with the 3D solver OpenFOAM - A comparison of different - - PowerPoint PPT Presentation

Electric welding arc modeling with 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

• Cont

ntext / / Motiv tivation: better understand the heat source

• So

Software Open enFOAM-1.6 .6.x .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

• n 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

A V  , B J E    , ,

Model: electromagnetic part

• 3D model with
• 2D axi-symmetric models:

Electric potential formulation Magnetic field formulation

IIW Doc.212-1189 -11

A V  , B J E    , ,

V

J E   ,

θ

B

Model: electromagnetic part

• 3D model with
• 2D axi-symmetric models:

Electric potential formulation Magnetic field formulation

IIW Doc.212-1189 -11

A V  , B J E    , ,

V

J E   ,

θ

B

θ

B

J E   ,

Model: electromagnetic part

• 3D model with
• 2D axi-symmetric models:

Electric potential formulation Magnetic field formulation

IIW Doc.212-1189 -11

A V  , B J E    , ,

V

J E   ,

θ

B

θ

B

J E   ,

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

s m u / 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).

dl l l J r B

r axial





θ

) ( μ

θ θ

) ( A B    

Magnetic field magnitude calculated with the electric potential formulation (left) and the 3D approach (right).

θ θ

) ( A B    

Why ?

dl l l J r B

r axial





θ

) ( μ

Maxwell’s equations

• Gauss’ law for magnetism:
• Ampère’s law:
• Faraday’s law:
• Gauss’ law:

+ charge conservation + generalized Ohm’ s law

• 

  B 

E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

J B E

t

   μ μ ε     

tot

q ε E

1 

   

IIW Doc.212-1189 -11

Assumptions (plasma core)

• local electro-neutrality
• quasi-steady electromagnetic

phenomena

• c

Debye

L m 

8

10 

IIW Doc.212-1189 -11

• Gauss’ law for magnetism:
• Ampère’s law:
• Faraday’s law:
• Gauss’ law:

+ charge conservation + generalized Ohm’ s law

• 

  B  E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

tot

q ε E

1 

    J μ B E μ ε

t

       

Maxwell’s equations

IIW Doc.212-1189 -11

• local electro-neutrality
• quasi-steady electromagnetic

phenomena

• c

Debye

L m 

8

10 

c c

t L ,

J E μ

t

   

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

• 

  B  E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

J μ B E μ ε

t

       

tot

q ε E

1 

   

Maxwell’s equations

IIW Doc.212-1189 -11

• local electro-neutrality
• quasi-steady electromagnetic

phenomena

• c

Debye

L m 

8

10  1 /  

coll Lar Lar

   E J B J n J

drift e Hall

     σ σ     

c c

t L ,

J E μ

t

   

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

• 

  B  E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

J μ B E μ ε

t

       

tot

q E

1

ε    

Maxwell’s equations

IIW Doc.212-1189 -11

• local electro-neutrality
• quasi-steady electromagnetic

phenomena

• c

Debye

L m 

8

10 

1 Re 

m

1 /  

coll Lar Lar

  

drift ind

J B u J        σ

c c

t L ,

J E μ

t

    E J B J n 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

• 

  B  E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

J μ B E μ ε

t

       

tot

q E

1

ε    

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

• 

  B  E B

t

           J qtot

t



ther diff Hall ind drift

J J J J J J           

J μ B E μ ε

t

       

tot

q E

1

ε    

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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       E  V 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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       E  V E   

 

σ     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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       E  V E    A B     

 

σ     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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       E  V E    A B     

V A       σ μ0 

 

σ     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

• 

  B       E    J 

E J J

drift

   σ  

J B   μ       E  V E    A B     

(1) with Lorentz gauge : (1)

V σ A    μ 

   A 

 

σ     V

Electromagnetic model for arc plasma core

IIW Doc.212-1189 -11

with

• V

E J     σ σ  

A B     

V A    σ μ0 

(1) (1) with Lorentz gauge :

