scaling laws in the welding arc
DESCRIPTION
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. Evolution in the Modeling of the Welding Arc. Outline. Description of the Welding Arc Modeling of the Arc Column Scaling of Arc Column - PowerPoint PPT PresentationTRANSCRIPT
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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.
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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
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3
Outline
• Description of the Welding Arc• Modeling of the Arc Column• Scaling of Arc Column• Comparison with Numerical Modeling• Improving the Estimations• Discussion
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Description of the Welding Arc
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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
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The Welding Arc
Flow Temperature
This talk
MetTrans 6/01
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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
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Modeling of the Arc Column
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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
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Rg
Tc
Ri
Ti
Tc Ti
radiation, conduction,
electron drift
Joule heating
radiation, conduction
Ti
Arc Column
unknowns
column gas
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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
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Plasma Properties
“ionization” temperature
Tampkin and Evans,1967
Ar
iRTR TTSS
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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
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Scaling of the Arc Column
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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
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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
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Comparison of OMS and Numerical Results
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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
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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
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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
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Improving the Estimations
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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
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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
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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
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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
%
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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
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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”
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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
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Conclusions
• Most important:
Method to provide closed-form solutions to the welding arc
• non-linear equations• variable properties
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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
%
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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%?!
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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
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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
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Parameters
},,,2
,,2
5,,,{}{ iRgg
bRTTT
T TSkIh
RI
e
kSkP
Plasma
System Gas
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Unknown Scaling Factors
},,{}{ gCiT RTRS
Cooling distance in gas
Arc radius
Arc temperature
Rg
Tc
Ti
Ri