Advanced Scaling Techniques for the Modeling of Materials Processing

Advanced Scaling Techniques for the Modeling of Materials Processing Karem E. Tello Colorado School of Mines Ustun Duman Novelis Patricio F. Mendez Director, Canadian Centre for Welding and Joining University of Alberta

Phenomena in Materials Processing • Transport processes play a central role • Heat transfer • Fluid Flow • Diffusion • Complex boundary conditions and volumetric factors: • Free surfaces • Marangoni • Vaporization • Electromagnetics • Chemical reactions • Phase transformations • Multiple phenomena are coupled

Example: Weld Pool at High Currents gouging region trailing region rim

Driving forces in the weld pool (12) Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Multiphysics in the Weld Pool electrode arc rgh solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Multiphysics in the Weld Pool electrode arc brghDT solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Multiphysics in the Weld Pool electrode arc J B B J×B solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Free surface Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Free surface Gas shear Multiphysics in the Weld Pool electrode arc t solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Free surface Gas shear Arc pressure Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Free surface Gas shear Arc pressure Marangoni Multiphysics in the Weld Pool electrode arc t solidified metal weld pool substrate

Driving forces in the weld pool (12) Inertial forces Viscous forces Hydrostatic Buoyancy Conduction Convection Electromagnetic Free surface Gas shear Arc pressure Marangoni Capillary Multiphysics in the Weld Pool electrode arc solidified metal weld pool substrate

Multicoupling in the Weld Pool • Inertial forces • Viscous forces Capillary Hydrostatic Buoyancy Marangoni • Conduction • Convection Arc pressure Gas shear Electromagnetic Free surface

Experiments cannot show under the surface Numerical simulations have convergence problems with a very deformed free surface Disagreement about dominant mechanism • Proposed explanations for very deformed weld pool • Ishizaki (1980): gas shear, experimental • Oreper (1983): Marangoni, numerical • Lin (1985): vortex, analytical • Choo (1991): Arc pressure, gas shear, numerical • Rokhlin (1993): electromagnetic, hydrodynamic, experimental • Weiss (1996): arc pressure, numerical

State of the Art in Understanding of Welding (and Materials) Processes • Questions that can be “easily” answered • For a given current, gas, and geometry, what is the maximum velocity of the molten metal? • For a given set of parameters, what are the temperatures, displacements, velocities, etc? • Questions more difficult to answer: • What mechanism is dominant in determining metal velocity? • If I am designing a weld, what current should I use to achieve a given penetration? • Can I alter one parameter and compensate with other parameters to keep the same result?

Scaling can help answer the “difficult” questions • Dimensional Analysis • Buckingham’s “Pi” theorem • “Informed” Dimensional Analysis • dimensionless groups based on knowledge about system • Inspectional Analysis • dimensionless groups from normalized equations • Ordering • Scaling laws from dominant terms in governing equations (e.g. Bejan, M M Chen, Dantzig and Tucker, Kline, Denn, Deen, Sides, Astarita, and more)

Typical ordering procedure • Write governing equations • Normalize the variables using their characteristic values. • Some characteristic values might be unknown. • This step results in differential expressions based on the normalized variables. • Replace normalized expressions into governing equations. • Normalize equations using the dominant coefficient • Solve for the unknown characteristic values • choose terms where they are present • make their coefficients equal to 1. • Verify that the terms not chosen are not larger than one. • If any term is larger than one, normalize equations again assuming different dominant terms.

Typical ordering procedure • Limitations • Approximation of differential expressions can be grossly inaccurate not true in important practical cases! • Higher order derivatives • Functions with high curvature

Typical ordering procedure • Limitations • Cannot perform manually balances for coupled problems with many equations • when making coefficients equal to 1, there maybe more than one unknown • impractical to check manually for all balances (there is no guaranteed unicity in ordering)

Order of Magnitude Scaling (OMS) • Addresses the drawbacks • Table of improved characteristic values • Linear algebra treatment • Mendez, P.F. Advanced Scaling Techniques for the Modeling of Materials Processing. Keynote paper in Sohn Symposium. August 27-31, 2006. San Diego, CA. p. 393-404.

OMS of a high current weld pool • Goals: • Estimate characteristic values: • velocity, thickness, temperature • Relate results to process parameters • materials properties, welding velocity, weld current • Capture all physics, simplifications in the math • Identify dominant phenomena: • gas shear? Marangoni? electromagnetic? arc pressure? velocity thickness

1. Governing Equations z’ x z w U

Boundary Conditions: 1. Governing Equations at free surface at solid-melt interface far from weld free surface solid-melt interface far from weld

Variables and Parameters independent variables (2) dependent variables (9) parameters (18) 1. Governing Equations with so many parameters Dimensional Analysis is not effective from other models, experiments

2. Normalization of variables unknown characteristic values (9):

3. Replace into governing equations governing equation

3. Replace into governing equations governing equation scaled variables OM(1)

output input input 4. Normalize equations governing equation scaled variables OM(1) normalized equation

output input input 5. Solve for unknowns two possible balances B1

output input input 5. Solve for unknowns two possible balances B1 B2

output input input 5. Solve for unknowns two possible balances balance B1 generates one algebraic equation: B1 B2

output input input 5. Solve for unknowns two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: B1 B2

output input input 6. Check for self-consistency two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 B1 B2

Shortcomings of manual approach two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

Shortcomings of manual approach ? two possible balances 1 equation 2 unknowns balance B1 generates one algebraic equation: ? ? ? 1 equation 3 unknowns balance B2 generates a different equation: ? self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES! • Each balance equation involves more than one unknown

Each balance equation involves more than one unknown A system of equations involves many thousands of possible balances Shortcomings of manual approach two possible balances balance B1 generates one algebraic equation: balance B2 generates a different equation: self-consistency: choose the balance that makes the neglected term less than 1 TWO BIG PROBLEMS FOR MATERIALS PROCESSES!

Shortcomings of manual approach all coefficients are power laws all terms in parenthesis expected to be OM(1)

Shortcomings of manual approach • Simple scaling approach involves 334098 possible combinations • There are 116 self-consistent solutions • there is no unicity of solution • we cannot stop at first self-consistent solution • self-consistent solutions are grouped into 55 classes (1- 6 solutions per class)

Automating iterative process • Power-law coefficients can be transformed into linear expressions using logarithms • Several power law equations can then be transformed into a linear system of equations • Normalizing an equation consists of subtracting rows

Matrix of Coefficients one row for each term of the equation 9 equations 6 BCs

Advanced Scaling Techniques for the Modeling of Materials Processing