Modeling of Welding Processes through Order of Magnitude Scaling

Modeling of Welding Processes throughOrder of Magnitude ScalingPatricio Mendez, Tom EagarWelding and Joining GroupMassachusetts Institute of TechnologyMMT-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 forcesWeld pool

What is Order of Magnitude Scaling?OMS is a method useful for analyzing systems with many driving forcesWeld poolArc

What is Order of Magnitude Scaling?OMS is a method useful for analyzing systems with many driving forcesWeld poolElectrode tipArc

OutlineContext of the problemSimple example of OMSApplications to WeldingDiscussion

Context of the Problem

Context of the ProblemPhilosophyArtsScienceEngineering

Context of the ProblemScienceEngineeringPhilosophyArtsScienceEngineeringPhilosophyArts~1700

Context of the ProblemScienceEngineeringPhilosophyArtsScienceEngineeringPhilosophyArtsEngineeringScienceFundamentalsApplications~1700~1900

Context of the ProblemScienceEngineeringPhilosophyArtsScienceEngineeringPhilosophyArtsScienceEngineeringGap is gettingtoo large!FundamentalsApplications~1700~1900~1980

Example: Modeling of an Electric ArcVery complex process:Fluid flow (Navier-Stokes)Heat transferElectromagnetism (Maxwell)It is very difficult toobtain general conclusions with too many parameters

Chart1

1

1

2

2

8

8

10

11

12

14

20

14

14

Squire 1951(analytical)

Maecker 1955 (approximate)

Shercliff 1969 (analytical)

Yas'ko 1969 (dimensional analysis)

Ramakrishnan 1978

Glickstein 1979

Hsu 1983

McKelliget 1986

Choo, 1990

Lee 1996

Kim 1997

Lowke 1997

year of publication

number of dimensionless groups (m)

Sheet1

19511Squire

19691Shercliff

19512Squire

19552Maecker

19788Ramakrishnan

19798Glickstein

196910Yas'ko

198311Hsu

198612McKelliget

199014Choo

199720Lowke

199714Kim

199614Lee

Sheet2

Sheet3

Example: Modeling of an Electric ArcComplexity of the physics increased substantially

Generalization of problems with OMSFundamentals

Generalization of problems with OMSFundamentalsDifferential equations

Generalization of problems with OMSFundamentalsDifferential equationsAsymptotic analysis(dominant balance)

Generalization of problems with OMSFundamentalsDifferential equationsAsymptotic analysis(dominant balance)Engineering

Generalization of problems with OMSFundamentalsDimensional analysisDifferential equationsAsymptotic analysis(dominant balance)Engineering

Generalization of problems with OMSMatrix algebraFundamentalsDimensional analysisDifferential equationsAsymptotic analysis(dominant balance)Engineering

Generalization of problems with OMSMatrix algebraFundamentalsDimensional analysisDifferential equationsAsymptotic analysis(dominant balance)EngineeringArtificial Intelligence

Generalization of problems with OMSMatrix algebraFundamentalsDimensional analysisDifferential equationsAsymptotic analysis(dominant balance)EngineeringOrder of Magnitude ReasoningArtificial Intelligence

Generalization of problems with OMSMatrix algebraFundamentalsDimensional analysisDifferential equationsAsymptotic analysis(dominant balance)EngineeringOrder of Magnitude ReasoningArtificial IntelligenceOrder of MagnitudeScaling

OMS: a simple example

X = unknown

P1, P2 = parameters (positive and constant)

Dimensional Analysis in OMS

There are two parameters: P1 and P2:n=2


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


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

Asymptotic regimes in OMS

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

Dominant balance in OMS

There are 6 possible balancesCombinations of 3 terms taken 2 at a time:


There are 6 possible balancesCombinations of 3 terms taken 2 at a time: One possible balance:


There are 6 possible balancesCombinations of 3 terms taken 2 at a time: One possible balance: P2/P1 0 in regime I


