a dynamic model from a gas metal electrode · 2011-12-12 · a dynamic model of drop detachment...

J. Phys. D: Appl. Phys. 31 (1998) 107–123. Printed in the UK PII: S0022-3727(98)83363-9

A dynamic model of drops detachingfrom a gas metal arc weldingelectrode

L A Jones †§, T W Eagar‡ and J H Lang †

† Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Room 10-176, 77 Massachusetts Avenue,Cambridge, MA 02139, USA‡ Department of Materials Science and Engineering, Massachusetts Institute ofTechnology, Room 8-309, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Received 11 April 1997, in final form 19 August 1997

Abstract. A dynamic model of drop detachment in gas metal arc welding ispresented for low and moderate welding currents in an argon-rich plasma.Simulations performed with this model are compared with extensive experimentalmeasurements of constant-current welding images and with limited experimentalmeasurements of pulsed-current welding images. The comparisons indicate thatthe experimental axial magnetic forces are much less potent than the calculatedaxial magnetic forces when welding-current transients are not present. To explainthis finding the hypothesis that internal flows are able to develop under therelatively quiescent conditions that exist during drop development inconstant-current welding is advanced.

1. Introduction

Once a neck begins to form on a drop of molten metal onthe end of a gas metal arc welding (GMAW) electrode, themagnetic forces increase substantially and they acceleratethe drop off of the end of the electrode. In the globular andspray modes of drop transfer, the time from neck formationto drop detachment is substantially less than the total dropformation time. This relatively brief time of transitionbetween a growing drop and a detached drop has been notedbefore [1], but the effect of magnetic forces on this timehas not been studied.

In this paper, the axisymmetrical geometrical shapesdeveloped in a companion paper [2] to model the shapesof necking drops are incorporated into a dynamic model ofdetaching drops. Such axisymmetrical shapes are typicallyobserved in argon-rich GMAW plasmas. The dynamicmodel is a lumped-parameter one in nature and this isevident in the computation and application of the forcesacting on the drop. Forces are computed on the basis ofthe instantaneous geometry of the drop, but then applied tothe centre of mass, rather than being applied in a continuumway to the distributed mass of the liquid drop. The centreof mass of the drop is moved in response to these forcesand a new drop geometry is then computed on the basis ofthe updated position of the centre of mass and an updateddrop volume (the volume growth rate simulates the meltingrate of the electrode).

§ Present address: Bose Corporation, MS 415, 1 New York Avenue,Framingham, MA 01701, USA.

A lumped-parameter, dynamic model of full (notpendent) liquid drops that is similar to the model in thispaper was developed in [3]. The primary concern in [3] waswith, among other things, inertial effects in the breakingup of drops in turbulent mixing vessels. A dynamic dropmodel was developed to interrelate the relevant physicalparameters in the drop break-up problem. The drops weremodelled not as deforming ellipsoids, but rather as cylindersdeforming such that cross sections of the cylinders wereellipses. A spring force was derived from the surfacetension and a damping force from viscous stresses. Thestudy suggested that, given a dynamic pressure impulseapplied to the centres of mass of drops, the conditionsfor drop break-up are a function of all of the variables ofthe problem. This behaviour betrays the complexity of thefluid motion in the drop, as described by the Navier–Stokesequation together with a free-surface boundary condition.

A simple, lumped-parameter, dynamic model of apendent water drop growing and detaching was developedin [4]. The geometry of the drop process was not explicitlyconsidered, rather deformations of the drop were modelledas a spring–mass system with a growing mass, such that,under certain conditions, the mass was suddenly reduced,corresponding to the detachment of a drop. The variousparameters of the model, such as the spring constant, wereadjusted to match experimental data at low drop growthrates. Interestingly, at higher growth rates the modelsuccessfully reproduced fluctuations in the drop detachmentfrequency observed in experiments. This type of model wasapplied to drops in gas metal arc welding in [5, 6], in which

0022-3727/98/010107+17$19.50 c© 1998 IOP Publishing Ltd 107

L A Jones et al

the various parameters were again chosen to make themodel agree with experimental measurements of metal dropdetachment frequencies and drop sizes. As with the resultsin [4], the model successfully reproduced the fluctuations inthe drop detachment frequency at higher drop growth rates,suggesting that a relatively simple model can successfullycapture some of the fundamental characteristics of dropdetachment in gas metal arc welding.

A distributed dynamic model of drops detaching ina welding arc, solved used finite-element methods, waspresented in [7]. Such an approach holds great promisebecause it may be possible to solve for the shapeof detaching drops in a self-consistent manner. Theshapes obtained in [7], however, matched poorly withexperimentally observed shapes.

The dynamic model developed in this study explicitlyconsiders the geometry of drops as they detach from theelectrode, thereby providing a detailed view of how theforces acting on the drops evolve. For example, whenthe geometry of a drop deforms away from a sphericalshape, its surface area and hence surface energy, increase,thereby developing a force which resists the deformation.As another example, the final collapse of a drop’s neckand the subsequent initial velocity of the detached droprepresent the imbalance between the primary forces actingon the drop just before detachment, namely magnetic forcesand surface tension, both of which are strongly dependenton the geometry of the drop.

In the first part of this paper, a dynamic model isdescribed for (i) drops attached to the electrode with noneck, (ii) drops attached to the electrode with a neck and(iii) drops which have detached from the electrode and aretraversing the arc. A drop growth and detachment cycle issimulated by smoothly switching between these three cases.A more detailed description of the model together with alisting of the computer code used to implement it was givenin [8].

In the second part of this paper, the effects ofplasma flow on detached drops in flight across thearc are studied via comparisons with experimentalmeasurements. Simulations performed with the dynamicmodel are then compared with extensive experimentalmeasurements of constant-current welding images andwith limited experimental measurements of pulsed-currentwelding images. The changes required in the dynamicmodel to make it match the experimental measurementsare reported and an attempt is made to understandwhat these changes indicate about the experimentalphenomena. A hypothesis to explain the major differencesobserved between the simulations and the experimentalmeasurements is advanced.

2. The dynamic model for attached drops

In the companion paper [2], the shapes of pendent dropsattached to a solid electrode without a neck were modelledusing axisymmetrical truncated ellipsoids. To simulate thedynamics of a drop on a welding electrode approximatedby truncated ellipsoids, the motion of the centre of massof truncated ellipsoids was computed forward in time in

′z

z

′x

x

zs s,ure

Vs

c

a

z

ze e,u

S

b

Figure 1. The truncated-ellipsoid dynamic model of awelding drop with no neck.

Time (sec)

12.38 12.40 12.42 12.44 12.46 12.48 12.50

Dro

p V

olum

e (c

m3 )

0.006

0.008

0.010

0.012

0.0240 cm3/s

Figure 2. The volume of a drop computed during animpulse response assuming axisymmetrical ellipsoidshapes. A linear fit of the data is also shown.

response to the various forces acting on it. All of the forcesdepend on the geometry of the drop. After each time step,there is a new centre-of-mass position and a new volume(the volume of the ellipsoid is increased with each timestep to simulate the melting of the electrode) for whichan appropriate ellipsoid must be found. In other words,given a volume and a centre of mass at each time step,it is necessary to find an appropriate truncated ellipsoidby computing a truncated-ellipsoid solution vector(a, c, ζ )

which satisfies the set of nonlinear equations

Vs− Vx = 0 (1a)

zs− zx = 0 (1b)

re− rx = 0 (1c)

wherere is the electrode radius, andVs andzs are the givenvalues of the volume and centre of mass, respectively,as shown in figure 1. Thex-subscripted variables arecomputed using the solution vector(a, c, ζ ) and equationsappropriate for a truncated ellipsoid.

