Scaling of the Cathode Region of a Long GTA Welding ArcP. F. Mendez, M. A. RamirezG. Trpaga, T. W. EagarMassachusetts Institute of TechnologyAugust 23, 2000

MotivationThe arc is an essential component of a math model of the welding processThe generalization of the numerical results is desirable for the interpretation and transmission of the findings

These two seemingly different motivations will be addressed simultaneously.Emphasis will be put on generalization instead of details of the numerical solutions

Evolution of Arc ModelingAs the complexity of the model increases, the characterization of the model requires exponentially more points

Evolution of Arc ModelingModeling restrictions have been removed graduallyThe increase in complexity focused more on the physics than on the geometryNot all the parameters that describe the problem are simultaneously relevantThe problem can be divided in different regimes in which different sets of parameters balance each otherIn each regime the description of the problem is very simple

Order of Magnitude Scaling (OMS)Determines dominant and balancing forces (without solving the PDEs)Determines the range of validity of a particular balanceProvides order of magnitude estimations of the solutionsRanks the relative importance of the dimensionless groupsPermits the construction of universal maps of a process

Formulation of the problemCathode region:e.m. forces generate pressurepressure generates momentumno interaction with anode regionlittle temperature variations

Temperature variations in the cathode regionHsu, 1983Ramirez, 1999200 A, 10 mm, Argon

Re

RC

ZS

Governing EquationscontinuityNavier-StokesMaxwell

_1011508849.unknown

_1016618351.unknown

_1019971059.unknown

_1019971077.unknown

_1011508912.unknown

_1011508768.unknown

Problem ParametersThe parameters that completely describe the problem as formulated are:

r : density of the plasma (at max. temperature) m : viscosity of the plasma (at max. temperature) m0 : permeability of vacuum Rc : cathode radius Jc : cathode density Ra : anode radius h : arc length

Boundary ConditionsFor the numerical model:IJ

AB

BI

IJ

JK

DK

AD

BJ

VZ

0

0

vz/z=0

vz/r=0

0

vz/r=0

-

VR

0

0

0

vr/z=0

0

0

-

T

4000

T/Z=0

1000

1000

2500

h/r=0

-

B

oIr/2Rc2

-

-

oI/2r

B/z=0

B=0

oI/2r

Boundary ConditionsFor Order of Magnitude Scaling:unknown characteristic values

A

B

C

D

B

0

0

_1012891791.unknown

_1012892065.unknown

E

F

G

H

VR

0

(-VRS

(0

0

VZ

(0

(0

(0

(VZS

Behavior of the solutionsBC for order of OMSReal values (200 A, 10 mm)EF407HG25685VZVZVZVZ

E

F

G

H

VZ

(0

(0

(0

(VZS

E

F

G

H

VZ

40

7

85

256

Scaling functionsFor the electromagnetic fieldunknown characteristic value

_1012233101.unknown

_1012233125.unknown

_1012892803.unknown

_1012233113.unknown

_1012233029.unknown

Scaling functionsFor the fluid flowunknown characteristic values

(14)

(15)

(16)

(17)

(18)

_1011535295.unknown

_1018186923.unknown

_1019664004.unknown

_1011535374.unknown

_1011534791.unknown

Modified Dominant BalanceIf:unknown functions vary smoothlynormalization is performedThe dominant and balancing forces can be determined without solving the differential equation

Modified Dominant BalanceDominant:Radial inertial forcesAxial inertial forcesBalancingRadial pressure variationRadial e.m. forcesAxial pressure variation

Estimations of the characteristic parameters Based on the balance obtained:

Power-law expressions

_1011674939.unknown

_1012308726.unknown

_1012309038.unknown

_1018090032.unknown

_1012287648.unknown

_1011674693.unknown

Estimations of the characteristic parameters

Dimensional AnalysisThe dimensionless groups that govern the model are(show both natural and imposed dim groups)

Range of ValiditySince the coefficients in the normalized equations are the ratio of secondary forces to the dominant, the boundary between limiting cases can be defined when they are 1

Corrections for the EstimationsThe differences between the estimations and numerical calculations depend only on the governing dimensionless groupsConsidering only the more relevant dimensionless groups usually is accurate enough

(23)

(24)

(25)

(26)

_1012319891.unknown

_1012630140.unknown

_1012630354.unknown

_1012319889.unknown

Corrections for the EstimationsPower laws are convenient expressions

The small exponents indicate that the estimations capture most of the behavior of the model

(27)

(28)

(29)

(30)

_1012663092.unknown

_1012667816.unknown

_1028305549.unknown

_1012663066.unknown

Corrections for the Estimations

Maximum errors: fZ 13 %, fVR 6 %, fVZ 3%, fP 8 %.