There are 6 possible balancesCombinations of 3 terms taken 2 at a time: One possible balance: P2/P1 0 in regime IX P1 in regime I


There are 6 possible balancesCombinations of 3 terms taken 2 at a time: One possible balance: P2/P1 0 in regime IX P1 in regime Inatural dimensionless group

Properties of the natural dimensionless groups (NDG)Each regime has a different set of NDGFor each regime there are m NDG All NDG are less than 1 in their regimeThe edge of the regimes can be defined by NDG=1The magnitude of the NDG is a measure of their importance

Estimations in OMS

For the balance of the example:

In regime I:estimation

Corrections in OMS CorrectionsDimensional analysis states:

correction function


Dominant balance states:

when P2/P10correction function


Dominant balance states:

Therefore:when P2/P10correction functionwhen P2/P10

Properties of the correction functionsProperties of the correction functionsThe correction function is 1 near the asymptotic caseThe correction function depends on the NDGThe less important NDG can be discarded with little loss of accuracyThe correction function can be estimated empirically by comparison with calculations or experiments

Generalization of OMSThe concepts above can be applied when:The system has many equationsThe terms have the form of a product of powersThe terms are functions instead of constantsIn this case the functions need to be normalized

Application of OMS to the Weld Pool at High CurrentDriving forces:Gas shearArc PressureElectromagnetic forcesHydrostatic pressureCapillary forcesMarangoni forcesBuoyancy forcesBalancing forcesInertialViscous

Application of OMS to the Weld Pool at High CurrentGoverning equations, 2-D model (9) :conservation of mass Navier-Stokes(2)conservation of energyMarangoniOhm (2)Ampere (2)conservation of charge

Application of OMS to the Weld Pool at High CurrentGoverning equations, 2-D model (9) :conservation of mass Navier-Stokes(2)conservation of energyMarangoniOhm (2)Ampere (2)conservation of chargeUnknowns (9):Thickness of weld poolFlow velocities (2)PressureTemperatureElectric potentialCurrent density (2)Magnetic induction

Application of OMS to the Weld Pool at High CurrentParameters (17):L, r, a, k, Qmax, Jmax, se, g, n, sT, s, Pmax, tmax, U, m0, b, wsReference Units (7):m, kg, s, K, A, J, VDimensionless Groups (10)Reynolds, Stokes, Elsasser, Grashoff, Peclet, Marangoni, Capillary, Poiseuille, geometric, ratio of diffusivity

Application of OMS to the Weld Pool at High CurrentEstimations (8):Thickness of weld poolFlow velocities (2)PressureTemperatureElectric potentialCurrent density in XMagnetic induction

Application of OMS to the Weld Pool at High Current

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

Application of OMS to the ArcDriving forces:Electromagnetic forcesRadialAxialBalancing forcesInertialViscous

Application of OMS to the ArcIsothermal, axisymmetric modelGoverning equations (6):conservation of massNavier-Stokes(2)Ampere (2)conservation of magnetic fieldUnknowns (6)Flow velocities (2)PressureCurrent density (2)Magnetic induction

Application of OMS to the ArcParameters (7): r, m, m0 , RC , JC , h, Ra Reference Units (4):m, kg, s, ADimensionless Groups (3)Reynoldsdimensionless arc lengthdimensionless anode radius

Application of OMS to the ArcEstimations (5):Length of cathode regionFlow velocities (2)PressureRadial current density

Application of OMS to the ArcVZP

Application of OMS to the ArcComparison with numerical simulations:

Re

RC

ZS

Application of OMS to the ArcCorrection functions

Application of OMS to the ArcVR(R,Z)/VRS200 A10 mm2160 A70 mm

ConclusionOMS is useful for:Problems with simple geometries and many driving forcesThe estimation of unknown characteristic valuesThe ranking of importance of different driving forcesThe determination of asymptotic regimesThe scaling of experimental or numerical data

Now its even hard to apply 1900s physics.We can model complex problems, but still cant understand them well.

Put them on the internet

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Modeling of Welding Processes through Order of Magnitude Scaling