The effect of surface tension on the motion of the drop’scentre of mass is modelled as a nonlinear spring with oneend of the spring attached to the centre of mass and the

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A dynamic model of drop detachment

other end anchored to the solid electrode at a positionze.The three-dimensional nature of the surface tension force isreduced to a dependence on the single verticalz dimensionvia

dE = γsadS ≈ −fsadzs (2)

⇒ fsa= −γsadS

dzs(3)

whereE is the surface energy,γsa is the surface-tensioncoefficient of the molten metal,S is the surface area ofthe truncated ellipsoid andfsa is the nonlinear spring forcedue to surface tension. The calculation of the derivativeon the right-hand side of (3) is greatly simplified if theinstantaneous volume of the drop is assumed to be constant.This assumption is based on the separation of time scalesbetween the slower thermal process of the drop growth rateand the faster mechanical process of drop oscillation. Onthe time scale of drop oscillations, the volume of the dropis approximately constant. Experimental measurements arepresented in the next section which support this assumption.

Motions of the drop are damped by loss mechanismsin the fluid such as viscous losses and Joule heatingfrom eddy currents induced by fluid motion through thewelding current’s magnetic field. For a drop attached tothe electrode, the damping is modelled as

fda= −kda(vs− ve) (4)

where ve is the velocity of the liquid/solid boundary ofthe electrode,vs is the velocity of the centre of mass ofthe drop andkda is a coefficient that was determined fromexperimental measurements.

Finally, the gravitational force is modelled by

fg = ρVg (5)

whereρ is the density of molten steel (6.25×103 kg m−3),V is the instantaneous volume of the drop andg is theacceleration due to gravity.

3. The drop impulse response

The parameters of the dynamic model described thus far– a drop attached to the electrode without a neck –were determined by comparing the impulse response ofthe model to experimentally measured impulse responsesof the drop. The welding arc was initially operated ata very low base current of 40 A with 330 A impulsessuperimposed at 5 Hz and a 2% duty cycle. Under suchconditions with a 1/16 inch (1.6 mm) diameter electrode,a drop was ejected from the electrode with each currentpulse. The peak current level was then briefly reducedto 260 A such that a drop was not ejected and a dropimpulse response was observed. These impulse responseswere captured with high-speed videography [9] and theresulting images of three drops were measured at 1 msintervals. The most reliable measurements were of thewidth of the attached drop at its equator, although additionalmeasurements were performed to compute an estimate ofthe growing volume of the drop. An example of these

Dro

p 1:

a

(cm

)

0.09

0.10

0.11

0.12

0.13

0.14

0.15

0.16

Time (ms)

0 20 40 60 80 100 120

Dro

p 1:

F

orce

(× 10

-3 N

)

-8

-6

-4

-2

0

2

4

Experiment

Simulation

Gravitational Force ( fg )

Damping Force ( fda )

Surface Tension Force ( fsa )

Figure 3. The drop impulse-response simulation.

volume computations is shown in figure 2. It is assumedthat the fluctuations of the computed drop volume – whichare at the frequency of the drop oscillation – are simplyan artefact of the application of axisymmetrical ellipsoidshapes to compute the drop volume, for it is highly unlikelythat the volume did not increase monotonically. Therefore,linear fits of the computed volume data were obtainedand used as the estimate of the drop growth rate. Thevolume of the drop in figure 2 increased by approximately0.000 34 cm3 during one period ('14 ms) of oscillation ofits equatorial diameter, which corresponds to a 3.4% changewith respect to the mean volume (0.0102 cm3) measuredover the entire impulse response (only the first portion ofthe impulse response is shown in figure 2). Although thedrop was measurably growing, on the time scale of the droposcillations, the growth rate was small, thus justifying theassumption of a constant drop volume with respect to thederivative of the surface area made in the previous section.

Simulations of the impulse response of the drop werecomputed using the dynamic drop model. The initialtruncated-ellipsoid volumes and subsequent growth rateswere taken from the linear fits of the volume computations.The magnetic force impulse on the model drop due to thecurrent pulse was computed as described in the companionpaper [2, section 7].

An example comparison of the measured drop’sequatorial radius and the simulated drop’s equatorial radiusis shown in the upper graph of figure 3. Of interest here arethe oscillation frequency and damping rate. The parametersused in the simulation to yield the match shown betweenthe simulated and the experimental oscillation frequencyand damping rate wereγsa = 0.66 N m−1, which is

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L A Jones et al

f k fm mu2 2=

fg2

fg1

fs2i fd2i

fs2n

m2(t)

m1(t)

Polynomial-VolumeWaist

Polynomial-Volume/

Truncated-Ellipsoid

IntersectionTruncated-EllipsoidEquator

f k fm1 m= 1 l

Figure 4. A representation of the forces included in thedynamic model of a necking drop.

about one half the surface-tension coefficient of moltensteel, andkda = 2.0× 10−3 kg s−1, which is a dampingcoefficient much greater than that which might be expectedfor viscous losses due to irrotational flow inside a deformingellipsoidal shell. A Fourier analysis of the experimentaland simulated data yielded fundamental frequencies of 70and 64 Hz, respectively. The surface tension, damping andgravitational forces generated by the model are shown inthe lower graph of figure 3; the nonlinear nature of thesurface tension model is apparent. Magnetic forces wereinsignificant during the oscillation because the base currentwas only 40 A during this time.

The results of simulations of the other two dropsmeasured are similar [8]. It was necessary to use a surface-tension coefficient approximately one half the known valuefor molten steel (γs ≈ 1.2 N m−1) and a relativelylarge damping coefficient. It is hypothesized that thesediscrepancies are due to fluid flow inside the drop and themagnitudes of the discrepancies suggest that the effects ofthis flow are significant. The excess viscous losses may bethe result of rotational flow in the drop driven by rotationalcomponents of magnetic body forces and Marangoni flow(flow driven by surface tension gradients). This rotationalflow would result in additional viscous losses. It wouldalso increase eddy current losses.

4. The dynamic model for necking drops

Two centres of mass were simulated for a necking dropin order to capture both common and differential modesof motion of the upper and lower parts of the drop. Arepresentation of the forces included in the model for anecking drop is shown in figure 4. Spring and damping

forces are modelled to act between the two centres of mass(fs2i andfd2), while the upper centre of mass is also pulledupwards by the surface tension force at the neck waist(fs2n). Gravitational and axial magnetic forces,fg andfm, respectively, act both on the lower and on the uppermass,m1(t) andm2(t), respectively. The computation ofthe axial magnetic forcesfml andfmu acting on the lowerand upper parts of the drop is described in the companionpaper [2, section 7]. These forces are applied to the lowerand upper centres of mass asfm1 andfm2, respectively. Thescaling constantsk1 andk2 are used in section 9 to matchthe simulation results with experimental measurements.

