Modeling of Welding Processes through Order of Magnitude Scaling

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Modeling of Welding Processes throughOrder of
Magnitude Scaling

• Patricio Mendez, Tom Eagar
• Welding and Joining Group
• Massachusetts Institute of Technology
• MMT-2000, Ariel, Israel, November 13-15, 2000

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What is Order of Magnitude Scaling?

• OMS is a method useful for analyzing systems with
  many driving forces

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What is Order of Magnitude Scaling?

• OMS is a method useful for analyzing systems with
  many driving forces

Weld pool
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What is Order of Magnitude Scaling?

• OMS is a method useful for analyzing systems with
  many driving forces

Weld pool
Arc
5
What is Order of Magnitude Scaling?

• OMS is a method useful for analyzing systems with
  many driving forces

Weld pool
Electrode tip
Arc
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Outline

• Context of the problem
• Simple example of OMS
• Applications to Welding
• Discussion

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Context of the Problem
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Context of the Problem
Engineering
Science
Arts
Philosophy
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Context of the Problem
Engineering
Engineering
1700
Science
Science
Arts
Philosophy
Arts
Philosophy
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Context of the Problem
Applications
Engineering
1900
Engineering
Engineering
1700
Science
Fundamentals
Science
Science
Arts
Philosophy
Arts
Philosophy
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Context of the Problem
Applications
1980
Engineering
Gap is getting too large!
1900
Engineering
Engineering
1700
Science
Science
Science
Arts
Philosophy
Arts
Philosophy
Fundamentals
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Example Modeling of an Electric Arc

• Very complex process
• Fluid flow (Navier-Stokes)
• Heat transfer
• Electromagnetism (Maxwell)

It is very difficult to obtain general
conclusions with too many parameters
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Example Modeling of an Electric Arc
Complexity of the physics increased substantially
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Generalization of problems with OMS
Fundamentals
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Generalization of problems with OMS
Differential equations
Fundamentals
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Generalization of problems with OMS
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Dimensional analysis
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Dimensional analysis
Matrix algebra
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Matrix algebra
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Order of Magnitude Reasoning
Matrix algebra
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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Generalization of problems with OMS
Engineering
Artificial Intelligence
Dimensional analysis
Order of Magnitude Reasoning
Matrix algebra
Order of Magnitude Scaling
Asymptotic analysis (dominant balance)
Differential equations
Fundamentals
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OMS a simple example

• X unknown
• P1, P2 parameters (positive and constant)

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Dimensional Analysis in OMS

• There are two parameters P1 and P2
• n2

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Dimensional Analysis in OMS

• There are two parameters P1 and P2
• n2
• Units of X, P1, and P2 are the same
• k1 (only one independent unit in the problem)

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Dimensional Analysis in OMS

• There are two parameters P1 and P2
• n2
• Units of X, P1, and P2 are the same
• k1 (only one independent unit in the problem)
• Number of dimensionless groups
• mn-k
• m1 (only one dimensionless group)
• PP2/P1 (arbitrary dimensionless group)

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Asymptotic regimes in OMS

• There are two asymptotic regimes
• Regime I P2/P1? 0
• Regime II P2/P1? ?

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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time

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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time
• One possible balance

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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time
• One possible balance

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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time
• One possible balance

P2/P1? 0 in regime I
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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time
• One possible balance

P2/P1? 0 in regime I
X ? P1 in regime I
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Dominant balance in OMS

• There are 6 possible balances
• Combinations of 3 terms taken 2 at a time
• One possible balance

natural dimensionless group
P2/P1? 0 in regime I
X ? P1 in regime I
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Properties of the natural dimensionless groups
(NDG)

• Each regime has a different set of NDG
• For each regime there are m NDG
• All NDG are less than 1 in their regime
• The edge of the regimes can be defined by NDG1
• The magnitude of the NDG is a measure of their
  importance

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Estimations in OMS

• For the balance of the example
• In regime I

estimation
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Corrections in OMS

