J. Phys. D: Appl. Phys. 31 (1998) 93–106. Printed in the UK PII: S0022-3727(98)83362-7

Magnetic forces acting on moltendrops in gas metal arc welding

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. In gas metal arc welding, magnetic forces arising from the interaction ofthe welding current with its own magnetic field play an important role in thedetachment of drops from the molten welding electrode. These forces drive thedynamic evolution of the drop and also depend on the instantaneous shape of thedrop. In this paper, experimentally observed manifestations of magnetic forces areshown and a technique for approximating the temporal evolution of the axialmagnetic force from experimentally measured drop shapes is reported. Thetechnique provides quantitative data illustrating the large increase in the magneticforces when a drop detaches from the electrode.

1. Introduction

Significant magnetic forces generated by the weldingcurrent act on drops detaching from a gas metal arcwelding (GMAW) electrode. At lower welding currents,the magnetic forces act to detach liquid drops from asolid electrode, which separates this process from the muchstudied dynamics of current-carrying liquid metal columns[1]. At very high welding currents, a column of liquidmetal forms and the analyses of conducting liquid columndynamics are applicable.

For lower currents, the seminal analyses of magneticforces acting on drops detaching from a welding electrodeare those of Maecker [2], published in 1955, Greene [3],published in 1960, and Amson [4], published in 1965.Maecker first computed the axial force acting on weldingdrops due to the divergence of current in the drop. Greenefirst calculated this force for spherical shapes with a uniformcurrent density emerging from the surface of the drop,whereas Amson showed how the calculation could beperformed for any drop shape and any surface-currentemission density. Amson obtained the same results asGreene did for the special case of spherical drops and auniform surface-current emission density. All subsequentstudies of drop detachment have used either the calculationsof Maecker, in which the axial magnetic force acting onthe drop was computed as the force acting on a frustumof current, or the calculations of Greene, in which thebottom of the frustum is spherical, following the bottomof a spherical drop.

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

Fluid motion inside the welding drop does not directlyaffect the configuration of the magnetic field. Only theshape of the drop and the current path are important, eventhough the interaction between the current and its ownmagnetic field generates fluid flow in the drop. On the basisof this result, appropriate shapes were fitted to experimentalimages of drops detaching during current pulses, and theaxial magnetic forces consistent with these shapes andconsistent with an approximate current path within the dropwere computed. This work is the first in which the generalresults in [4] have been applied to geometries directlymeasured from experimental images, thereby yielding amore accurate account of the axial magnetic forces.

In a companion paper [5], the application of axialmagnetic forces in adynamic model is detailed. Thedynamic model includes the formation of a neck duringdrop detachment – a phenomenon that substantiallyincreases the magnetic forces acting on the drop. Resultsfrom this dynamic drop-detachment model are comparedwith various experiments and are used to assess theeffectiveness of axial magnetic forces on detaching dropsunder various conditions.

2. Effects in GMAW

Over a wide range of conditions, the effects of magneticforces may be seen acting on drops detaching from awelding electrode. A drop detaching from a gas metalarc welding electrode at 260 A and 29 V is shown infigure 1. For the 1/16 inch (1.6 mm) diameter ER70S-3electrode wire shown in an Ar–2% O2 plasma, this currentis near the upper end of the globular transfer region. The

0022-3727/98/010093+14$19.50 c© 1998 IOP Publishing Ltd 93

L A Jones et al

15.42100 15.42200 15.42300 15.42400

15.42500 15.42600 15.42700 15.42800

Figure 1. Drop detachment with constant current 260 Aand 29 V electrode positive. The electrode is 1/16 inch(1.6 mm) diameter ER70S-3 wire in an Ar–2% O2 plasma.Note the vertically flattened (oblate) shape of the dropduring and after detachment. The recording time inseconds is shown above each image.

distinctly flattened (oblate) shape of the drop is a result ofthe magnetic forces acting on the drop. When a drop ofapproximately the same volume detaches under very low-current conditions, the drop is largely spherical and notflattened; see figure 16 later, in which the current at themoment of detachment is 40 A. Also, although it is notapparent from the images in figure 1, the drops accelerateoff the end of the electrode at substantially more than theacceleration due to gravity. This excess acceleration is dueto magnetic forces and measurements of it are presented inthe companion paper [5].