In the companion paper, the shapes of neckingdrops were modelled using axisymmetrical polynomialvolumes and truncated ellipsoids [2, section 5]. Givena truncated ellipsoid(a, c, ζ ) and an electrode positionze, the connecting polynomial-volume neck can then bereconstructed. The geometry of the centres of mass of theassociated dynamic model is shown in figure 5. Given avolume and the positions of the upper and lower centresof mass at each time step, a solution vector (a, c, ζ ) iscomputed which satisfies the set of nonlinear equations

V − Vx = 0 (6a)

z2− (z1− z′1x+ z′2x) = 0 (6b)

γx − z′2x = 0 (6c)

where V , z1 and z2 are the given values of the volumeand centre-of-mass positions, respectively, and thex-subscripted variables are computed using the solution vector(a, c, ζ ) in equations appropriate for a truncated ellipsoidand a polynomial volume. The valueγ is the position ofthe truncation of the ellipsoid relative to the equator of theellipsoid (thex ′–z′ axes),z′1 is the position relative to theellipsoid’s equator of the centre of mass of the volume ofthe ellipsoid below its equator (Vsl), and z′2 is the positionrelative to the ellipsoid’s equator of the centre of mass ofthe volume of the ellipsoid above its equator and the portionof polynomial volume below its waist (Vsu+ V4l).

Equations (6) were carefully formulated to yield robustconvergence to a unique solution without over-constrainingthe problem, because difficulties were encountered usingother approaches. For example, the mass of the polynomialvolume above the polynomial waist (the volumeVpu abovez′′ = δw in figure 5) is ignored, because adding a thirdcentre of mass to track this volume often caused theproblem to converge poorly, if at all. Also, uniqueness issatisfied and convergence is greatly enhanced by arbitrarilyrequiring the centre of mass of the upper part of the drop (atz2 in figure 5) to be at the boundary between the truncatedellipsoid and the polynomial volume. Equation (6c)embodies this constraint.

The surface tension neck forcefs2n acting at thepolynomial-volume waist of a drop is approximated by

fs2n= 2πrwγs2n (7)

whererw is the radius of the polynomial-volume waist andthe parameterγs2n is some fraction of the fluid’s surface-tension constantγs (similar to the Harkins and Browncorrection of Tate’s law [10]).

110


′′z

z

′x

x

rw

Vsl

Vpu

Vpl

′′x

′z

′z2 Vsu

ze

z1

z2

re

′z1

γ

dw

V V V V= + +s su pl l

c

a

z

Figure 5. The truncated-ellipsoid/polynomial-volumedynamic model of a necking drop.

∆z1

S z z1 2,b g S z z z1 1 2+ D ,b g

V V

fs2ifs2i

Surface-tangentangle and radius

held constantat the polynomial-

volume waist

Figure 6. The computation of the internal surface tensionforce fs2i.

The force fs2i is computed by ‘squeezing’ the dropwhile holding the volumeV constant, as shown in figure 6,and using an approximation of the derivative,

fs2i = −γs2idS

d(z2− z1)(8)

where the parameterγs2i is also some fraction of the

′z

z

′x

x

z2 2, u

z1 1,u

a

c

V

S

Figure 7. The full-ellipsoid dynamic model of a detacheddrop.

fluid’s surface-tension constantγs and S is the surfacearea. The idea behind this approach to computingfs2i

is to test the differential-mode stiffness between the twocentres of mass by ‘pushing’ on the lower centre of masswhile holding the upper centre of mass fixed. Decouplingfrom the electrode when computing this internal surfacetension force is necessary because coupling with the solidelectrode results in common mode forces acting on bothcentres of mass. Values forγs2n andγs2i were determined bycomparing dynamic model simulations with experimentalmeasurements, as was a value ofkd2i for use in anexpression forfd2i similar to (4).

5. The dynamic model for free drops

For a detached drop in free flight across the arc, an exactsolution of a full ellipsoid model is easily obtained. Giventhe volumeV and centres-of-mass positionsz1 and z2

shown in figure 7,

c = 43|z2− z1| (9a)

a = 3V

4πc. (9b)

The modelled forces acting on the full ellipsoid includegravity, surface tension, damping and plasma flow, thelatter of which is considered in section 7. Magneticforces are not included because experimental observations

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L A Jones et al

Time (ms)

0 5 10 15 20 25

Axi

al D

ista

nce

from

the

Po

int o

f Det

achm

en

t (cm

)

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

IncreasingCurrent

Figure 8. Measured flight trajectories in constant-currentGMAW, 29 V, Ar–2% O2 for two consecutive drops at eachcurrent level. Beginning at the far right-hand side andproceeding to the left-hand side, the symbols at the end ofeach trajectory correspond to 180 A (◦), 190 A (�),200 A (M), 210 A (O), 220 A (�), 230 A (hexagon),240 A (◦), 250 A (�), 260 A (M), 270 A (O) and 280 A (�).The closest dotted version of a symbol corresponds to thetrajectory of the second drop.

presented in [2, section 6] indicated that no current flowedthrough free drops as they traversed the arc and thus nomagnetic body forces acted on the drops.

The surface tension force resulting from the differencein motion of the centres of mass of a free drop is modelledas

fsf = −γsfdS

d(z2− z1)(10)

and the viscous damping is modelled as

fdf = −kdf(v2− v1) (11)

where v2 and v1 are the velocities of the upper and lowercentres of mass, respectively. For a low-viscosity fluid withirrotational flow in the interior of the drop,kdf is a verysmall number. However, it was observed experimentallythat, after detachment, the motion of free drops damps outmuch faster than the model predicts using a small value forkdf, suggesting that the flow in the interior of the drop isnot irrotational.

6. Shape switching in the model

At the end of each time step in the dynamic simulationof drop growth and detachment, a decision must be madeconcerning which shape or shapes to use to model thedrop: a truncated ellipsoid, a truncated ellipsoid and apolynomial volume, or a full ellipsoid. Truncated ellipsoidsare used initially and when the balance of forces at theliquid/solid boundary (the truncation point of the ellipsoid)

Time (ms)

0 5 10 15 20 25

Vel

ocity

Aw

ay fr

om th

e E

lect

rode

(cm

s-1)

-140

-120

-100

-80

-60

-40

-20

0

IncreasingCurrent

Figure 9. Fitted drop flight velocities in constant-currentGMAW, 29 V, Ar–2% O2. The symbols have the samemeanings as those in figure 8.

becomes less than zero, truncated ellipsoids and polynomialvolumes are then used, which allows the formation of aneck. The approximate force balance used to trigger atransition between model shapes is

fst− fsa− fda+ fmu−Kpr. (12)

The retaining surface tension forcefst is computed with

fst = 2πreγs sinβ (13)

whereβ is the tangent angle of the truncated ellipsoid atthe electrode as shown in figure 1. The forcesfsa andfda

are the spring and damping forces, (3) and (4), respectively.The spring forcefsa represents the resistance of the dropto being stretched by gravity. The axial magnetic forcefmu is that acting on the ellipsoid above its equator andwas shown in [2, section 8] to be negative in argon-richplasmas. Finally, the termKpr is an adjustable parameterthat was found to be necessary to match the momentof neck formation with that observed in experiments. Itis believed that this parameter represents the unmodelledexcess pressure in the drop due to curvature of the drop’ssurface† and unmodelled radial magnetic pressure whichaids the formation of a drop neck. This parameter was setto Kpr = −11fmu for all of the simulations reported belowin sections 8 and 9.

