Modeling of Materials Processes using Dimensional Analysis and Asymptotic Considerations Patricio Mendez, Tom Eagar Welding and Joining Group Massachusetts Institute of Technology Thermec’2000, Las Vegas, NV, December 4-8, 2000

Outline Evolution of engineering problems Our approach to the latest challenges Foundations Scope Output Conclusions

Evolution of Engineering Problems Philosophy Arts Science Engineering

Evolution of Engineering Problems Science Engineering Philosophy Arts Science Engineering Philosophy Arts ~1700

Engineering Evolution of Engineering Problems Science Engineering Philosophy Arts Science Engineering Philosophy Arts Science ~1700 ~1900

Engineering Evolution of Engineering Problems Science Engineering Philosophy Arts Science Engineering Philosophy Arts Fundamentals Applications ~1700 ~1900 ~1980 Science

Very complex process: Fluid flow (Navier-Stokes) Heat transfer Electromagnetism (Maxwell) Evolution of the Modeling of an Electric Arc It is very difficult to obtain general conclusions with too many parameters

Our Approach: Order of Magnitude Scaling OMS is a method useful for analyzing systems with simple geometry and complex physics

Our Approach: Order of Magnitude Scaling OMS is a method useful for analyzing systems with simple geometry and complex physics Weld pool

Our Approach: Order of Magnitude Scaling OMS is a method useful for analyzing systems with simple geometry and complex physics Weld poolArc

Our Approach: Order of Magnitude Scaling OMS is a method useful for analyzing systems with simple geometry and complex physics Weld poolElectrode tipArc

Fundamentals Foundations of OMS

Fundamentals Foundations of OMS Differential equations

Fundamentals Foundations of OMS Differential equations Asymptotic analysis (dominant balance)

Fundamentals Foundations of OMS Differential equations Asymptotic analysis (dominant balance) Applications

Fundamentals Foundations of OMS Dimensional analysis Differential equations Asymptotic analysis (dominant balance) Applications

Matrix algebra Fundamentals Foundations of OMS Dimensional analysis Differential equations Asymptotic analysis (dominant balance) Applications

Matrix algebra Fundamentals Foundations of OMS Dimensional analysis Differential equations Asymptotic analysis (dominant balance) ApplicationsArtificial Intelligence

Matrix algebra Fundamentals Foundations of OMS Dimensional analysis Differential equations Asymptotic analysis (dominant balance) Applications Order of Magnitude Reasoning Artificial Intelligence

Matrix algebra Fundamentals Foundations of OMS Dimensional analysis Differential equations Asymptotic analysis (dominant balance) Applications Order of Magnitude Reasoning Artificial Intelligence Order of Magnitude Scaling

Differences with Related Areas Difference with Dimensional Analysis DA: arbitrary dimensionless groups OMS: natural dimensionless groups Difference with Asymptotic Analysis: No need to solve the equations Discrete variations (not necessarily very small) Difference with Artificial Intelligence Use of dimensional analysis Emphasis on physics over algorithms

Scope of Order of Magnitude Scaling 1951, 1969 number of dimensionless groups associated with geometry ( m g ) number of dimensionless groups associated with the physics (m p ) , , , 1996, 1997 availability of digital computers 1997 analytical models numerical models induction/deduction, numerical/analytical deduction through order of magnitude scaling induction through numerical analysis generalization very difficult simple physics, simple geometries complex physics, simple geometries complex physics, complex geometries simple physics, complex geometries evolution of arc modeling

Output of OMS 1.Estimations of unknown characteristic values 2.Ranking of importance of different driving forces 3.Determination of asymptotic regimes 4.Scaling of experimental or numerical data

1. Estimations of Unknown Characteristic Values (Weld Pool at High Current) U*U* ** **

2. Ranking of Importance of Different Driving Forces (Weld Pool at High Current) arc pressure / viscous electromagnetic / viscous hydrostatic / viscous capillary / viscous Marangoni / gas shearbuoyancy / viscous gas shear / viscous convection / conduction inertial / viscous diff. = /diff. 

