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
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