Modeling of welding processes through order of magnitude scaling
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Modeling of Welding Processes through Order 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?.

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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

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

What is Order of Magnitude Scaling?

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


What is order of magnitude scaling1

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 scaling2

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 scaling3

What is Order of Magnitude Scaling?

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

Weld pool

Arc

Electrode tip


Outline

Outline

  • Context of the problem

  • Simple example of OMS

  • Applications to Welding

  • Discussion


Context of the problem

Context of the Problem


Context of the problem1

Context of the Problem

Engineering

Science

Arts

Philosophy


Context of the problem2

Context of the Problem

Engineering

Engineering

~1700

Science

Science

Arts

Philosophy

Arts

Philosophy


Context of the problem3

Context of the Problem

Applications

Engineering

~1900

Engineering

Engineering

~1700

Science

Fundamentals

Science

Science

Arts

Philosophy

Arts

Philosophy


Context of the problem4

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

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 arc1

Example: Modeling of an Electric Arc

Complexity of the physics increased substantially


Generalization of problems with oms

Generalization of problems with OMS

Fundamentals


Generalization of problems with oms1

Generalization of problems with OMS

Differential equations

Fundamentals


Generalization of problems with oms2

Generalization of problems with OMS

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals


Generalization of problems with oms3

Generalization of problems with OMS

Engineering

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals


Generalization of problems with oms4

Generalization of problems with OMS

Engineering

Dimensional analysis

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals


Generalization of problems with oms5

Generalization of problems with OMS

Engineering

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals


Generalization of problems with oms6

Generalization of problems with OMS

Engineering

Artificial Intelligence

Dimensional analysis

Matrix algebra

Asymptotic analysis

(dominant balance)

Differential equations

Fundamentals


Generalization of problems with oms7

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 oms8

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

OMS: a simple example

  • X = unknown

  • P1, P2 = parameters (positive and constant)


Dimensional analysis in oms

Dimensional Analysis in OMS

  • There are two parameters: P1and P2:

    • n=2


Dimensional analysis in oms1

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 oms2

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

Asymptotic regimes in OMS

  • There are two asymptotic regimes:

    • Regime I: P2/P1 0

    • Regime II: P2/P1 


Dominant balance in oms

Dominant balance in OMS

  • There are 6 possible balances

    • Combinations of 3 terms taken 2 at a time:


Dominant balance in oms1

balancing

dominant

secondary

Dominant balance in OMS

  • There are 6 possible balances

    • Combinations of 3 terms taken 2 at a time:

  • One possible balance:


Dominant balance in oms2

balancing

dominant

secondary

Dominant balance in OMS

  • There are 6 possible balances

    • Combinations of 3 terms taken 2 at a time:

  • One possible balance:


Dominant balance in oms3

balancing

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 balance in oms4

balancing

dominant

secondary

Dominant balance in OMS

  • There are 6 possible balances

    • Combinations of 3 terms taken 2 at a time:

  • One possible balance:

XP1 in regime I

P2/P1 0 in regime I


Dominant balance in oms5

balancing

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

XP1 in regime I

P2/P1 0 in regime I


Properties of the natural dimensionless groups ndg

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


Estimations in oms

Estimations in OMS

  • For the balance of the example:

  • In regime I:

estimation


Corrections in oms

Corrections in OMS

Corrections

  • Dimensional analysis states:

correction function


Corrections in oms1

Corrections in OMS

Corrections

  • Dimensional analysis states:

  • Dominant balance states:

correction function

when P2/P10


Corrections in oms2

Corrections in OMS

Corrections

  • Dimensional analysis states:

  • Dominant balance states:

  • Therefore:

correction function

when P2/P10

when P2/P10


Properties of the correction functions

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

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

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 current1

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 current2

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 current3

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 current4

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


Application of oms to the weld pool at high current5

T*

d*

U*

Application of OMS to the Weld Pool at High Current


Application of oms to the weld pool at high current6

gas shear / viscous

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 Current

Relevance of NDG (Natural Dimensionless Groups)


Application of oms to the arc

Application of OMS to the Arc

  • Driving forces:

    • Electromagnetic forces

      • Radial

      • Axial

  • Balancing forces

    • Inertial

    • Viscous


Application of oms to the arc1

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 arc2

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 arc3

Application of OMS to the Arc

  • Estimations (5):

    • Length of cathode region

    • Flow velocities (2)

    • Pressure

    • Radial current density


Application of oms to the arc4

Application of OMS to the Arc

P

VZ


Application of oms to the arc5

Application of OMS to the Arc

  • Comparison with numerical simulations:


Application of oms to the arc6

Application of OMS to the Arc

  • Correction functions


Application of oms to the arc7

Application of OMS to the Arc

VR(R,Z)/VRS

200 A

10 mm

2160 A

70 mm


Conclusion

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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