The magnetic forces arise from the interaction of thewelding current with its own magnetic field. If the currentdiverges in the drop, then downward forces act on the fluidin the drop, whereas if the current converges in the drop,then upward forces act on the fluid in the drop. Notethat, unlike gravity, which acts uniformly in the verticaldirection on the fluid (assuming that the density of thefluid is spatially uniform) and is an irrotational force,the magnetic force does not act uniformly and there is arotational component of force acting on the fluid.

3. The independence from fluid flow

In a drop of metal in GMAW, the magnetic forces acting onit depend only on the geometry of the current path and noton the velocity of the fluid in the drop at each moment intime. This decoupling from the fluid velocity dramaticallysimplifies the calculation of the magnetic forces because itis then only necessary to measure the shape of the current

path at each moment in time in order to calculate the totalmagnetic force acting on the drop. However, this force isspatially distributed in the fluid and computing the resultingfluid motion is still a difficult task.

To show that the magnetic forces depend only on theshape of the current path and not on the velocity of thefluid in gas metal arc welding, it is appropriate to beginwith an induction equation which may be derived directlyfrom the magneto-quasi-static approximations of Maxwell’sequations [6]. If the electrical conductivityσ and themagnetic permeabilityµ are constant and uniform, thenthe magnetic fluxB and fluid velocityv may be related by

1

µσ∇2B = ∂B

∂t−∇ × (v ×B). (1)

Writing this equation in terms of the dimensionlessvariables (t , v, x, y, z),

t = tτ v = vu (x, y, z) = (x, y, z)` (2)whereτ , u and` are a characteristic system time constant,velocity and length, respectively, yields

∇2B = τmτ

∂B

∂t− Rm∇ × (v ×B). (3)

The time constantτm = µσ`2 (4)

is the magnetic diffusion time and the dimensionless value

Rm = µσu` (5)is the magnetic Reynolds number. The ratioτm/τ in (3)indicates the speed of magnetic diffusion relative to thetime scale of interest. The magnetic Reynolds numberRmprovides a measure of the relative importance of convectionversus magnetic diffusion. In some physical processes, onemight expect either or both of these measures to be muchless than unity; that is, the magnetic diffusion time is veryshort with respect to the time scale of interest and/or thetime scale of the material’s motion.

The ratioτm/τ and the magnetic Reynolds numberRmfor a drop of molten steel in a GMAW arc may be calculatedusing the following values:

µ = 4π × 10−7 H m−1 (6a)σ = 1.4× 106 −1 m−1 (6b)` = 1.6× 10−3 m (6c)u = 0.63 m s−1 (6d)τ = 0.01 s. (6e)

The characteristic length̀ is the electrode diameter andthe characteristic velocityu is the peak velocity of a1 mm amplitude, 100 Hz sine wave, which is a sinewave in excess of the maximum oscillation amplitude andfrequency of drops observed on the given electrode size.The characteristic timeτ is the period of 100 Hz. Theresults are

τm/τ = 0.0005 (7a)Rm = 0.0018 (7b)

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Magnetic forces acting on molten drops

and therefore both terms on the right-hand side in (3) maybe ignored.

On the time scale of interest, the diffusion of theB field throughout the drop is essentially instantaneousand the very small magnetic Reynolds number indicatesthat the magnetic diffusion process is much faster thanfluid convection in the drop. For uniform permeabilityµthroughout the material (as had previously been assumed),(3) reduces to

∇2H = 0 (8)whereH is the magnetic field intensity. Therefore, in a gasmetal arc welding electrode the magnetic field is unaffectedby the fluid velocity in the drop. The distribution ofH isdetermined by the instantaneous geometry of the currentpath in the drop and, assuming uniform fluid conductivity(as had previously been assumed), the current path isdetermined by factors other than the fluid velocity.

4. The magnetic force on a generalized dropshape

In [4], the magnetic stress tensor was used to calculate themagnetic force acting on a generalized axisymmetrical dropshape. In particular, the total axial magnetic force due tocurrent divergence was calculated. The motion of the fluidin the drop interacts with the magnetic field only in thatit affects the resulting shape of the drop and hence theshape of the current path. Therefore, if the shape of thecurrent path is known at each instant, the magnetic stresstensor may be used to calculate the total magnetic force onthe drop (the sum of the irrotational and rotational forces).The derivation in [4] is briefly recounted here to establishnotation, introduce intermediate results and prepare for theapplication of the results to shapes other than spheroids.