At low currents, where the drops are large and themagnetic forces are relatively small, the force balance in(12), which triggers the transition from truncated ellipsoidsto truncated ellipsoids and polynomial volumes, is notcritical since the drop usually stays on the electrode forsome time after the transition. The forces acting on thetruncated ellipsoids and polynomial volumes are what then

† Excess pressure was not modelled because it has little meaning in thecontext of the constrained – rather than free – surface shapes used here.

112


Current (Amps)

180 200 220 240 260 280

Fre

e D

rop

Acc

eler

atio

n (m

s-2)

-60

-50

-40

-30

-20

-10

0

Total Acceleration(Plasma + Gravity)

Plasma Acceleration (ap)

Figure 10. Fitted drop flight accelerations inconstant-current GMAW, 29 V, Ar–2% O2. Each point is theaverage acceleration for the two drop flights measured ateach current. The total acceleration data are from the curvefits to the image data in figure 8. The plasma accelerationdata are less the acceleration due to gravity (9.81 m s−2).

Current (Amps)

180 200 220 240 260 280

For

ce (×

10-3

N)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Plasma Force (fp)

Gravitational Force

Figure 11. Computed plasma flow force (◦) andgravitational force (�) acting on the drops measured infigure 8. Each point is the average force for the two dropsmeasured at each current.

determine the time at which the drop detaches from theelectrode. However, at higher currents in the globulartransfer region, where the drops are small and the magneticforces are relatively large, the drop detaches soon after thetransition from truncated ellipsoids to truncated ellipsoidsand polynomial volumes, and the time of the transition,triggered by the force balance in (12), becomes moreimportant.

A drop is considered to detach from the electrode whenits neck becomes sufficiently narrow (rw < 0.1re). Thevolume below the neck waist forms a free drop whose upperand lower centres of mass maintain continuity of positionand velocity with the previous shapes. The volume of fluidin the portion of the polynomial volume above its waist(Vpu) at the moment of detachment forms a new growingdrop on the end of the electrode.

Under some conditions, the formation of a drop neckdoes not always lead to detachment of the drop, rather thedrop is pulled back onto the electrode by surface tension.Accordingly, the dynamic model allows for switchingfrom truncated ellipsoids and polynomial volumes back totruncated ellipsoids.

13.09150 13.09175 13.09200 13.09225 13.09250 13.09275

13.09300 13.09325 13.09350 13.09375 13.09400 13.09425

13.09450 13.09475 13.09500 13.09525 13.09550 13.09575

13.09600 13.09625 13.09650 13.09675 13.09700 13.09725

13.09750 13.09775 13.09800 13.09825 13.09850 13.09875

Figure 12. Drop detachment with a 290 A current pulse inan Ar–2% O2 plasma. The 4 ms pulse is first visuallydetectable in the second image of the first row. The currenthas returned to its base level of 40 A in the first picture ofthe fourth row.

7. Effects of plasma flow

After the welding current emerges from the end of theelectrode, its flow through the plasma generates radial andaxial pressure via interaction with its own magnetic field,just as it did in the electrode. The plasma has a very lowdensity (' 5×10−2 kg m−3 at 10 000 K [11, figure 2.1]) andresponds vigorously to this magnetic pressure by flowingtowards the base plate. Additionally, in an argon-richplasma it is readily observed that the plasma expands asit approaches the base plate. This expansion cools theplasma and causes a reduction in the column pressure,thereby also promoting an axial flow [12]. Both effectsresult in substantial plasma flow velocities which acceleratedetached metal drops toward the base plate.

Few published measurements of plasma flow areavailable for gas metal arc welding. The results fromthe available experimental studies of plasma flow werecorrelated in [11] and, with respect to the acceleration ofmetal drops in welding arcs, the following, rather weak,conclusions may be drawn:

(i) At 100 A, the on-axis plasma flow velocity 7 mmfrom the end of the electrode has been measured in variousstudies with various types of arcs to be approximately100 m s−1.

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L A Jones et al

9.92000

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9.92125

9.91125

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9.91975

9.92175

Figure 13. A failure of the drop to detach with a 280 Acurrent pulse in an Ar–2% O2 plasma. The current pulsewas insufficient to detach the drop. Subsequent currentpulses also fail to detach the growing drop.

(ii) The plasma flow rate increases with the current,although the function by which it increases is not known(linear, quadratic and so on).

(iii) Along the axis of the electrode, the flow ratedecreases with the distance from the electrode, althoughthe function by which it decreases is not known.

Even if the plasma flow field were known exactly, anadditional problem would be to determine the drag forceof the plasma flow on the drops traversing the arc.

It has been hypothesized that the magnetic pressure inthe arc generates only a very small upward reaction forceon the bottom of a drop because the static pressure is largelyannulled by the dynamic pressure of the plasma flow [13].In pulsed-current welding, plasma flow has been estimatedfrom experimental measurements to develop in a time of theorder of 0.2 ms [14] and so, except for a brief moment atthe beginning of each current pulse, there is little magneticpressure pushing upward on a drop. Therefore, no upwardplasma reaction force was included in the dynamic drop-detachment model in this study.

In several recent studies of drop detachment [5, 15, 16],it has been suggested that the plasma flow exerts a detachingforce on pendent drops on the electrode. Such a force wassuggested in [17] (published in 1960), but then discountedvia qualitative arguments in later studies [18, 11].

A simple analytical model of the plasma flow – whichyields unrealistically large flows – is submerged-jet flow[11]. Submerged-jet flow is the flow resulting from a pointsource of forcefp acting on a viscous fluid of infinite extent.

Figure 14. A simulation of a 290 A, 4 ms current pulse inAr–2% O2 shown at 1.5 ms intervals. Horizontal ticks areat 0.10 cm intervals and vertical ticks are at 0.05 cmintervals. The full line in figure 16 indicates the drivingcurrent waveform.

If fp is assumed to act at a point below the bottom of adrop, simple computations reveal that the magnitude of flowbehind the point source and around the shoulders of a dropon the electrode is of the order of 1 m s−1 [8], which,as noted above, is probably an unrealistically high value.Nevertheless, this velocity is very low compared with themeasured value of 100 m s−1 7 mm below the end of theelectrode. Most of the flow in the jet is drawn in fromthe sides of the arc rather than from behind the jet and thussuch flow does not exert an appreciable axial force on dropsattached to the electrode.

