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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

What is Order of Magnitude Scaling?

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

What is Order of Magnitude Scaling?

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

Weld pool

What is Order of Magnitude Scaling?

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

Weld pool

Arc

What is Order of Magnitude Scaling?

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

Weld pool

Arc

Electrode tip

Outline

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

Context of the Problem

Applications

Engineering

~1900

Engineering

Engineering

~1700

Science

Fundamentals

Science

Science

Arts

Philosophy

Arts

Philosophy

Context of the Problem

Applications

~1980

Engineering

Gap is getting

too large!

~1900

Engineering

Engineering

~1700

Science

Science

Science

Arts

Philosophy

Arts

Philosophy

Fundamentals

Example: Modeling of an Electric Arc

It is very difficult to

obtain general conclusions

with too many parameters

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

Example: Modeling of an Electric Arc

Complexity of the physics increased substantially

Generalization of problems with OMS

Fundamentals

Generalization of problems with OMS

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Generalization of problems with OMS

Engineering

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Generalization of problems with OMS

Engineering

Dimensional analysis

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Generalization of problems with OMS

Engineering

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Generalization of problems with OMS

Engineering

Artificial Intelligence

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Generalization of problems with OMS

Engineering

Artificial Intelligence

Dimensional analysis

Order of Magnitude Reasoning

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

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

OMS: a simple example

- X = unknown
- P1, P2 = parameters (positive and constant)

Dimensional Analysis in OMS

- There are two parameters: P1and P2:
- n=2

Dimensional Analysis in OMS

- There are two parameters: P1and P2:
- n=2
- Units of X, P1, and P2 are the same:
- k=1 (only one independent unit in the problem)

Dimensional Analysis in OMS

- There are two parameters: P1and P2:
- n=2
- Units of X, P1, and P2 are the same:
- k=1 (only one independent unit in the problem)
- Number of dimensionless groups:
- m=n-k
- m=1 (only one dimensionless group)
- P=P2/P1 (arbitrary dimensionless group)

Asymptotic regimes in OMS

- There are two asymptotic regimes:
- Regime I: P2/P1 0
- Regime II: P2/P1

Dominant balance in OMS

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

dominant

secondary

Dominant balance in OMS- There are 6 possible balances
- Combinations of 3 terms taken 2 at a time:
- One possible balance:

dominant

secondary

Dominant balance in OMS- There are 6 possible balances
- Combinations of 3 terms taken 2 at a time:
- One possible balance:

dominant

secondary

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

dominant

secondary

Dominant balance in OMS- There are 6 possible balances
- Combinations of 3 terms taken 2 at a time:
- One possible balance:

XP1 in regime I

P2/P1 0 in regime I

dominant

secondary

Dominant balance in OMS- There are 6 possible balances
- Combinations of 3 terms taken 2 at a time:
- One possible balance:

“natural” dimensionless group

XP1 in regime I

P2/P1 0 in regime I

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 NDG=1
- The magnitude of the NDG is a measure of their importance

Corrections in OMS

Corrections

- Dimensional analysis states:
- Dominant balance states:

correction function

when P2/P10

Corrections in OMS

Corrections

- Dimensional analysis states:
- Dominant balance states:
- Therefore:

correction function

when P2/P10

when P2/P10

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

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

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

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

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

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

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

inertial / viscous

electromagnetic / viscous

convection / conduction

Marangoni / gas shear

arc pressure / viscous

hydrostatic / viscous

buoyancy / viscous

capillary / viscous

diff.=/diff.^

Application of OMS to the Weld Pool at High CurrentRelevance of NDG (Natural Dimensionless Groups)

Application of OMS to the Arc

- Driving forces:
- Electromagnetic forces
- Radial
- Axial
- Balancing forces
- Inertial
- Viscous

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

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

Application of OMS to the Arc

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

Application of OMS to the Arc

- Comparison with numerical simulations:

Application of OMS to the Arc

- Correction functions

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

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