Modeling of Welding Processes through Order of Magnitude Scaling

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Modeling of Welding Processes through Order of Magnitude Scaling

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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

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

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

Weld pool

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

Weld pool

Arc

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

Weld pool

Arc

Electrode tip

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

Engineering

Science

Arts

Philosophy

Engineering

Engineering

~1700

Science

Science

Arts

Philosophy

Arts

Philosophy

Applications

Engineering

~1900

Engineering

Engineering

~1700

Science

Fundamentals

Science

Science

Arts

Philosophy

Arts

Philosophy

Applications

~1980

Engineering

Gap is getting

too large!

~1900

Engineering

Engineering

~1700

Science

Science

Science

Arts

Philosophy

Arts

Philosophy

Fundamentals

It is very difficult to

obtain general conclusions

with too many parameters

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

Complexity of the physics increased substantially

Fundamentals

Differential equations

Fundamentals

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Dimensional analysis

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Artificial Intelligence

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Artificial Intelligence

Dimensional analysis

Order of Magnitude Reasoning

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

Engineering

Artificial Intelligence

Dimensional analysis

Order of Magnitude Reasoning

Matrix algebra

Order of

Magnitude

Scaling

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals

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

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

- 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)

- 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)

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

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

balancing

dominant

secondary

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

- One possible balance:

balancing

dominant

secondary

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

- One possible balance:

balancing

dominant

secondary

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

- One possible balance:

P2/P1 0 in regime I

balancing

dominant

secondary

- 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

balancing

dominant

secondary

- 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

- 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

- For the balance of the example:
- In regime I:

estimation

Corrections

- Dimensional analysis states:

correction function

Corrections

- Dimensional analysis states:
- Dominant balance states:

correction function

when P2/P10

Corrections

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

correction function

when P2/P10

when P2/P10

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

- 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

- Driving forces:
- Gas shear
- Arc Pressure
- Electromagnetic forces
- Hydrostatic pressure
- Capillary forces
- Marangoni forces
- Buoyancy forces

- Balancing forces
- Inertial
- Viscous

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

- 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

- 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

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

T*

d*

U*

gas shear / viscous

inertial / viscous

electromagnetic / viscous

convection / conduction

Marangoni / gas shear

arc pressure / viscous

hydrostatic / viscous

buoyancy / viscous

capillary / viscous

diff.=/diff.^

Relevance of NDG (Natural Dimensionless Groups)

- Driving forces:
- Electromagnetic forces
- Radial
- Axial

- Electromagnetic forces
- Balancing forces
- Inertial
- Viscous

- 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

- 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

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

P

VZ

- Comparison with numerical simulations:

- Correction functions

VR(R,Z)/VRS

200 A

10 mm

2160 A

70 mm

- 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