The plasma flow force acting on a free drop traversingthe arc, however, is appreciable. This force may beapproximated by [19]

fp = −CDAP

(ρpv

2pz

2

)(14)

whereAP is the projected area of the drop in thex–y plane,CD is the coefficient of drag which is of the order of 0.80for the physical properties found in a welding arc [8],ρp

is the density of the plasma andvpz is the plasma velocityaround the drop. Given the range of temperatures foundin a welding arc, an appropriate value (or set of values) ofthe plasma density is difficult to determine. Also, there ismuch uncertainty in the drag coefficient and the plasma flowvelocity. These factors strongly suggest that, for computingthe plasma flow force on free drops, it would be better

114


Figure 15. Simulation of a 280 A, 4 ms current pulse inAr–2% O2 shown at 1.5 ms intervals. The horizontal ticksare at 0.10 cm intervals and the vertical ticks are at0.05 cm intervals. The broken line in figure 16 indicates thedriving current waveform.

to lean heavily on experimental data measured from dropstraversing the arc.

The flight trajectories of two consecutive drops weremeasured over the range 180–280 A in 10 A increments inconstant current GMAW at each current level. The centresof mass of the drops were measured from video imagesfrom the moment the drops detached from the electrodeuntil the moment they made contact with the weld pool.The measured trajectories are shown in figure 8. Theshielding gas was Ar–2% O2 flowing at 50 cfh (1.4 m3 h−1)through a 3

4 inch (19.1 mm) diameter gas cup and theelectrode was 1/16 inch (1.6 mm) diameter ER70S-3 wire.At all currents, a control system maintained the wire feedspeed such that the contact-tube-to-base-plate voltage was29 V. The contact tube was mounted flush with the bottomof the gas cup and the gas cup was 1 inch (25.4 mm)above the base plate. Therefore, the electrode stickoutmay be determined in each experiment shown in figure 8by subtracting the flight distance of the drop from 1 inch(25.4 mm).

Constant acceleration (quadratic) curves were fitted tothe measurements shown in figure 8. The fit was alwayswithin the range of measurement error over the entire lengthof the drop flight, suggesting that the drops had undergoneessentially constant acceleration. This result is at odds withthe submerged-jet model, whereby the axial plasma flowvelocity falls off as 1/r and the plasma force falls off as1/r2. The first derivative of the fitted quadratic curves

Time (ms)

0 5 10 15 20

Cur

rent

(A

)

0

50

100

150

200

250

300

Figure 16. Simulated current pulses used to find thedetachment/no-detachment boundary.

Time (ms)

0 5 10 15 20

z-di

rect

ed M

agne

tic F

orce

(× 10

-3 N

)

-6

-5

-4

-3

-2

-1

0

Sum

Drop DetachesForce on LowerCenter of Mass ( fm1 )

Force on UpperCenter of Mass ( fm2 )

Figure 17. Magnetic forces acting in the model drop infigure 14.

yielded the downward (negative) drop velocities shown infigure 9. The second derivative of the fitted quadraticcurves, the downward (negative) drop acceleration, isshown in figure 10 as the ‘total acceleration’. Theaccelerations were remarkably constant at lower currentsand began increasing at 240 A. After acceleration due togravity (9.81 m s−2) had been subtracted, the accelerationdue to plasma flow (ap) was found to be significant evenat lower currents, for which about one half of the totalacceleration was due to plasma flow and this accelerationwas essentially constant as a drop traversed the arc.

To determine the force of the plasma flow on the drops,the mass of the measured drops must be known. Forthe drops whose flight trajectories were measured, volumemeasurements from the images were performed assumingthat the drops were axisymmetrical ellipsoids. A materialdensity ofρ = 6.25× 103 kg m−3 was assumed and theplasma force acting on the drops was computed using

fp = ρV ap. (15)

The results are shown in figure 11 together with theforce due to gravity. The latter force effectively showsthe drop volumes (since the gravitational acceleration isconstant) and, as might be expected, the volume is inverselyproportional to the current. The apparent decrease involume below 200 A is most likely to have arisen from

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L A Jones et al

Current (Amps)

180 200 220 240 260 280

Dro

p F

requ

ency

(H

z)

0

10

20

30

40

50

60

Figure 18. Drop frequencies in GMAW measured from theexperiments over a range of constant currents at 29 V with1/16 inch (1.6 mm) diameter ER70S-3 electrode wire andAr–2% O2 shielding gas. Each data point is the meandetachment frequency measured for 100 drops at theindicated current.

Current (Amps)

180 200 220 240 260 280

Dro

p V

olum

e (c

m3 )

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Figure 19. Drop volumes in GMAW measured from theexperiments over a range of constant currents at 29 V with1/16 inch (1.6 mm) diameter ER70S-3 electrode wire andAr–2% O2 shielding gas. Each data point is the meanvolume of ten consecutive drops within the set of 100 usedto measured the drop frequency shown in figure 18.

measurement error, for below 200 A the shape of thelarge drops deviated significantly from an axisymmetricalellipsoid. Assuming that this was indeed the case, figure 11shows that, at low currents ('200 A), the plasma force waslarge because the metal drops were large and presented alarge area to the plasma flow. At high currents ('270 A)the drops were much smaller and presented a much smallersurface area to the flow but, owing to the increasingvelocity of the flow, the plasma force was again large. Atintermediate currents the magnitude of the plasma forcegoes through a minimum.

Equation (14) may be inverted to obtain the plasmavelocity as a function of the plasma flow force, that is,

vpz = −(

2fp

ρpCDAP

)1/2

. (16)

However, care must be taken in interpreting results obtainedfrom this equation because the true value of the plasmadensity ρp is not well known and neither is the dragcoefficientCD, which is an empirical value. Therefore,

Current (Amps)

180 200 220 240 260 280

Ele

ctro

de F

eed

Ra

te (

cm s

-1)

2

3

4

5

6

7

Computed

Measured

Figure 20. Electrode feed rates measured directly andcomputed from video measurements of drops.

the experimental measurements were not extended to(16); rather, an appropriate plasma flow force model wasdeveloped for the dynamic drop-detachment model suchthat the drops traversing the arc in the model matchedthe trajectories shown in figure 8. The flow force modelrequired to do this is discussed in section 9 and comparedwith the forces shown in figure 11.

8. Pulsed-current detachment

The dynamic drop-detachment model was first used tosimulate the response shown in figure 16 of the companionpaper [2], repeated here for convenience in figure 12. Adrop is shown detaching in response to a 4 ms, 290 Acurrent pulse occurring at 5 Hz. Before and after the pulse,the arc is sustained with a 40 A base current. Under theseconditions, 290 A is the minimum pulse magnitude forwhich a drop detaches with each pulse. If the magnitudeof the pulse is reduced to 280 A, a drop does not detachwith each pulse, as shown in figure 13.

The initial volume and centre of mass of thedrop at time 13.091 50 s in figure 12 were computedfrom measurements described in the companion paper[2, section 8]. The electrode feed rate in the simulationwas set to the mean of 5 s of feed speed data prior tothe particular drop being ejected from the electrode. Allsurface-tension related coefficients were set to the surfacetension of molten steel, namely 1.2 N m−1.