Universal Process MapsCharacteristic values are scaled and corrected universal maps of the process can be buildbased only on the problem parametersno need for empirical measurements (e.g. to get maximum velocity from numerical model or experiment)

Universal Process MapsVR(R,Z)/VRS200 A10 mm2160 A70 mm

DiscussionThe estimations obtained are comparable to those available in the literature (Maecker 1955)Accuracy of the estimations can be increased by usingnumerical results or experimentsrelevant dimensionless groupsThe effect of simplifications (e.g. constant properties appear as error in the correction functions)The contour maps suggest ways of improving the numerical modelIn sharper electrodes the e.m. forces also generate momentum without increase in pressureapplication to GMAW?

DiscussionIs there a unique solution to the dominant balance?Can dominant balance be fooled?Does normalization solve potential paradoxes in dominant balance?

ConclusionsViscous effects are smallElectromagnetic forces create pressure, which is balanced by inertial forcesthermal expansion is secondaryPower-law form equations, based only on the parameters of the problem provide:properties of the fluid flowcorrection functionsUniversal maps of the arc can be generatedthey can be scaled to a wide range of arcs

1=RF Discharges(nat. convection).2=Welding (Laminar flow).3=EAF (turbulent flow).2A=B.C.(anode and cathode modeling)2B=Coupled arc and weld pool (welding)2C=Geometry effects in welding.2AA=ANODE REGION2AB= ANODE AND CATHODE2AC=CATHODE REGIONState of the Art in Arc Modeling

I want to share with you some aspects of our ongoing work on modeling at the W&J group at MIT. In particular, a technique we are developing for which we see much promise as an auxiliary tool for the analysis of numerical models. Ill use the modeling of the cathode region of the arc as an example.Weld properties -->heat transfer--> heat sourceThese are only a few selected references relative to arc modelingAs we see, theres no shortage of knowledge about the arc.The geometric complexity didnt increase but the description of the physics involves many parameters.

Squire 1951: point force, point heat sourceShercliff:1969: point current sourceMaecker 1955: cathode spotRamakrishnan 1978, Glickstein 1979: 1D models of heat and mass transferHsu 1983: 2D modelMcKelliget: 2D, anode conditionsChoo 1990, Kim 1997: 2D, deformed free surfaceLee 1996: 2D cathode tip angle

Property of engineering systems: can be divided in regimes with a particular balance in each.

E.g. pressure can be balanced by inertial forces or viscous forcesWe developed a special technique for obtaining this balance of forcesUses dimensional analysis to obtain generalityIs similar to the perturbation method of applied math.Important difference: No need to solve PDEsFrom the complete numerical model we can extract the temperature profileWe see that the temperature variations within the cathode region are small compared to the variation elsewhere in the arcPoint out the different terms in the equationsRc can be deduced from the welding current.All the parameters are controllable experimentallyexcept Ra.The numerical model includes the anode model used by McKelligetImportant: we can identify that the velocities have characteristic values, even if we dont know what those values are.The functions show characteristic valuesThe variation between characteristic values is smoothThe expressions DO NOT assume a parabolic profile. They assume a parabolic behavior only at the axis, which can be demonstrated analytically.The expressions DO NOT assume a parabolic profile. They assume a parabolic behavior only at the axis, which can be demonstrated analytically.

Through dimensional analysis is possible to demonstrate that the expressions obtained will always be of the form of a power lawSince the coefficients depend on the dimensionless groups, a map can be constructed

This is similar to perturbation analysisPut them on the internet

Put them on the internetone dimensionless group (Ra/Rc) could not be estimated with this isothermal model

one dimensionless group (Ra/Rc) could not be estimated with this isothermal model

As we see, theres no shortage of knowledge about the arc.Modeling restrictions have been removed gradually.Lets see how these models evolved in complexity