Scaling of the cathode region of a long gta welding arc
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Scaling of the Cathode Region of a Long GTA Welding Arc. P. F. Mendez, M. A. Ramirez G. Trápaga, T. W. Eagar Massachusetts Institute of Technology August 23, 2000. Motivation. The arc is an essential component of a math model of the welding process

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Scaling of the Cathode Region of a Long GTA Welding Arc

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Scaling of the cathode region of a long gta welding arc

Scaling of the Cathode Region of a Long GTA Welding Arc

P. F. Mendez, M. A. Ramirez

G. Trápaga, T. W. Eagar

Massachusetts Institute of Technology

August 23, 2000


Motivation

Motivation

  • The arc is an essential component of a math model of the welding process

  • The 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 modeling

Evolution of Arc Modeling

  • As the complexity of the model increases, the characterization of the model requires exponentially more points


Evolution of arc modeling1

Evolution of Arc Modeling

  • Modeling restrictions have been removed gradually

  • The increase in complexity focused more on the physics than on the geometry

  • Not all the parameters that describe the problem are simultaneously relevant

    • The problem can be divided in different regimes in which different sets of parameters balance each other

    • In each regime the description of the problem is very simple


Order of magnitude scaling oms

Order of Magnitude Scaling (OMS)

  • Determines dominant and balancing forces (without solving the PDE’s)

  • Determines the range of validity of a particular balance

  • Provides order of magnitude estimations of the solutions

  • Ranks the relative importance of the dimensionless groups

  • Permits the construction of universal maps of a process


Formulation of the problem

Formulation of the problem

  • Cathode region:

    • e.m. forces generate pressure

    • pressure generates momentum

    • no interaction with anode region

    • little temperature variations


Temperature variations in the cathode region

Temperature variations in the cathode region

Hsu, 1983

Ramirez, 1999

200 A, 10 mm, Argon


Governing equations

Unknown functions:

Governing Equations

continuity

Navier-Stokes

Maxwell


Problem parameters

Problem Parameters

  • The 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 conditions

Boundary Conditions

  • For the numerical model:

I

J


Boundary conditions1

Boundary Conditions

  • For Order of Magnitude Scaling:

unknown characteristic values


Behavior of the solutions

7

F

E

40

H

G

85

256

Behavior of the solutions

BC for order of OMS

Real values (200 A, 10 mm)

VZ

VZ

VZ

40

7

E

F

256

VZ

85

H

G


Scaling functions

Scaling functions

  • For the electromagnetic field

unknown characteristic value


Scaling functions1

Scaling functions

  • For the fluid flow

unknown characteristic values


Modified dominant balance

Modified Dominant Balance

  • If:

    • unknown functions vary smoothly

    • normalization is performed

  • The dominant and balancing forces can be determined without solving the differential equation


Modified dominant balance1

Modified Dominant Balance

  • Dominant:

    • Radial inertial forces

    • Axial inertial forces

  • Balancing

    • Radial pressure variation

    • Radial e.m. forces

    • Axial pressure variation


Estimations of the characteristic parameters

Estimations of the characteristic parameters

  • Based on the balance obtained:

Power-law expressions


Estimations of the characteristic parameters1

Estimations of the characteristic parameters


Dimensional analysis

Dimensional Analysis

  • The dimensionless groups that govern the model are(show both natural and imposed dim groups)


Range of validity

h/Rc

Typical welding arc

10

1

102

Re

Range of Validity

  • Since 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 estimations

Corrections for the Estimations

  • The differences between the estimations and numerical calculations depend only on the governing dimensionless groups

  • Considering only the more relevant dimensionless groups usually is accurate enough


Corrections for the estimations1

Corrections for the Estimations

  • Power laws are convenient expressions

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


Corrections for the estimations2

correction function

actual difference

Corrections for the Estimations


Universal process maps

Universal Process Maps

  • Characteristic values are scaled and corrected  universal maps of the process can be build

    • based only on the problem parameters

    • no need for empirical measurements (e.g. to get maximum velocity from numerical model or experiment)


Universal process maps1

Universal Process Maps

VR(R,Z)/VRS

200 A

10 mm

2160 A

70 mm


Discussion

Discussion

  • The estimations obtained are comparable to those available in the literature (Maecker 1955)

  • Accuracy of the estimations can be increased by using

    • numerical results or experiments

    • relevant dimensionless groups

  • The effect of simplifications (e.g. constant properties appear as error in the correction functions)

  • The contour maps suggest ways of improving the numerical model

  • In sharper electrodes the e.m. forces also generate momentum without increase in pressure

    • application to GMAW?


Discussion1

Discussion

  • Is there a unique solution to the dominant balance?

    • Can dominant balance be fooled?