Under these conditions, the drop didnot detach in thesimulation. The detachment/no-detachment boundary wasinstead found to lie between 390 and 400 A. However,with one change to the model the boundary was movedto between 280 and 290 A. The surface-tension coefficientγs2n in (7) (see the forcefs2n in figure 4) determines the rateat which the surface tension force of the neck retaining thedrop diminishes as the drop detaches. When this coefficientwas set to

γs2n= 0.45γs (17)

whereγs = 1.2 N m−1, the model accurately predicted thedetachment/no-detachment boundary to be between 280 and290 A. Figures 14 and 15 show simulations of the drop’sresponse on either side of the boundary and figure 16 shows

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Figure 21. A simulation of a 180 A constant-current casein Ar–2% O2 shown at 4 ms intervals. The horizontal ticksare at 0.10 cm intervals and the vertical ticks are at0.05 cm intervals. The feed rate supplied to the model wasthe experimentally measured feed speed.

the driving current waveforms used in the simulations.The value in (17) forγs2n is similar to the value ofγsa

determined in section 3 to match drop impulse responsesimulations to experimental measurements.

On comparing the simulated geometries in figures 14and 15 with the experimental geometries in figures 12 and13, respectively, it is apparent that, unlike the model drop,the physical drop compresses very little towards an oblateshape in response to the current pulse. Plots of the axialmagnetic forces acting on the model drop are shown infigure 17. The total axial magnetic force in the model(fm1+fm2) peaks at a greater value than the axial magneticforce computed from measurements of the experimentaldrop shape (the diamond-marked plot in figure 18 of [2]).This is a result of the greater drop distortion in the dynamicmodel and the narrower neck which forms while the currentis at its peak value. In addition, the dynamic-model droplingers on the electrode longer than does the physical drop.

It is hypothesized that the physical drop resistsflattening towards an oblate shape because of an increase inthe isotropic fluid pressure caused by the radially directedcomponent of the magnetic force acting on the drop.The effect is to ‘stiffen’ the drop. This force may beapproximated byµ0I

2/(8π) and in [18] it wasaddedto thez-directed force acting on a frustum of current. Increasingthe total axial magnetic force acting on the drop by addinga radial magnetic pressure component to thez-directedmagnetic force computed in the companion paper [2] might

22.93200 22.93500 22.93800 22.94100

22.94400 22.94700 22.95000 22.95300

Figure 22. Constant-current GMAW at 180 A and 27 V inAr–2% O2. The electrode is 1/16 inch (1.6 mm) diameterER70S-3 wire and the interval between images is 3 ms.

alleviate the need to reduce the surface-tension constantγs2n

as in (17). However, it is shown in the next section that,when using constant welding current, a significantlysmalleraxial magnetic force than that which was computed in [2]is required to model the experiments dynamically, not alarger magnetic force.

9. Constant-current detachment

Simulations using the dynamic model were comparedwith measurements of constant-current GMAW obtainedfrom video images for a range of welding currents.Constant-current drop detachment differs from pulsed-current operation in that inertial effects – excited bythe harmonic content of current pulses – are much lessimportant. With a constant current, drop detachment andthe subsequent initial velocity of the drop are a result ofthe delicate balance between the magnetic force and theretaining surface tension force of the drop’s neck.

The experiments considered were the same set as thatused to study plasma flow in section 7. These experimentscovered the welding current range 180–280 A in 10 Aincrements. A custom-built, transistor-regulated, linearpower supply (as opposed to a switched, inverter-typesupply or an SCR supply) with virtually no current ripplewas used. At all currents, a control system maintained thewire feed speed such that the contact-to-base-plate voltagewas 29 V. The shielding gas was Ar–2% O2 flowing at50 cfh (1.4 m3 h−1) through a3

4 inch (19.1 mm) diametergas cup and the electrode was 1/16 inch (1.6 mm) diameterER70S-3 wire. The contact tube was mounted flush withthe bottom of the gas cup and the gas cup was always

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L A Jones et al

Figure 23. A simulation of a 220 A constant-current casein Ar–2% O2 shown at 1.5 ms intervals. The horizontalticks are at 0.10 cm intervals and the vertical ticks are at0.05 cm intervals. The feed rate supplied to the model wasthe experimentally measured feed speed.

1 inch (25.4 mm) above the base plate. The base platewas 3

8 inch (9.5 mm) mild steel and its travel speed was (arather slow) 5 inch s−1 (127 mm s−1) for the current range180–210 A and 10 inch s−1 (254 mm s−1) for the currentrange 220–280 A.

The lower limit of the current range, 180 A, was thelowest current at which an arc could be stabilized witha 1/16 inch (1.6 mm) diameter electrode in Ar–2% O2.The upper limit of the current range, 280 A, was thehighest current before the drop size bifurcated. At 290and 300 A, two distinct drop sizes were observed with thedetachments alternating between several drops larger thanthe diameter of the electrode followed by several dropssmaller than the diameter of the electrode. At 310 A thedrops were observed to be all of the smaller size and,at higher currents still, strong tapering of the electrodewas observed. The present model is most suited to dropsizes which are somewhat larger than the diameter of theelectrode.

The following process features were studied:

(i) the drop detachment frequency,(ii) the drop volume,(iii) the drop necking time,(iv) the initial detachment velocity and(v) the free-flight acceleration.

The drop detachment frequency could be accuratelymeasured from the experiments with relative ease. From

15.21000 15.21300 15.21600 15.21900

15.22200 15.22500 15.22800 15.23100


high-speed video images, the time necessary for 100 dropsto form and detach was measured to obtain a mean dropperiod. The measured mean drop detachment frequencies(1/period) are shown in figure 18.

Within the set of 100 drops used to measure theformation and detachment frequencies at each current level,the volumes of ten consecutive free drops traversing thearc were computed, assuming them to be axisymmetricalellipsoids. The mean volume of each set of ten drops isshown in figure 19. As expected, the drop volume wasinversely proportional to the current up to 280 A and thedecrease in volume below 200 A was most probably dueto measurement error when the shape of the large dropsdeviated significantly from an axisymmetrical ellipsoid.The veracity of the drop volume measurements was checkedby multiplying the volume measurements by the measureddrop frequencies and dividing by the electrode’s crosssectional area to obtain a computed electrode feed rate.This computed rate is compared with the directly measuredelectrode feed rate in figure 20. The computed feed ratesare generally low, but the agreement with the measuredelectrode feed speeds is nevertheless quite good.

There is no melting rate (thermal) model in the dynamicsimulation. Instead, the rate of increase in volume of thedrops is computed directly from a specified electrode feedrate. Both the measured and the computed feed rates shownin figure 20 were supplied to the model and results for bothinputs are reported below.

Examples of simulations using the dynamic drop-detachment model together with the equivalent experimen-tal images are shown in figures 21–26. The large, experi-mentally observed drops for 180 A operation in figure 22had a classic pendent drop shape which the model could not

118


Figure 25. A simulation of a 280 A constant-current casein Ar–2% O2 shown at 0.5 ms intervals. The horizontalticks are at 0.10 cm intervals and the vertical ticks are at0.05 cm intervals. The feed rate supplied to the model wasthe experimentally measured feed speed.

reproduce. However, the simulated geometries for 220 Aoperation in figure 23 compare well with the experimen-tally observed shapes in figure 24. In particular, the modelreproduced the oblate shape of the drop as it detached fromthe electrode. Finally, the experimental images for 280 Aoperation in figure 26 show the sides of the solid electrodejust beginning to taper – a phenomenon that the modeldoes not reproduce (the diameter of the solid electrode inthe model is constant).

