electric welding arc modeling with the 3d solver …hani/pdf_files/slides-choquet-iiw...electric...
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Electric welding arc modelingwith the 3D solver OpenFOAM
- A comparison of different electromagnetic models -
Isabelle Choquet,¹ Alireza J. Shirvan,¹ and Håkan Nilsson²
¹University West, Dep. of Engineering Science, Trollhättan, Sweden
²Chalmers University of Technology, Dep. of Applied Mechanics, Gothenburg, Sweden
IIW Doc.212-1189 -11
IIW Annual Assembly 2011 SG212: the physics of welding
• Context / Motivation:
better understand the heat source
• Software OpenFOAM-1.6.x
- open source CFD software
- C++ library of object-oriented classes
for implementing solvers for continuum mechanics
IIW Doc.212-1189 -11
IIW intermediate meeting, Trollhättan, Doc. XII-2017-11
” Numerical simulation of Ar-x%CO₂ shielding gas and its effect on an electric welding arc”
Influence of BC on anode an cathode Influence of gas composition
Here focus on a comparison of different electromagnetic models IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions (plasma core):
• one-fluid model
• local thermal equilibrium
• mechanically incompressible and thermally expansible
• steady flow
• laminar flow(assuming laminar shielding gas inlet)
IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions (plasma core):
• one-fluid model
• local thermal equilibrium
• mechanically incompressible and thermally expansible
• steady flow
• laminar flow(assuming laminar shielding gas inlet)
IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions (plasma core):
• one-fluid model
• local thermal equilibrium
• mechanically incompressible and thermally expansible
• steady flow
• laminar flow(assuming laminar shielding gas inlet)
IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions (plasma core):
• one-fluid model
• local thermal equilibrium
• mechanically incompressible, and thermally expansible
• steady flow
• laminar flow(assuming laminar shielding gas inlet)
)(ρ T
IIW Doc.212-1189 -11
Model: thermal fluid part
Argon plasma density as function of temperature.
IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions:
• one-fluid model
• local thermal equilibrium
• mechanically incompressible, and thermally expansible
• plasma optically thin
• steady
• laminar flow(assuming laminar shielding gas inlet)
)(T
IIW Doc.212-1189 -11
Model: thermal fluid part
Main assumptions:
• one-fluid model
• local thermal equilibrium
• mechanically incompressible, and thermally expansible
• plasma optically thin
• steady laminar flow
• laminar flow(assuming laminar shielding gas inlet)
)(T
IIW Doc.212-1189 -11
Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
V JE
, θB
Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
V JE
, θB
θB JE
,
Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
V JE
, θB
θB JE
,
M.A. Ramírez , G. Trapaga and J. McKelliget (2003). A comparison between two different numerical formulations of welding arc simulation. Modelling Simul. Mater. Sci. Eng, 11, pp. 675-695.
Sketch of the cross section of a TIG torch
Test case: Tungsten Inert Gas welding
Picture of a TIG torch
Shielding gas inlet
Ar shielding gas inlet smu /36.2
Applied current: I=200A
IIW Doc.212-1189 -11
Magnetic field magnitude calculated with the electric potential formulation (left) and the 3D approach (right).
dlllJr
B
r
axial
0
0θ )(
μθθ )( AB
Magnetic field magnitude calculated with the electric potential formulation (left) and the 3D approach (right).
θθ )( AB
Why ?