  • Does normalization solve potential paradoxes in dominant balance?


Conclusions

Conclusions

  • Viscous effects are small

  • Electromagnetic forces create pressure, which is balanced by inertial forces

    • thermal expansion is secondary

  • Power-law form equations, based only on the parameters of the problem provide:

    • properties of the fluid flow

    • correction functions

  • Universal maps of the arc can be generated

    • they can be scaled to a wide range of arcs


State of the art in arc modeling

State of the Art in Arc Modeling

ELECTRODE BEHAVIOR

Morrow & Lowke.1993. 1D theory for

Delalondre & Simonin.1990. 1D.

Chen, David, Zacharia, 1997

CATHODE AND ANODE FALLS

the electric sheats of electric arcs. (anode

modeling high intensity arcs inclu-

model interaction between the

and cathode falls).

ding non equilibrium description of

and the weld pool in GTA. Free

the cathode sheath.

surface changing with time

2AA

Zhu & Lowke.1992. Treats cathode

Kim, Fan, Na. 1997. 2D

boundary layer and arc column as

Auttukhov, 1983

Chen & Zacharia, 1991.

GTA, cathode influence

unified system.

Current thermionic emission

Analysis of the electrode tip

and free surface. No

angle and geometry of the

assumption on Jc.

Tekriwal, Mazunder.1988.2D analytical

2AC

GTA weld pool.

Mckelliget & Szekely.1986

Cram, 1983. Focussing

model for heat source (pointed tip).

Choo, 1990. Couple between

cathode and anode

on the energy balance of

Convection at anode by means of heat

arc and weld pool. Deformation

development

the electrode

transfer coefficient, properties constant

Sui & Kou, 1990. 2D,

of the pool.

Effect of the tip geometry

2AB

Hsu & Pfender,1983. Detailed

shielding gas, nozzle,

model for the cathode region

Westhoff, 1989. 2D arc model

2A

specifying current density at anode.

2C

deformation of weld pool. Small

changes in Jc changes T fields.

Dawson, Bendzak, Mueller, 1997

Pradip, Yogadra, Rama, 1995

Fluid flow and Heat Transfer in a

3D, heat transfer, fluid flow in GTAW

twin cathode DC furnace. Exp.

with non-axisymetric b.c.’s. Maxwell

2B

messurments in lab. modeling furnace

equations, uses buoyancy, surface tension

and electromagnetic forces.

ARC COLUMN MODELING

Qian, Farouk, Mathasaran, 1995

McKelliget & Szekely, 1986. 2D, DC approximate

BASED ON PHYSICAL PRINC.

Fluid flow and heat transfer in

boundary conditions. Jc=65MA/m

2

B.C.’S APPROXIMATED OR

EAF, 2D, DC

EAF

WELDING

EXP. DETERMINED.

Goldak & Moore. 1986. Finite ele-

ment method. Describes the source.

McKelliget & Szekely, 1983

Kovitya & Cram, 1986. 2D, LTE

2D arc model, coupling pool

MHD, boundary conditions assumed.

Lowke 1980. 2D continuity

and arc models.

Kovitya & Lowke, 1985. 2D, uses

energy, naturla convection.

properties theoretically calculated.

Hsu, Etemadi, Pfender, 1983. 2D

Jc=100MA/m

. Extends Lowke’s

2

2

MHD eqs. with b.c’s experimentally

McKelliget & Swzekely, 1981.

RF DISCHARGES

,odel to incorporate Lorentz’s for-

determined. Anode and cathode

Heat transfer & Fluid flow in

ces and electron drift enthalpy.

excluded. Jc assumed with a gaussian

a EAF

shape.

Allum, 1981. 2D Assumes current

Ramakrishan & Nou. 1980.

and velocity in gaussian profiles.

Dinulesca & Pfender, 1980

lowke, 1979. 2d momentum and en

3

1

2D, semianalytical model.

Includes magnetic, viscous and

Analysis of the boundary

ergy eqs. Natural convection.

Radial vel. field assumed.

gravitational forces.

layer in high intensity arcs

Low currents.

Lowke, 1979. 1D, analytical

Chang, Eagar, Szekely. 1979. Velo-

model for arc voltage, electric

city fields calculated analytically

Ushio, Szekely, Chang, 1980

field and plasma velocity.

using Lorentz’s forces.

2D model, assuming parabolic

Glickstein, 1979. 1D, analytical. Radial

current density distribution,

variations of temperature and J. No plas-

k-e

turbulent model.

ma flow.

Squire 51: isothermal, point force

Maecker 55: e.m. force approx

Shercliff 69: point current

Yas’ko 69: dimensional analysis

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 REGION

2AB= ANODE AND

CATHODE

2AC=CATHODE REGION


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