Scaling Laws in the Welding Arc

P.F. Mendez, M.A. Ramírez

G. Trapaga, and T.W. EagarMIT, Cambridge, MA, USA

October 1st, 2001, Graz, Austria.

2

Evolution in the Modeling of the Welding Arc

Squ

ire

1951

(an

alyt

ical

)S

herc

liff

196

9 (a

naly

tica

l)

Lowke 1997

num

ber

of d

imen

sion

less

gro

ups

asso

ciat

ed w

ith

geom

etry

(mg)

number of dimensionless groups associated with the physics (mp)

1 2 3 4 5 6

1

2

3

4

5S

quir

e 19

51 (

anal

ytic

al)

Mae

cker

195

5 (a

ppro

xim

ate)

7

Ramakrishnan 1978Glickstein 1979

Hsu

198

3

McK

elli

get 1

986

Cho

o 19

90L

ee 1

996

Kim

199

7

availability ofdigital computers

3

Outline

• Description of the Welding Arc• Modeling of the Arc Column• Scaling of Arc Column• Comparison with Numerical Modeling• Improving the Estimations• Discussion

Description of the Welding Arc

5

The Welding Arc

cathodeboundary layer

cathode

cathode region

column

anode region

anode boundarylayer

anode

e.m. forces,inertial forces,viscous forces

joule heating

electron driftconvection, radiation,

conductionconvection, radiation,

conduction

6

The Welding Arc

Flow Temperature

This talk

MetTrans 6/01

7

continuity

Navier-Stokes

Maxwell

Governing Equations

01

Z

VRV

RRZ

R

BJZ

VRV

RRRR

P

Z

VV

R

VV Z

RR

RZ

RR

2

21

BJZ

V

R

VR

RRZ

P

Z

VV

R

VV R

ZZZZ

ZR

2

21

Z

TJ

R

TJ

e

kS

JJ

Z

T

R

TR

RRk

Z

TV

R

TVC ZR

bR

ZRZRp 2

51 22

2

2

Z

BJ R

0

RBRR

J Z 1

0

01

2

2

RBRRRZ

B

energy

Unknown functions:),( ZRVR ),( ZRVZ

),( ZRP

),( ZRJ R ),( ZRJ Z),( ZRB

),( ZRT

Modeling of the Arc Column

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

Tem

pera

ture

(K

)

Distance from cathode (mm)

0 2 4 6 8 10

5000

10000

15000

20000

25000

Hsu et. al. (Numerical)Present study (Numerical)

column

10

Rg

Tc

Ri

Ti

Tc Ti

radiation, conduction,

electron drift

Joule heating

radiation, conduction

Ti

Arc Column

unknowns

column gas

11

Simplified Governing Equations

Energy in plasma

R

Tk

R

Tk gp

02

51 22

R

TJ

e

kS

JJ

R

TR

RRk R

bpR

ZRp

01

gRg SR

TR

RRk

Z

BJ R

0

RBRR

J Z 1

0

01

2

2

RBRRRZ

B

Maxwell

Energy in gas

“Interface” plasma-gas

coefficient OM(1)

g

gp

gpg

ig

rr

rrR

Tk 12

parameters

unknown scaling factor

Normalization

12

Plasma Properties

“ionization” temperature

Tampkin and Evans,1967

Ar

iRTR TTSS

13

Plasma Properties

Bou

los,

Fau

chai

s, P

fend

er, 1

994

Ar

iT TTkk

Ar

iT TT

Bou

los,

Fau

chai

s, P

fend

er, 1

994

Scaling of the Arc Column

15

Order of Magnitude Scaling (OMS)

• Matrix of Coefficients• Balance 2 terms for equation• Check-self consistency

term

s

parameters unknowns

inte

rfac

e g

as p

lasm

a

exponents

16

Estimations from OMS

• Matrix of Estimations• In this case: 10 iterations• E.g.:

parameters

unkn

owns

exponents

1.01.01.04.0

2.02.02.0 2ˆ

iRGgRTTTi TSkISkR

Comparison of OMS and Numerical Results

18

Cases Analyzed

gas I h[A] [m]

Ar 200 0.01Ar 200 0.02Ar 300 0.0063Ar 300 0.01Ar 300 0.02Air 520 0.07Air 1150 0.07Air 2160 0.07

19

Arc Radius

1.E-04

1.E-03

1.E-02

1.E-01

0 500 1000 1500 2000 2500

welding current I [A]

Ri [

m]

numerical

estimation

within order of magnitude

20

Arc Temperature and Gradient in Gas

1.E+02

1.E+03

1.E+04

1.E+05

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

estimation1.E-04

1.E-03

1.E-02

1.E-01

0 500 1000 1500 2000 2500

welding current I [A]

de

lta

Rg

[m

]

numerical

estimation

Ti Rg

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

1.E-03

1.E-02

1.E-01

1.E+00

0 500 1000 1500 2000 2500

welding current I [A]

conduction termJoule radialelectron drift

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

24

Other Dimensionless Groups: Ri/h

fRi= -0.0207Ri/h + 0.6533

fdeltaRg = -0.0934Ri/h + 1.4127

fTc = 0.7041Ri/h + 0.7449

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ri/h

num

eri

cal/e

stim

atio

n

Ri

Tc

deltaRg1

•Account for factors not considered in the governing equations•In this case: aspect ratio

<<1Correction functions

25

Corrected Estimation of Arc Radius

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 500 1000 1500 2000 2500

welding current I [A]

Ri

[m]

numerical

prediction

erro

r<10

%

26

Corrected Estimation of Arc Temperature and Gradient in Gas

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

prediction

error50%?!

0

2000

4000

6000

8000

10000

12000

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

prediction

erro

r10

%

TiRg

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

30

31

0

2000

4000

6000

8000

10000

12000

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

prediction

Corrected Estimation of Arc Temperature

erro

r10

%

32

Corrected Estimation of Gradient in the Gas

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

prediction

error50%?!

33

Arc Temperature

1.E+02

1.E+03

1.E+04

1.E+05

0 500 1000 1500 2000 2500

welding current I [A]

Tc

[K]

numerical

estimation

34

Gradient in the Gas

1.E-04

1.E-03

1.E-02

1.E-01

0 500 1000 1500 2000 2500

welding current I [A]

de

lta

Rg

[m

]

numerical

estimation

35

Parameters

},,,2

,,2

5,,,{}{ iRgg

bRTTT

T TSkIh

RI

e

kSkP

Plasma

System Gas

36

Unknown Scaling Factors

},,{}{ gCiT RTRS

Cooling distance in gas

Arc radius

Arc temperature

Rg

Tc

Ti

Ri