1. 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

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

3. Evolution of Engineering Problems Philosophy Arts Science Engineering

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

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

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

7. 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

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

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

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

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

12. Fundamentals Foundations of OMS

13. Fundamentals Foundations of OMS Differential equations

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

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

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

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

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

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

20. 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

21. 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

22. 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 ) 12 34 56 1 2 3 4 5 1951, 1955 7 1978, 1979 19831986 1990, 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

23. 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

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

25. 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. 

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

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

28. 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

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

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

32. 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)

33. 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)

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

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

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

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

38. 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

39. 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

40. 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

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

42. Corrections in OMS Corrections Dimensional analysis states that: correction function

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

44. 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

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

46. 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

47. 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

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

49. Application of OMS to Arc Modeling VZVZ P

50. Comparison with numerical simulations:

51. Application of OMS to Arc Modeling Correction functions

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. VT T , V 