dlllJr
B
r
axial
0
0θ )(
μ
Maxwell’s equations
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
JBEt
000 μμε
totqεE 1
0
IIW Doc.212-1189 -11
Assumptions (plasma core)
local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
0
0
0 0
totqεE 1
0
JμBEμεt
000
Maxwell’s equations
IIW Doc.212-1189 -11
local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
cctL , JEμ
t
0
Assumptions (plasma core)
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
0
0
0
0
0 0
JμBEμεt
000
totqεE 1
0
Maxwell’s equations
IIW Doc.212-1189 -11
local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
1/ collLarLar EJBJn
J drift
e
Hall
σ
σ
cctL , JEμ
t
0
Assumptions (plasma core)
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
0
0
0
0
0 0 0
JμBEμεt
000
totqE 1
0ε
Maxwell’s equations
IIW Doc.212-1189 -11
local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
1Re m
1/ collLarLar
driftind JBuJ
σ
cctL , JEμ
t
0
EJBJn
J drift
e
Hall
σ
σ
Assumptions (plasma core)
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
0
0
0
0
00 0 0
JμBEμεt
000
totqE 1
0ε
Maxwell´s equations
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
EBt
0 Jqtott
therdiffHallinddrift JJJJJJ
0
0
0
0
00 0 0
JμBEμεt
000
totqE 1
0ε
Maxwell´s equations
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
VE
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
VE
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
VE
AB
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
VE
AB
VA σμ0
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Gauss’ law for magnetism:
Ampère’s law:
Faraday’s law:
Gauss’ law:
+ charge conservation
+ generalized Ohm’ s law
•
0 B
0
E
0 J
EJJ drift
σ
JB
0μ
0 E
VE
AB
(1) with Lorentz gauge :
(1)
VσA 0
μ
0 A
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
with
•
•
•
VEJ σσ
AB
VA σμ0
(1)
(1) with Lorentz gauge : 0 A
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
with
•
•
•
VEJ σσ
AB
VA σμ0
(1)
(1) with Lorentz gauge : 0 A
0σ V
Electromagnetic model for arc plasma core
IIW Doc.212-1189 -11
Argon plasma electric conductivity as function of temperature
with
•
•
•
VEJ σσ
AB
VA σμ0
(1) with Lorentz gauge : 0 A
0σ V
IIW Doc.212-1189 -11
2D axi-symmetric case
•
•
•
2D axi-symmetric case
IIW Doc.212-1189 -11
(1)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
0
θ
IIW Doc.212-1189 -11
(1)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
IIW Doc.212-1189 -11
(1)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
) , 0 , ( zr AAA
),(),,( zrAzrA zr
IIW Doc.212-1189 -11
(1)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
) , 0 ,( zr JJJ
) , 0 , ( zr AAA
),(),,( zrAzrA zr
),(),,( zrJzrJ zr
IIW Doc.212-1189 -11
(1)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
) , 0 ,( zr JJJ
) , 0 , ( zr AAA
),(),,( zrAzrA zr
),( zrBθ) 0 , 0, ( θBB
),(),,( zrJzrJ zr
IIW Doc.212-1189 -11
(1)
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1)
(1) with Lorentz gauge :
0rB
r
Vσμ
r
AA
r
Ar
rr
rrr
022
2
z
1
z
VA
r
Ar
rr
zz
σμ02
2
z
1
z
VEJ zz
σσ0θJ
r
A
z
AB zr
θ
0zB
:J
:B
0 A
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1)
(1) with Lorentz gauge :
0rB
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
z
VEJ zz
σσ0θJ
r
A
z
AB zr
θ
0zB
:J
:B
0 A
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1)
(1) with Lorentz gauge :
0rB
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
z
VEJ zz
σσ0θJ
r
A
z
AB zr
θ
0zB
:J
:B
0 A
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case(1)
(1) with Lorentz gauge :
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0 A
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case(1)
(1) with Lorentz gauge :
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0 A
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
•
•
•
IIW Doc.212-1189 -11
2D axi-symmetric case(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
VV rzzr
2
,
2
,
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
Induction diffusion equation
(2)
IIW Doc.212-1189 -11
2D axi-symmetric case - equivalent formulations
(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
IIW Doc.212-1189 -11
2D axi-symmetric case - equivalent formulations
(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
F1
F2
F3
IIW Doc.212-1189 -11
2D axi-symmetric case - equivalent formulations
(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