3. Determination of Asymptotic Regimes (non-isothermal boundary layer) regime II regime I ln(Re) ln(Pr) VT T , V  T V

4. Scaling of Experimental or Numerical Data (Arc Modeling) V R (R,Z)/V RS 200 A 10 mm 2160 A 70 mm

Conclusion OMS is useful for: Problems with simple geometries and many driving forces The estimation of unknown characteristic values The ranking of importance of different driving forces The determination of asymptotic regimes The scaling of experimental or numerical data

OMS: basic concepts X = unknown P 1, P 2 = parameters (positive and constant)

Dimensional Analysis in OMS There are two parameters: P 1 and P 2 : n=2

Dimensional Analysis in OMS There are two parameters: P 1 and P 2 : n=2 Units of X, P 1, and P 2 are the same: k=1 (only one independent unit in the problem)

Dimensional Analysis in OMS There are two parameters: P 1 and P 2 : n=2 Units of X, P 1, and P 2 are the same: k=1 (only one independent unit in the problem) Number of dimensionless groups: m=n-k=1 (only one dimensionless group)

Asymptotic regimes in OMS There are two asymptotic regimes: Regime I: P 2 /P 1  Regime II: P 2 /P 1 

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time:

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time: One possible balance: balancingdominantsecondary

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time: One possible balance: balancingdominantsecondary

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time: One possible balance: balancingdominantsecondary P 2 /P 1  0 in regime I

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time: One possible balance: balancingdominantsecondary P 2 /P 1  0 in regime IX  P 1 in regime I

Dominant balance in OMS There are 6 possible balances Combinations of 3 terms taken 2 at a time: One possible balance: balancingdominantsecondary P 2 /P 1  0 in regime IX  P 1 in regime I “natural” dimensionless group

Estimations in OMS For the balance of the example: In regime I: estimation

Corrections in OMS Corrections Dimensional analysis states that: correction function

Corrections in OMS Corrections Dimensional analysis states that: Dominant balance states that: when P 2 /P 1  0 correction function

Corrections in OMS Corrections Dimensional analysis states that: Dominant balance states that: Therefore: when P 2 /P 1  0 correction function when P 2 /P 1  0

Application of OMS to Arc Modeling Driving forces: Electromagnetic forces Radial Axial Balancing forces Inertial Viscous

Application of OMS to Arc Modeling Isothermal, axisymmetric model Governing equations (6): conservation of mass Navier-Stokes(2) Ampere (2) conservation of magnetic field Unknowns (6) Flow velocities (2) Pressure Current density (2) Magnetic induction

Application of OMS to Arc Modeling Parameters (7): , ,  0, R C, J C, h, R a Reference Units (4): m, kg, s, A Dimensionless Groups (3) Reynolds dimensionless arc length dimensionless anode radius

Application of OMS to Arc Modeling Estimations (5): Length of cathode region Flow velocities (2) Pressure Radial current density

Application of OMS to Arc Modeling VZVZ P

Comparison with numerical simulations:

Application of OMS to Arc Modeling Correction functions

Application of OMS to the Weld Pool at High Current Driving forces: Gas shear Arc Pressure Electromagnetic forces Hydrostatic pressure Capillary forces Marangoni forces Buoyancy forces Balancing forces Inertial Viscous

Application of OMS to the Weld Pool at High Current Governing equations, 2-D model (9) : conservation of mass Navier-Stokes(2) conservation of energy Marangoni Ohm (2) Ampere (2) conservation of charge

Application of OMS to the Weld Pool at High Current Governing equations, 2-D model (9) : conservation of mass Navier-Stokes(2) conservation of energy Marangoni Ohm (2) Ampere (2) conservation of charge Unknowns (9): Thickness of weld pool Flow velocities (2) Pressure Temperature Electric potential Current density (2) Magnetic induction

Application of OMS to the Weld Pool at High Current Parameters (17): L, , , k, Q max, J max,  e, g,,  T, , P max,  max, U ,  0, ,  s Reference Units (7): m, kg, s, K, A, J, V Dimensionless Groups (10) Reynolds, Stokes, Elsasser, Grashoff, Peclet, Marangoni, Capillary, Poiseuille, geometric, ratio of diffusivity

Application of OMS to the Weld Pool at High Current Estimations (8): Thickness of weld pool Flow velocities (2) Pressure Temperature Electric potential Current density in X Magnetic induction

Generalization of OMS The concepts above can be applied when: The system has many equations The terms have the form of a product of powers The terms are functions instead of constants In this case the functions need to be normalized

Properties of the natural dimensionless groups (NDG) Each regime has a different set of NDG For each regime there are m NDG All NDG are less than 1 in their regime The edge of the regimes can be defined by NDG=1 The magnitude of the NDG is a measure of their importance

Properties of the correction functions The correction function is  1 near the asymptotic case The correction function depends on the NDG The less important NDG can be discarded with little loss of accuracy The correction function can be estimated empirically by comparison with calculations or experiments

VT T , V 