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
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• 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
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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
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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
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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
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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
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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
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Model: thermal fluid part
Argon plasma density as function of temperature.
IIW Doc.212-1189 -11
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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
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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
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Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
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Model: electromagnetic part
• 3D model with
• 2D axi-symmetric models:
Electric potential formulation
Magnetic field formulation
IIW Doc.212-1189 -11
AV
, BJE
,,
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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
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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
,
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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.
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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
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Magnetic field magnitude calculated with the electric potential formulation (left) and the 3D approach (right).
dlllJr
B
r
axial
0
0θ )(
μθθ )( AB
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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
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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
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Assumptions (plasma core)
local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
IIW Doc.212-1189 -11
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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
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local electro-neutrality
quasi-steady electromagnetic phenomena
cDebye Lm 810
cctL , JEμ
t
0
Assumptions (plasma core)
IIW Doc.212-1189 -11
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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
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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
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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
![Page 26: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/26.jpg)
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
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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
![Page 28: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/28.jpg)
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
![Page 29: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/29.jpg)
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
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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
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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
![Page 32: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/32.jpg)
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
![Page 33: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/33.jpg)
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
![Page 34: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/34.jpg)
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
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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
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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
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with
•
•
•
VEJ σσ
AB
VA σμ0
(1) with Lorentz gauge : 0 A
0σ V
IIW Doc.212-1189 -11
2D axi-symmetric case
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•
•
•
2D axi-symmetric case
IIW Doc.212-1189 -11
(1)
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•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
0
θ
IIW Doc.212-1189 -11
(1)
![Page 40: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/40.jpg)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
IIW Doc.212-1189 -11
(1)
![Page 41: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/41.jpg)
•
•
•
2D axi-symmetric case
Then
•
with
• with
with
•
),( zrV
0
θ
) , 0 , ( zr AAA
),(),,( zrAzrA zr
IIW Doc.212-1189 -11
(1)
![Page 42: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/42.jpg)
•
•
•
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)
![Page 43: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/43.jpg)
•
•
•
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)
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•
•
•
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
![Page 45: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/45.jpg)
•
•
•
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
![Page 46: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/46.jpg)
•
•
•
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
![Page 47: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/47.jpg)
•
•
•
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
![Page 48: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/48.jpg)
•
•
•
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
![Page 49: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/49.jpg)
•
•
•
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)
![Page 50: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/50.jpg)
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
![Page 51: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/51.jpg)
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
![Page 52: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/52.jpg)
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
![Page 53: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/53.jpg)
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
![Page 54: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/54.jpg)
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
![Page 55: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/55.jpg)
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
![Page 56: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/56.jpg)
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
![Page 57: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/57.jpg)
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
![Page 58: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/58.jpg)
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
![Page 59: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/59.jpg)
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
![Page 60: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/60.jpg)
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
![Page 61: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/61.jpg)
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVAIIW Doc.212-1189 -11
![Page 62: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/62.jpg)
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVA
]/[, 2mAJJaxial
][mr
axialJ
JJaxial
J
Current density:
IIW Doc.212-1189 -11
![Page 63: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/63.jpg)
Test case: infinite conducting rod
)./(2700 mVA
)./(10 5 mVA
Analytic solution:
IIW Doc.212-1189 -11
![Page 64: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/64.jpg)
Azimuthal component of the magnetic fieldalong the radial direction (r₀= 10ˉ³m)
IIW Doc.212-1189 -11
![Page 65: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/65.jpg)
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
![Page 66: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/66.jpg)
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
![Page 67: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/67.jpg)
Magnetic field magnitude calculated with the axi-symmetric approach.
θθ )( AB
![Page 68: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/68.jpg)
JdensityCurrent
![Page 69: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/69.jpg)
rz JJ
JdensityCurrent
![Page 70: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/70.jpg)
JdensityCurrent
zr JJ
rz JJ
![Page 71: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/71.jpg)
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
![Page 72: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/72.jpg)
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
,
![Page 73: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/73.jpg)
• 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
![Page 74: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/74.jpg)
• 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 ,,
![Page 75: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/75.jpg)
• 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
,
![Page 76: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/76.jpg)
• 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
![Page 77: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/77.jpg)
• 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
![Page 78: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/78.jpg)
• 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
![Page 79: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/79.jpg)
• 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
![Page 80: Electric welding arc modeling with the 3D solver …hani/pdf_files/Slides-Choquet-IIW...Electric welding arc modeling with the 3D solver OpenFOAM-A comparison of different electromagnetic](https://reader035.vdocuments.site/reader035/viewer/2022070704/5e88c0e5a661df18676974bf/html5/thumbnails/80.jpg)
• 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
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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.