The simulated drop frequencies are shown in figure 27and the simulated drop volumes in figure 28. To obtain therelatively good agreement between the simulations and theexperimental measurements, the model had to be modifiedfrom that used for pulsed current. The necessary changesto the model were

γs2n= [−0.002(IDC− 180)+ 0.2]γs (18a)

fm1 = 0.2fml (18b)

fm2 = 0.2fmu. (18c)

The neck surface-tension coefficientγs2n in (7) was madea linear function of the constant welding currentIDC, witha value of 0.2γs at 180 A and 0 at 280 A. In addition, theaxial forces of magnetic origin applied to the lower andupper centres of mass,fm1 andfm2, respectively, utilizedonly a fraction of the axial forces computed to act on thelower and upper parts of the necking drop,fml and fmu,respectively.

15.79800 15.79900 15.80000 15.80100

15.80200 15.80300 15.80400 15.80500


Just like with the simulations using pulsed current, it ishypothesized that the required reduction ofγs2n relative tothe physical surface-tension coefficientγs accounts for theunmodelled radial magnetic pressure acting on the drop’sneck when the drop detaches. However, such a hypothesissuggests that the magnetic forces in the model should belarger than the axial forcesfml andfmu and yet, as indicatedin (18b) and (18c), the axial magnetic force in the modelhad to be substantially reduced.

Once a detachment begins, the momentum imparted tothe drop is determined by the time required for detachmentand by the forces acting on the drop over this period.The modifications in (18) were also necessary to matchthe simulated drop necking times with the experimentallymeasured necking times – with the necking time definedas the time during which a concavity at the liquid/solidinterface on the electrode is present – and also to match theexperimentally measured drop velocities at the moment ofdetachment.

The simulated and measured necking times are shownin figure 29. The decrease in the measured necking timesat low currents (≤200 A) was most probably due to diffi-culties in measuring the necking time. At lower currents,many small concavities at the liquid/solid interface wouldbriefly appear and then disappear. In these cases, the neck-ing time was defined as the difference between the timewhen the concavity appeared that led directly to a drop de-tachment and the time when the drop detached. The modelwas matched via (18) to necking times at higher currents,for which the measured times were unambiguous. The mea-sured and simulated detached-drop velocities are shown infigure 30. The measured velocities at detachment are fromthe fitted velocity profiles at times 0 and 10 ms in figure 9.

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L A Jones et al

Current (Amps)

180 200 220 240 260 280

Dro

p F

requ

ency

(H

z)

0

10

20

30

40

50

60

Model

Experiment

Figure 27. Drop-detachment frequencies simulated on thebasis of the measured feed rate (M) and the feed ratecomputed from video image measurements (O). Theexperimental measurements from figure 18 are also shown(◦).

Current (Amps)

180 200 220 240 260 280

Dro

p V

olum

e (c

m3 )

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Model

Experiment

Figure 28. Drop volumes simulated on the basis of themeasured feed rate (M) and the feed rate computed fromvideo image measurements (O). The experimentalmeasurements from figure 19 are also shown (◦).

Equations (18) were obtained by iteratively adjustingthe parametersγs2n, fm1 andfm2 over the range of currentsin order to match, simultaneously, the measured volumedata shown in figure 28, the measured necking times shownin figure 29 and the measured initial velocities in figure 30.The experimentally measured necking times were found tobe much longer than had initially been predicted by themodel and, although the drop volumes and initial velocitiescould be matched by adjusting therelative magnitudes ofthe surface-tension neck force and the magnetic forces,allof the forces had to be drastically reduced to the valuesin (18) in order to match the long necking times.

During detachment of a drop, the waist radiusrw (seefigure 5) goes to zero andfs2n, computed using (7), fallsto zero linearly withrw. In contrast, the downwardz-directed magnetic force acting on the upper part of the dropincreases approximately logarithmically towards infinity asrw approaches zero. Ignoring the gravitational force, thesum force,fs2n+ fmu, applied to the drop is that shownin figure 31. The area A under the sum-force curve is thechange in momentum which may be measured via the massand initial velocity of a free drop. However, this change in

Current (Amps)

180 200 220 240 260 280

Nec

king

Tim

e (m

s)

0

5

10

15

20

25

30

35

Model

Experiment

Figure 29. Measured (◦) and simulated necking timesbased on the measured feed rate (M) and the feed ratecomputed from video image measurements (O).

Current (Amps)

180 200 220 240 260 280

Fre

e D

rop

Vel

ocity

(cm

s-1)

-140

-120

-100

-80

-60

-40

-20

0

Model

Model

Velocities 10 ms Later

Velocities at Detachment

Figure 30. Free-drop velocities simulated on the basis ofthe measured feed rate (full lines) and the feed ratecomputed from video image measurements (broken lines).The experimental measurements from figure 9 at times 0and 10 ms are also shown using the same symbols asthose in figure 9.

momentum may be achieved via the rapid application of alarge sum force (figure 31(a)) or the longer application ofa small sum force (figure 31(b)). The long experimentallymeasured necking times strongly suggest the latter case.

After detachment, the drop is accelerated in the arc bythe plasma flow. The experimentally measured accelerationof the drop was shown in section 7 to be virtually constant.Therefore, the difference in the velocities at detachment and10 ms later, shown in figure 30, divided by 10 ms yieldsthe accelerations of the drops due to the plasma flow andgravity. In the dynamic drop-detachment model, the flowvelocity was modelled as being independent of the axialposition and linearly dependent on the arc current,

vpz = −100I (19)

such that, for I = 100 A, vpz = −10 000 cm s−1

(−100 m s−1). The drag force (14) acting on the drop wasthen varied by using an adjustable constantkfp to match theexperimentally measured accelerations, that is,

fp = −kfpCDAP

(ρpv

2pz

2

). (20)

120


(b)

(a)

fs2n

fmu

f fs2n mu+

tA

fs2n

fmu

f fs2n mu+

tA

Figure 31. The momentum change A due to fs2n + fmu. In(a) the momentum change is achieved via a sum of largeforces whereas in (b) the same momentum change isachieved with a sum of smaller forces acting over a longertime.

Current (Amps)

180 200 220 240 260 280

Fre

e D

rop

Acc

eler

atio

n (m

s-2)

-60

-50

-40

-30

-20

-10

0

Plasma accelerationModel

Experiment

Total acceleration(Plasma + Gravity) Experiment

Model

Figure 32. Free-drop accelerations simulated on the basisof the measured feed rate (M) and the feed rate computedfrom video image measurements (O). The data fromfigure 10, computed from experimental measurements, arealso shown.

This axial force on the crown of the drop tends to make thedrop oblate, but is countered by the axial forces near theequator of the drop which tend to make it prolate. Thesecompeting effects were modelled by applying half of theplasma flow force to the upper centre of mass of a freedrop and half of the plasma flow force to the lower centreof mass.