In cylindrical coordinates(r, φ, z), consider a general-ized pendent drop shape, as shown in figure 2. A moltendrop that is axisymmetrical about the verticalz axis is pen-dent on a solid electrode. The surface of the drop is P andthe liquid/solid interface between the electrode and dropis modelled as a spatially discrete boundary at surface Q.The welding currentI flows axisymmetrically downwardsin the solid electrode and some or all of it continues into thedrop (Id) and emerges from surface P of the liquid drop. Ifnot all of the current continues into the drop, the remainder(I − Id) emerges from the surface of the solid electrode.

Because the current is modelled as always flowingaxisymmetrically, the magnetic field has only an azimuthalcomponent and is easily determined from Ampère’s law tobe

H = I (r, z)2πr

ι̂φ (9)

where I (r, z) is the current bounded by a hoop whichpasses through point(r, z). The z-directed component ofthe magnetic force is

fz =∫

Qτz da +

∫Pτz da (10)

whereτz is thez-directed component of the ‘traction’ vectorτ . The magnitude of the traction vector is

|τ | = 12µ0|H|2 = µ0

8π2I (r, z)2

r2. (11)

The magnetic field at the drop’s surface is azimuthal andthus always tangential to the surface and, since the magneticfield vector bisects the angle between the traction andsurface normal vectors [7], the traction vector is alwaysperpendicular to the surface and opposite to the surfacenormal vectorn, as shown in figure 2. Thez-directedcomponent of the traction may therefore be determinedfrom the surface geometry, as shown in figure 3. Forsurface Q,

τz = −drdu|τ | (12)

and a differential element of surface area is

da = 2πr(u) du. (13)

Similarly, for surface P,

τz = drds|τ | (14)

andda = 2πr(s) ds. (15)

Substituting (11) into (12) and (14), and the resultingexpressions into (10), yields

fz = −µ0π∫ U

0

(I (u)

2π

)2 drdu

du

r(u)

+µ0π∫ S

0

(I (s)

2π

)2 drds

ds

r(s). (16)

The current bounded by a hoop of radiusr(u) or r(s) onsurface Q or P, respectively, is equal to that emerging fromthe surface up to that point. Using (13) and (15),

I (u) =∫ u

0j(u′)

2πr(u′)

du′ (17)

I (s) =∫ s

0j(s ′)

2πr(s ′)

ds ′ (18)

where j(u′)

and j (s ′) are the surface-current emissiondensity functions along surfaces Q and P, respectively.Substituting (17) and (18) into (16) yields

fz = −µ0π∫ U

0

(∫ u0j (u′)r(u′) du′

)2 drdu

du

r(u)

+µ0π∫ S

0

(∫ s0j (s ′)r(s ′) ds ′

)2 drds

ds

r(s)(19)

which gives thez-directed force on the drop in terms ofany liquid/solid boundary Q, any drop surface profile Pand any surface-current emission density functions alongthese surfaces.

For any portion of the surface P where the surface-current emission density is zero, the contribution of thatportion to thez-directed magnetic force is independent ofthe surface profile and the current path, as illustrated in

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L A Jones et al

u = Us = S

s = 0

u=0

z = Z

r(s)

r(u)

Surface P

Surface Q

UnmeltedElectrodeWire

MeltingRegion

MoltenDrop

z = 0

z

s

u

re

nP

nQ

tQ

tP

Id

I

Figure 2. A generalized pendent drop shape. Adaptedfrom [4, figure 1].

s drdzd

sudzdu

dr

t z tQ

nQ

nP

tP t zQ P

Figure 3. The z -directed components of the traction.

figure 4. If the lower boundary of the portion not emittingcurrent is atz = ` and the upper boundary is atz = u, thesecond term of (19) for this portion of surfaceS reduces to

fzlu = µ0I2d

4πln

(ru

rl

)(20)

where Id is the current emerging below the liquid/solidboundary. This expression is independent of the dropsurface profile and the current path in the region betweenz = ` andz = u. See [4] for a proof of this result.

If surface Q in figure 2 is modelled as a cone and thesurface-current emission density functionj (u) is assumedto be uniform†, then the first term of (19) reduces to

fz3 = − µ016π

I 2d . (21)

This result is independent of the angle of the cone. Thedetails of the liquid/solid boundary inside the electrode are

† Surface Q lies entirely inside the electrode which is made of a high-conductivity material. The potential along surface Q is relatively uniformand thus the current densityj (u) across Q is relatively uniform.

