1. Scaling Laws in the Welding Arc P.F. Mendez, M.A. Ramírez G. Trapaga, and T.W. Eagar MIT, Cambridge, MA, USA October 1 st, 2001, Graz, Austria.

2. 2 Evolution in the Modeling of the Welding Arc

3. 3 Outline Description of the Welding Arc Modeling of the Arc Column Scaling of Arc Column Comparison with Numerical Modeling Improving the Estimations Discussion

4. Description of the Welding Arc

5. 5 The Welding Arc

6. 6 FlowTemperature This talk MetTrans 6/01

7. 7 continuity Navier-Stokes Maxwell Governing Equations energy Unknown functions:

8. Modeling of the Arc Column

9. 9 Assumptions Axisymmetric, steady state, optically thin, LTE, etc. Convection unimportant in column –Prandtl of plasma <1 –Elenbaas-Heller equation –Temperature distribution ~uniform in column length Temperature (K) Distance from cathode (mm) 0246810 5000 10000 15000 20000 25000 Hsu et. al. (Numerical) Present study (Numerical) column

10. 10 RgRg TcTc RiRi TiTi TcTc TiTi radiation, conduction, electron drift Joule heating radiation, conduction TiTi Arc Column unknowns column gas

11. 11 Simplified Governing Equations Energy in plasma Maxwell Energy in gas “Interface” plasma-gas coefficient OM(1) parameters unknown scaling factor Normalization

12. 12 Plasma Properties “ionization” temperature Tampkin and Evans,1967 Ar

13. 13 Plasma Properties Boulos, Fauchais, Pfender, 1994 Ar Boulos, Fauchais, Pfender, 1994

14. Scaling of the Arc Column

15. 15 Order of Magnitude Scaling (OMS) Matrix of Coefficients Balance 2 terms for equation Check-self consistency terms parameters unknowns interface gas plasma exponents

16. 16 Estimations from OMS Matrix of Estimations In this case: 10 iterations E.g.: parameters unknowns exponents

17. Comparison of OMS and Numerical Results

18. 18 Cases Analyzed

19. 19 Arc Radius within order of magnitude

20. 20 Arc Temperature and Gradient in Gas TiTi RgRg

21. Improving the Estimations

22. 22 How can we improve the accuracy of the estimations? Traditionally: constant “fudge” factor OMS: relates difference to –Natural dimensionless groups (endogenous factors) obtained systematically –Other dimensionless groups (exogenous factors) obtained by analysis of problem

23. 23 Natural Dimensionless Groups Indicate “how asymptotic” the model is Very small in welding arc We will not use them

24. 24 Other Dimensionless Groups: Ri/h 11 Account for factors not considered in the governing equations In this case: aspect ratio <<1 Correction functions

25. 25 Corrected Estimation of Arc Radius error<10%

26. 26 Corrected Estimation of Arc Temperature and Gradient in Gas error  50%?! error  10% TiTi RgRg

27. 27 Discussion Arc radius: predictions are very good Arc temperature: predictions could be improved: –effect of convection (modeled as endo. or exo.) Gradient in the gas: not important to know –sensitive to the definition of “ionization temperature”

28. 28 Conclusions Important parameters of the arc can be predicted accurately with closed-form expressions: –temperature, radius, velocity, length of cathode spot –for any gas and current in regime Energy in column: –axial Joule heating=radiation losses Energy in gas: –conduction=radiation losses

29. 29 Conclusions Most important: Method to provide closed-form solutions to the welding arc non-linear equations variable properties

30. 30

31. 31 Corrected Estimation of Arc Temperature error  10%

32. 32 Corrected Estimation of Gradient in the Gas error  50%?!

33. 33 Arc Temperature

34. 34 Gradient in the Gas

35. 35 Parameters Plasma System Gas

36. 36 Unknown Scaling Factors Cooling distance in gas Arc radius Arc temperature RgRg TcTc TiTi RiRi