   A 

 

σ     V

Electromagnetic model for arc plasma core

IIW Doc.212-1189 -11

with

• V

E J     σ σ  

A B     

V A    σ μ0 

(1) (1) with Lorentz gauge :

   A 

 

σ     V

Electromagnetic model for arc plasma core

IIW Doc.212-1189 -11

Argon plasma electric conductivity as function of temperature

with

• V

E J     σ σ  

A B     

V A    σ μ0 

(1) with Lorentz gauge :

   A 

 

σ     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

• θ



IIW Doc.212-1189 -11

(1)

• 2D axi-symmetric case

Then

• with
• with

with

• )

, ( z r V

θ



IIW Doc.212-1189 -11

(1)

• 2D axi-symmetric case

Then

• with
• with

with

• )

, ( z r V

θ



) , , (

z r

A A A   ) , ( ), , ( z r A z r A

z r

IIW Doc.212-1189 -11

(1)

• 2D axi-symmetric case

Then

• with
• with

with

• )

, ( z r V

θ



) , , (

z r

J J J   ) , , (

z r

A A A   ) , ( ), , ( z r A z r A

z r

) , ( ), , ( z r J z r J

z r

IIW Doc.212-1189 -11

(1)

• 2D axi-symmetric case

Then

• with
• with

with

• )

, ( z r V

θ



) , , (

z r

J J J   ) , , (

z r

A A A   ) , ( ), , ( z r A z r A

z r

) , ( z r Bθ

) , 0, (

θ

B B   ) , ( ), , ( z r J z r J

z r

IIW Doc.212-1189 -11

(1)

• IIW Doc.212-1189 -11

2D axi-symmetric case

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) (1) with Lorentz gauge :



r

B r V σ μ r A A r A r r r

r r r

                

2 2 2

z 1 z V A r A r r r

z z

                σ μ0

2 2

z 1 z V E J

z z

     σ σ 

θ

J r A z A B

z r θ

     



z

B

: J  : B 

   A 

• IIW Doc.212-1189 -11

2D axi-symmetric case

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) (1) with Lorentz gauge :



r

B r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1 z V E J

z z

     σ σ 

θ

J r A z A B

z r θ

     



z

B

: J  : B 

   A 

• IIW Doc.212-1189 -11

2D axi-symmetric case

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) (1) with Lorentz gauge :



r

B r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1 z V E J

z z

     σ σ 

θ

J r A z A B

z r θ

     



z

B

: J  : B 

   A 

• IIW Doc.212-1189 -11

2D axi-symmetric case

(1) (1) with Lorentz gauge :

r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

   A 

• IIW Doc.212-1189 -11

2D axi-symmetric case

(1) (1) with Lorentz gauge :

r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

   A 

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

θ)

( 1       

• IIW Doc.212-1189 -11

2D axi-symmetric case

(1) (1) with Lorentz gauge : (2) as

r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A  V V

r z z r 2 , 2 ,

  

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

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 V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

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 V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

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 V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

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 V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

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 V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

θ)

( 1       

(2)

Induction diffusion equation F1 F3 F2

IIW Doc.212-1189 -11

Magnetic field formulation

(1) with Lorentz gauge : (2) as

r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1

 

1 1

θ θ

                      z B σ z r rB σr r

   A 

(2)

V V

r z z r 2 , 2 ,

  



 

 

z l z l z

z r B dl l r J z r B ) , ( ) , ( ) , (

θ θ

(1)

r

J μ r V σ μ z B

θ

      

z

J μ z V σ μ r rB r

θ)

( 1       

(2)

Induction diffusion equation

IIW Doc.212-1189 -11

2D axi-symmetric case

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) (1) with Lorentz gauge :

r V σ μ r A z A r A r r r

r r r

                