Id

Zerocurrentemission

Non-zerocurrent emission

ru

rl

Magnetic forceindependent of

the currentpath.

Magnetic forceindependent of

the surfaceshape.

Magnetic forcedependent on the

surface profile and thesurface current density.

z = l

z = u

I

Figure 4. The z -directed magnetic forces on regions thatemit and do not emit current.

fortuitously not required and the liquid/solid boundary maybe modelled simply as a disc in thez plane. Therefore, inorder to calculate the magnetic force acting on a drop it isonly necessary to know the external drop surface profile (theprofile of surface P) over which current is emitted and thesurface-current emission density function along this profile.

To evaluate the second term of (19) when the surface-current emission density is not zero, the region where thecurrent emerges from the drop must be defined. Considerthe special case shown in figure 5. The drop is modelledas a truncated sphere with radiusa. The outer surfaceof the drop (surface P) emits current only up to angle8(surface①). Above this point, current is not emitted and sothe shape of the current path (surface②) is not important.Its contribution to thez-directed magnetic force may becalculated using (20),

fz2 = µ0I2d

4πln( rea sin8

). (22)

The contribution from the liquid/solid boundary (surface③)is given by (21). If the surface-current emission densityj (s) is uniform along surface①, then the contribution fromsurface①, expressed in terms of angleφ, is

fz1 = µ0π∫ 8

0

(∫ φ0

Id

2πa2(1− cos8)a2 sinφ′dφ′

)2× cotφ dφ (23)

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Magnetic forces acting on molten drops

r a( ) sinf f=

dr a d= cosf f

Id

I

z

a

re

r( )f 1

2

3

F

f

ConductionZone

S

x

Surface P

Surface Q

Figure 5. The current path in a spherical drop.

which, upon integration, yields

fz1 = µ0I2d

4π

[1

2+ 1

1− cos8 +2

(1− cos8)2

× ln(

1+ cos82

)]. (24)

The expressions for surfaces① (24), ② (22) and③ (21)must be summed to obtain thez-directed component of themagnetic force:

fz = fz1+ fz2+ fz3 = µ0I2d

4π

[1

4− ln

(a sin8

re

)+ 1

1− cos8 −2

(1− cos8)2 ln(

2

1+ cos8)]. (25)

This expression for the special case of a spherical drop anduniform surface-current emission density was obtained bothin [3] and in [4] (both in [3] and in [4] the negative of (25)was obtained because the absolute value of thez-directedmagnetic force was calculated) and it is commonly used inthe literature to compute the axial magnetic force acting ona welding drop.

5. Neck shapes

When a drop attempts to detach from a solid electrode,a neck forms. The current density in the drop’s neckincreases and the divergence of the current increases. Bothphenomena cause the magnetic force acting on the drop to

¢z

z

¢x

x

c

aVsu

Vsl

z

g

re

Figure 6. The truncated-ellipsoid model of a pendent drop.

increase and result in a measurable acceleration of the dropupon detachment. The time during which a neck initiatesand collapses is short compared with the total growth timeof the drop, but it is during this brief time that the magneticforces are most important.

In the previous section, it was shown that tocompute the magnetic force acting on the drop it isonly necessary to know the external drop surface profileand the surface-current emission density function alongthis profile. Measurement of the surface-current emissiondensity function along the drop surface profile is anextremely difficult task and is discussed in section 6. Thedrop surface profile, however, can be easily measured fromimages of the drop. In this section, a set of shapes used tomodel necking drops is introduced.

The surface profile of a drop where a neck has notyet formed may be modelled by using truncated ellipsoids.Figure 6 illustrates a drop profile constructed from anaxisymmetrical ellipsoid that is truncated at the top in thez plane by the cylindrical solid electrode. The unprimedx–z frame is the stationary laboratory frame and theprimed x ′–z′ frame is fixed to the geometrical centre ofthe ellipsoid, which is always located on the plane of theellipsoid’s equator. The ellipsoid is uniquely specified byits horizontal and vertical semiaxes,a ≥ 0 and c ≥ 0,respectively, and the angleζ at which the ellipsoid istruncated by the solid electrode.