The simulation results and associated experimentalmeasurements of drop acceleration are shown in figure 32.The value ofkfp in (20) was adjusted to match the dropaccelerations at 180 and 280 A and the appropriate valuewas found to bekfp = 0.2. In the middle of thecurrent range, the match between the simulations and the

Current (Amps)

180 200 220 240 260 280

For

ce (×

10-3

N)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Gravitational Force

ExperimentModel

Plasma ForceModel Experiment

Figure 33. Plasma and gravitational forces simulated onthe basis of the measured feed rate (M) and the feed ratecomputed from video image measurements (O). The datafrom figure 11, computed from experimentalmeasurements, are also shown.

f(z)

ru

rl

Id

z = δ

z = 0

r

Figure 34. The axisymmetrical shape of a uniform materialcarrying a current Id.

experiments was poor and this was primarily a result of apoor free-drop volume match in the middle of the currentrange, as can be seen in figure 28. The forces which causedthe accelerations in figure 32 are shown in figure 33. Themodel fails to predict the plasma force peak at 270 Adue to the combination of the model drop volume beinglow (figure 28) and the magnitude of the model dropacceleration also being low (figure 32). If the simulatedfree-drop volumes in figure 28 more closely tracked theexperimentally measured volumes in the middle currentrange, the simulation results in figures 32 and 33 wouldprobably match the measured data quite closely. Thisstrongly suggests that the plasma flow velocity is verynearly a linear function of the current. Also, the smallvalue of kfp required in (20) suggests that, in an argon-rich plasma, the drag coefficientCD is much lower thanexpected: 0.16 rather than 0.80, suggested in section 7.

10. Discussion

For simulations using both pulsed current and constantcurrent, it was found necessary to reduce the magnitude

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L A Jones et al

Io

r h s, ,

r

Figure 35. The discharge of current I0 from a point on aplate bounding a semi-infinite, viscous, incompressible,conducting fluid, as considered in [20]. An example of theresulting fluid flow lines is superimposed.

of the surface tension necking force drastically via theparameterγs2n. It was hypothesized that this adjustmentreflects the radially directed component of the magneticforce that is not included in the model, but which helpscollapse the neck of the drop during detachment. Astriking difference between the dynamic simulations ofpulsed-current and constant-current detachment is thatthe simulations using pulsed current utilized the fullmagnitude of the computedz-directed magnetic force,whereas the simulations using constant current matchedthe experimental measurements only when 20% of thecomputedz-directed magnetic force was applied to thedrop’s centres of mass. Why does the axial magnetic forceseem to be so much less effective at detaching drops inconstant-current welding? It is hypothesized that significantinternal fluid flow develops in the drop during constantcurrent welding, butnot during short-duty-cycle pulsed-current welding. This internal flow annuls the rotationalcomponent of the magnetic force, part of which isz-directed.

Consider an axisymmetrical shape of uniform materialwhich has a constant currentId flowing down through it,as shown in figure 34. The magnetic field inside the shapeis

H = − Idr

2πf 2(z)ιφ (21)

and the current density is

Jf = ∇ ×H = − Idr

πf 3(z)

df (z)

dzιr − Id

πf 2(z)ιz. (22)

Therefore, the magnetic force density acting at each pointin this volume is

F = Jf × µ0H = − µ0I2d r

2π2f 4(z)ιr + µ0I

2d r

2

2π2f 5(z)

df (z)

dzιz.

(23)The z-directed magnetic force on an axisymmetrical bodythat does not emit current, as computed in the companionpaper [2, section 4] using traction vectors, is precisely theιz component in (23) integrated over the volume of the

shape†. For the vector field defined by theιz componentin (23), it is easy to verify that

∇ × (Fzιz) = −∂Fz∂rιφ 6= 0 (24)

and so the z-directed component of the magneticforce density contains both irrotational and rotationalcomponents. In a solid body, these rotational componentscontribute to the totalz-directed magnetic force, or possiblycancel each other out, but in a fluid, they give rise torotational flows.

Until the flow develops, the rotational component ofthe z-directed magnetic force contributes to the total forceon the drop. An approximation of the time requiredfor the flow to develop was given in [20], in which thetemporal evolution of the flow field due to a discharge ofcurrentI0 from a point on a plate bounding a semi-infinite,viscous, incompressible, conducting fluid was computed.The geometry is shown in figure 35. It was found thatsteady-state flow (fully developed flow) within a distancer from the discharge is practically established by the timet = r2/ν, whereν is the kinematic viscosity of the fluid. Byusing this expression, it was found in [14] that steady-stateflow of the arc plasma is achieved within approximately0.2 ms. In contrast, for a welding drop with propertiesρ = 6.25× 103 kg m−3, η = 2.38× 10−3 N s m−2 andr = 0.79 mm (1/32 inch), the timet is approximately1.7 s, which indicates that steady-state flow in the dropis achieved relatively slowly. The current pulses usedin the experiments described in section 8 had relativelyshort lengths of 4 ms. In contrast, for a 180 A constantcurrent, the time between drop detachments (figure 27) wasapproximately 200 ms. Therefore, it is hypothesized thatthe reduction in the calculatedz-directed magnetic forcerequired for simulating a constant welding current is a resultof rotational flow partially developing in the drop. When ashort-duty-cycle pulsed current is used, this flow does nothave time to develop and thus thez-directed magnetic forcewhich acts to detach drops is substantially more effective.

11. Summary

A dynamic model of drop detachment in gas metal arcwelding has been presented for low and moderate weldingcurrents in an argon-rich plasma. When simulationsof an attached drop with no neck were comparedwith experimentally measured drop oscillations, theexperimentally measured damping rates were unexpectedlyhigh. The experimentally measured drop oscillationfrequencies were substantially lower than the frequenciespredicted by the untuned model. Simulations of dropdetachment were compared with extensive experimentalmeasurements of constant-current welding images andwith limited experimental measurements of pulsed-current

† For example, integrating theιz component of (23) over the volume infigure 34 yields equation (20) in [2],

fzlu =∫ δ

0

∫ f (z)

0

µ0I2d r

2

2π2f 5(z)

df (z)

dz2πr dr dz = µ0I

2d

4πln

(ru

rl

).

122


welding images. The comparisons indicated that theaxial magnetic forces calculated using magnetic stresstensors were substantially too high when using constantcurrent – andonly when using constant current. Ahypothesis proposed to explain this result is that, duringthe relatively quiescent development of a drop in constant-current GMAW, rotational flows partially develop in thedrop. These internal flows annul the rotational componentof the magnetic force, part of which is axially directed.With a short-duty-cycle pulsed current, these flows do nothave time to develop and thus the drop responds to theaxial magnetic force much like a solid body. Since it isnot known how to observe the flows inside a welding drop,this hypothesis will have to be confirmed by finite-elementsimulations of the flow field.

Acknowledgments

Support for this work was provided by the United StatesDepartment of Energy, Office of Basic Energy Sciences.This paper was extracted from chapters 4 and 5 of the firstauthor’s doctoral thesis [8] at the Department of ElectricalEngineering and Computer Science at the MassachusettsInstitute of Technology.

References

[1] Amson J C 1965 Lorentz force in the molten tip of an arcelectrodeBr. J. Appl. Phys.16 1169–79

[2] Jones L A, Eagar T W and Lang J H 1998 Magnetic forcesacting on molten drops in gas metal arc weldingJ. Phys.D: Appl. Phys.31 93–106

[3] Clark M M 1988 Drop breakup in a turbulent flow – I.Conceptual and modeling considerationsChem. Eng. Sci.43 671–9

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a dynamic model from a gas metal electrode · 2011-12-12 · a dynamic model of drop detachment...

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