2 2 2

1 z V z A r A r r r

z z

                σ μ0

2 2

1 z V E J

z z

     σ σ r A z A B

z r θ

      : J  : B 

   A 

F1 and

IIW Doc.212-1189 -11

2D axi-symmetric case

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) (1) with Lorentz gauge :

: J 

   A 

 

1 1

θ θ

                      z B σ z r rB σr r

Induction diffusion equation F2 and

z V E J

z z

     σ σ

IIW Doc.212-1189 -11

Electric potential formulation

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) with Lorentz gauge :

: J 

   A 

 

1 1

θ θ

                      z B σ z r rB σr r

Induction diffusion equation F2 and

z V E J

z z

     σ σ

IIW Doc.212-1189 -11

Electric potential formulation

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) with Lorentz gauge :

: J 

   A 



 

 

r l r l z

r B dl l J l r r B ) ( ) ( μ ) (

θ θ

z V E J

z z

     σ σ

and

IIW Doc.212-1189 -11

Electric potential formulation

1                       z V z r V r r r σ σ

with

r V E J

r r

     σ σ

(1) with Lorentz gauge :

: J 

   A 

z V E J

z z

     σ σ

<<



 

 

r l r l z

r B dl l J l r r B ) ( ) ( μ ) (

θ θ

Test case: infinite conducting rod

) . /( 2700 m V A   ) . /( 10 5 m V A



 

IIW Doc.212-1189 -11

Test case: infinite conducting rod

) . /( 2700 m V A   ) . /( 10 5 m V A



 

] / [ ,

2

m A J Jaxial



] [m r

axial

J



 J Jaxial



J

Current density:

IIW Doc.212-1189 -11

Test case: infinite conducting rod

) . /( 2700 m V A   ) . /( 10 5 m V A



 

Analytic solution:

IIW Doc.212-1189 -11

Azimuthal component of the magnetic field along 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

s m u / 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).

dl l l J r r B

r axial θ



 ) ( μ ) (

θ θ

) ( A B    

Magnetic field magnitude calculated with the axi- symmetric approach.

θ θ

) ( A B    

J density Current 

r z

J J 

J density Current 

J density Current 

z r

J J 

r z

J J 

Magnetic field magnitude calculated with the electric potential formulation (left) and the axi-symmetric approach (right).

dl l l J r r B

r axial θ



 ) ( μ ) (

θ θ

) ( A B    

electromagnetic model / conclusion

• 3D model with
• 2D axi-symmetric models:

Electric potential formulation Magnetic field formulation

IIW Doc.212-1189 -11

A V  , B J E    , ,

V

J E   ,

θ

B

θ

B

J E   ,

• 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

A V  , B J E    , ,

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

A V  , B J E    , ,

Conclusions

z r A

A V , ,

θ

B J E , ,  

• 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

A V  , B J E    , ,

Conclusions

z r A

A V , ,

θ

B J E , ,  

θ

B V ,

J E   ,

• 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

A V  , B J E    , , J E   ,

θ

B

Conclusions

z r A

A V , ,

θ

B J E , ,  

θ

B V ,

J E   ,

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

A V  , B J E    , , J E   ,

θ

B

Conclusions

z r A

A V , ,

θ

B J E , ,  

θ

B V ,

J E   ,

V

z r

J J 

• 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

A V  , B J E    , , J E   ,

θ

B

Conclusions

z r A

A V , ,

θ

B J E , ,  

θ

B V ,

J E   ,

V

z r

J J 

• 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

A V  , B J E    , , J E   ,

θ

B

Conclusions

z r A

A V , ,

θ

B J E , ,  

θ

B V ,

J E   ,

V

z r

J J 

• Acknowledgements

To Pro rofs

• fs. Ja

Jacques Aubreton and Marie-Françoise Elch lchinger for the data tables of thermodynamic and transport properties. To KK KK-foundation, ESA SAB, and the Su Susta tainable Pro roducti tion Ini niti tiative at t 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 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 33, pp. 2089-2098.