• Corrections
• Dimensional analysis states

correction function
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Corrections in OMS

• Corrections
• Dimensional analysis states
• Dominant balance states

correction function
when P2/P1?0
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Corrections in OMS

• Corrections
• Dimensional analysis states
• Dominant balance states
• Therefore

correction function
when P2/P1?0
when P2/P1?0
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Properties of the correction functions

• Properties of the correction functions
• The correction function is ? 1 near the
  asymptotic case
• The correction function depends on the NDG
• The less important NDG can be discarded with
  little loss of accuracy
• The correction function can be estimated
  empirically by comparison with calculations or
  experiments

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Generalization of OMS

• The concepts above can be applied when
• The system has many equations
• The terms have the form of a product of powers
• The terms are functions instead of constants
• In this case the functions need to be normalized

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Application of OMS to the Weld Pool at High
Current

• Driving forces
• Gas shear
• Arc Pressure
• Electromagnetic forces
• Hydrostatic pressure
• Capillary forces
• Marangoni forces
• Buoyancy forces
• Balancing forces
• Inertial
• Viscous

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Application of OMS to the Weld Pool at High
Current

• Governing equations, 2-D model (9)
• conservation of mass
• Navier-Stokes(2)
• conservation of energy
• Marangoni
• Ohm (2)
• Ampere (2)
• conservation of charge

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Application of OMS to the Weld Pool at High
Current

• Governing equations, 2-D model (9)
• conservation of mass
• Navier-Stokes(2)
• conservation of energy
• Marangoni
• Ohm (2)
• Ampere (2)
• conservation of charge
• Unknowns (9)
• Thickness of weld pool
• Flow velocities (2)
• Pressure
• Temperature
• Electric potential
• Current density (2)
• Magnetic induction

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Application of OMS to the Weld Pool at High
Current

• Parameters (17)
• L, r, a, k, Qmax, Jmax, se, g, n, sT, s, Pmax,
  tmax, U?, m0, b, ws
• Reference Units (7)
• m, kg, s, K, A, J, V
• Dimensionless Groups (10)
• Reynolds, Stokes, Elsasser, Grashoff, Peclet,
  Marangoni, Capillary, Poiseuille, geometric,
  ratio of diffusivity

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Application of OMS to the Weld Pool at High
Current

• Estimations (8)
• Thickness of weld pool
• Flow velocities (2)
• Pressure
• Temperature
• Electric potential
• Current density in X
• Magnetic induction

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Application of OMS to the Weld Pool at High
Current
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Application of OMS to the Weld Pool at High
Current
Relevance of NDG (Natural Dimensionless Groups)
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Application of OMS to the Arc

• Driving forces
• Electromagnetic forces
• Radial
• Axial
• Balancing forces
• Inertial
• Viscous

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Application of OMS to the Arc

• Isothermal, axisymmetric model
• Governing equations (6)
• conservation of mass
• Navier-Stokes(2)
• Ampere (2)
• conservation of magnetic field
• Unknowns (6)
• Flow velocities (2)
• Pressure
• Current density (2)
• Magnetic induction

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Application of OMS to the Arc

• Parameters (7)
• r, m, m0 , RC , JC , h, Ra
• Reference Units (4)
• m, kg, s, A
• Dimensionless Groups (3)
• Reynolds
• dimensionless arc length
• dimensionless anode radius

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Application of OMS to the Arc

• Estimations (5)
• Length of cathode region
• Flow velocities (2)
• Pressure
• Radial current density

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Application of OMS to the Arc
P
VZ
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Application of OMS to the Arc

• Comparison with numerical simulations

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Application of OMS to the Arc

• Correction functions

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Application of OMS to the Arc
VR(R,Z)/VRS
200 A 10 mm
2160 A 70 mm
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Conclusion

• OMS is useful for
• Problems with simple geometries and many driving
  forces
• The estimation of unknown characteristic values
• The ranking of importance of different driving
  forces
• The determination of asymptotic regimes
• The scaling of experimental or numerical data