Scaling Laws in the Welding Arc

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

2. Evolution in the Modeling of the Welding Arc

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. The Welding Arc

6. The Welding Arc MetTrans 6/01 This talk Flow Temperature

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

8. Modeling of the Arc Column

9. 25000 Hsu et. al. (Numerical) Present study (Numerical) 20000 Temperature (K) 15000 10000 5000 0 2 4 6 8 10 Distance from cathode (mm) 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 column

10. Arc Column Ti column gas Joule heating radiation, conduction, electron drift radiation, conduction Ti Tc Tc unknowns Ti Rg Ri

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

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

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

14. Scaling of the Arc Column

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

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

17. Comparison of OMS and Numerical Results

18. Cases Analyzed

19. Arc Radius within order of magnitude

20. Arc Temperature and Gradient in Gas Ti Rg

21. Improving the Estimations

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

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

25. Corrected Estimation of Arc Radius error<10%

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

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

31. Corrected Estimation of Arc Temperature error10%

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

33. Arc Temperature

34. Gradient in the Gas

35. System Parameters Plasma Gas

36. Unknown Scaling Factors Arc radius Arc temperature Cooling distance in gas Tc Ti Ri Rg