F1
F2
F3
IIW Doc.212-1189 -11
2D axi-symmetric case - equivalent formulations
(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
F1
F3
F2
IIW Doc.212-1189 -11
2D axi-symmetric case - equivalent formulations
(1)
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
F1
F3
F2
IIW Doc.212-1189 -11
Magnetic field formulation
(1) with Lorentz gauge :
(2) as
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
0
11 θθ
z
B
σzr
rB
σrr
0 A
(2)
VV rzzr
2
,
2
,
zl
zlz zrBdllrJzrB
0
),(),(),( 0θθ
(1) rJμ
r
Vσμ
z
B00
θ
zJμz
Vσμ
r
rB
r00
θ )(1
(2)
Induction diffusion equation
IIW Doc.212-1189 -11
2D axi-symmetric case
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1)
(1) with Lorentz gauge :
r
Vσμ
r
A
z
A
r
Ar
rr
rrr
022
21
z
V
z
A
r
Ar
rr
zz
σμ02
21
z
VEJ zz
σσ
r
A
z
AB zr
θ
:J
:B
0 A
F1
and
IIW Doc.212-1189 -11
2D axi-symmetric case
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1)
(1) with Lorentz gauge :
:J
0 A
0
11 θθ
z
B
σzr
rB
σrr
Induction diffusion equation
F2
andz
VEJ zz
σσ
IIW Doc.212-1189 -11
Electric potential formulation
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1) with Lorentz gauge :
:J
0 A
0
11 θθ
z
B
σzr
rB
σrr
Induction diffusion equation
F2
0
andz
VEJ zz
σσ
IIW Doc.212-1189 -11
Electric potential formulation
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1) with Lorentz gauge :
:J
0 A
rl
rlz rBdllJl
rrB
0
)()(μ
)( 0θ0
θ
z
VEJ zz
σσand
IIW Doc.212-1189 -11
Electric potential formulation
01
z
V
zr
Vr
rrσσ
with
r
VEJ rr
σσ
(1) with Lorentz gauge :
:J
0 A
z
VEJ zz
σσ<<
rl
rlz rBdllJl
rrB
0
)()(μ
)( 0θ0
θ
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVAIIW Doc.212-1189 -11
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVA
]/[, 2mAJJaxial
][mr
axialJ
JJaxial
J
Current density:
IIW Doc.212-1189 -11
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVA
Analytic solution:
IIW Doc.212-1189 -11
Azimuthal component of the magnetic fieldalong the radial direction (r₀= 10ˉ³m)
IIW Doc.212-1189 -11
Sketch of the cross section of a TIG torch
Test case: Tungsten Inert Gas welding
Picture of a TIG torch
Shielding gas inlet
Ar shielding gas inlet smu /36.2
Applied current: I=200A
IIW Doc.212-1189 -11
Magnetic field magnitude calculated with the electric potential formulation (left) and the axi-symmetric approach (right).
dlllJr
rB
r
axialθ
0
0 )(μ
)( θθ )( AB
Magnetic field magnitude calculated with the axi-symmetric approach.
θθ )( AB
JdensityCurrent
rz JJ
JdensityCurrent
JdensityCurrent
zr JJ
rz JJ
Magnetic field magnitude calculated with the electric potential formulation (left) and the axi-symmetric approach (right).
dlllJr
rB
r
axialθ
0
0 )(μ
)( θθ )( AB
electromagnetic model / conclusion
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
V JE
, θB
θB JE
,
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
Conclusions
• 3D model with
• 2D axi-symmetric models:
Electric potential if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
Conclusions
zr AAV ,,θBJE ,,
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
Conclusions
zr AAV ,,θBJE ,,
θBV , JE
,
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
JE
, θB
Conclusions
zr AAV ,,θBJE ,,
θBV , JE
,
V
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
JE
, θB
Conclusions
zr AAV ,,θBJE ,,
θBV , JE
,
V zr JJ
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
JE
, θB
Conclusions
zr AAV ,,θBJE ,,
θBV , JE
,
V zr JJ
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation if
Magnetic field formulation has an ”additional degree of freedom”
IIW Doc.212-1189 -11
AV
, BJE
,,
JE
, θB
Conclusions
zr AAV ,,θBJE ,,
θBV , JE
,
V zr JJ
• Acknowledgements
To Profs. Jacques Aubreton and Marie-Françoise Elchinger
for the data tables of thermodynamic and transport properties.
To KK-foundation,
ESAB, and
the Sustainable Production Initiative at Chalmers
for their support.
Thank you for your attention !
IIW Doc.212-1189 -11
Measured temperature profile for a current intensity 200 A and 2 mm long arc.
Figure from: G.N. Haddad and A.J.D. Farmer (1985) .Temperature measurements in gas tungsten arcs, Welding J, 64, pp. 339-342.
Boundary conditions:M.C. Tsai, and Sindo Kou (1990). Heat transfer and fluid flow in welding arcs produced by sharpened and flat electrodes, Int. J. Heat Mass Transfer, 33, pp. 2089-2098.