there still a place/role/opportunity for the meaningful use of analytical solutions?

5. growth of an initially spherical seed in an under cooled alloy In fact for the problem shown there is a rich source of available analytical solutions Carslaw and Jaeger, Conduction of Heat in Solids, (1959). (CJ) Rubinstein, The Stefan Problem (1971) (R) Alexiades and Solomon, Mathematical Modeling of Melting and Freezing Processes, (1984). (AS) Dantzig and Rappaz, Solidification (2009) (DR) Two Examples

6. One –D Solidification of a supper heated Binary Alloy-with a planar front (R, AS, DR) solid liquid alloy T<T equ Front movement Conc. History (kappa = 0.1) Temp. HistorySymbols Numerical Lines analytical Constitutional undercooling Calls into question planar assumption conc. profile

7. TmTm Solidification of a spherical seed in an under cooled PURE melt Analytical Solutions in Carslaw and Jaeger –(also solutions for planar and cylindrical case) Dantzig and Rappaz -(considers Surface Tension in limit of zero Stefan number)

8. Here we will demonstrate how these solutions can be coupled to model the solidification of A spherical seed in in an UNDER COOLED BINARY ALLOY With assumption of no- surface under cooling and no growth anisotropy. solid liquid alloy T<T equ TmTm

9. Temperature Conc. SolidLiquid Dimensionless Governing Equations Heat Concen. sensible/latent Stefan No. thermal/mass Lewis No. fixed values in solid

10. Temperature Conc. SolidLiquid Similarity Solution Assume Can then show that value of Lambda follows from solving the following set of equations Liquid temp and con. Then given by

11. Solution can tell us something about the nature of the Lewis Le number and Verify Numerical Algorithms for coupling of solute and thermal fields in crystal growth codes conc. profile Numerical (enthalpy) symbols Lines analytical Le ~1 similar thickness of Solute and thermal layers Le >>1 much thinner solute layer

12. Outline of Enthalpy Solution in Cylindrical Coordinates Assume an arbitrary thin diffuse interface where liquid fraction Define Throughout Domain a single governing Eq For a PURE material Numerical Solution Very Straight-forward

13. Initially seed Set Transition: When An explicit solution If Update Liquid fraction Update Temperature

14. R(t) Excellent agreement with analytical when predicting growth R(t)

15. Can extend to the case of a binary alloy by defining a mixture solute as Explicitly solving With Liquidus line If Update Liquid fraction Quad eq. in Update Temperature

16. 2-D enthalpy solutions Of cy. seed growth in an undercooled pure melt Similarity Solution Each one based on a different seed geometry and front update ALL are wrong—Since there is no imposed anisotropy Conclusions Analytical models of solidification phenomena are important tools in advancing our understanding of solidification processes. There is a rich source of available analytical solutions that can be adapted to provide meaningful solutions for a variety of solidification process and phenomena of current interest e.g. Coupling of thermal and solute fields in crystal growth ---Beyond this they can be used to bench-marking the predictive performance of large multi-scale, general numerical solidification process models. These solutions are useful -- In the first instance they allow for a clear and direct understanding of the behavior and interaction of key elements in a solidification system e.g. role of Lewis number And Grid Anisotropy