For cases in which a neck has formed, the lower partof the drop is well modelled by using truncated ellipsoids,but a shape is needed to model the neck which connectsthe lower part of the drop to the solid electrode. Volumesformed by rotating a third-order polynomial about thez axiswere chosen because these shapes are completely definedby the boundary conditions at the top of the truncatedellipsoid and the bottom of the electrode. An exampleof an experimentally observed necking drop, modelled by

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L A Jones et al

CubicPolynomial

TruncatedEllipse

Figure 7. An experimentally observed necking drop andthe truncated-ellipsoid/polynomial-volume model. (The dropimage is from figure 16, time 13.096 00 s.)

using a truncated ellipsoid and a connecting polynomial-volume neck, is shown in figure 7. The geometry of thepolynomial-volume neck is shown in figure 8.

Four boundary conditions exist such that the fourcoefficients of the cubic polynomial are uniquely deter-mined. At the top of the truncated ellipsoid (z′′ = 0), thepolynomial is matched to the radius, slope and curvatureof the truncated ellipsoid, yielding three coefficients†. Atthe bottom of the solid electrode (z′′ = δ), the value ofthe polynomial is matched to the electrode radius, yieldingthe fourth coefficient. Thus, given a truncated ellipsoid andthe electrode position, the connecting neck can always bereconstructed.

6. Current paths

Although the shape profiles of drops detaching from aGMAW electrode can be easily measured from imagesof the electrode, measuring the surface-current emissiondensity functions along these profiles is a difficult task.No experimental measurements of the current density ona GMAW electrode are available in the literature due tothe difficulty of making such measurements in the harshenvironment of the arc next to the free surface of a drop.The average current density on a 1/16 inch (1.6 mm)diameter welding electrode is of the order of 107 A m−2,but a more precise description of where the current emergesfrom the drop and how the current density is distributedover the drop is lacking.

The problem is complicated by the fact that, eventhough significant brightness about the drop is observedunder certain conditions, this brightness is more an indicator

† Matching the radius and curvature of the ellipse and polynomial attheir junction mimics matching the excess pressure there; however, excesspressure has little meaning in the context of the constrained – rather thanfree – surface shapes used here.

′′z

z

′x

x

rw

Vsl

Vpu

Vpl

′′x′z

γVsu

re

rt

Truncated Ellipsoid

Polynomial Volume

¢¢ =z 0

¢ =z 0

z = 0

(Truncated-Ellipsoid/ Polynomial-Volume Intersection)

(Arbitrary Fixed Location)

(Truncated-Ellipsoid Equator)

z z= e(Liquid/Solid Boundary)

(Waist)

d

a

c

Figure 8. The truncated-ellipsoid/polynomial-volume modelof a necking drop.

of the arc temperature and composition than it is of theenvelope of current flow. In [8], the current density andtemperature in a tungsten arc were thoroughly measured andit was found that the current-carrying region near the anodeextended substantially beyond the arc boundary suggestedby the envelope of brightness. The brightness envelope inphotographs was found to correspond approximately to the10 000◦C isotherm.

The region over which current emerges from a GMAWdrop is greatly affected by the shielding gas used. If CO2is used as the shielding gas, the anode spot is denselyconcentrated on the bottom of the electrode over a widecurrent range. If Ar–2% O2 is used, then typically the entiredrop is immersed in a bright portion of the plasma. In [9],this well-known observation was explained by performinga static-geometry finite-element analysis of a GMAW arc.It was found that the higher values of the heat capacity,heat conductivity and radiating capacity of CO2 resultedin a plasma with smaller lateral dimensions, higher currentdensity and higher electric field strengths than when Ar–2%O2 was used as a shielding gas.

For the present work, no current density measurementswere attempted and plausible current paths were chosenfor the shapes shown in figures 6 and 8 on the basisof general observations of the plasma brightness over thecourse of numerous experiments using Ar–2% O2 andER70S-3 electrode wire. Although the arc brightness isonly a marginal guide to current flow in the plasma, severalsituations were commonly observed:

(i) The plasma brightness, indicating areas of densercurrent flow, appeared to envelop drops completely under

98

Magnetic forces acting on molten drops

(a) (b)

Figure 9. The plasma coverage of a drop forming in anAr–2% O2 plasma at (a) 220 A and (b) 240 A.

(a) (b)

Figure 10. The plasma coverage during necking of a dropin an Ar–2% O2 plasma at 240 A. Two different drops areshown in (a) and (b).

a variety of conditions. Typical drops are shown in figure 9.Although the brightness was somewhat diminished near thesolid electrode, it was otherwise quite uniform. The plasmawas somewhat brighter at the bottom of the drops, possiblybecause of current emitted into the plasma from the upperpart of the drops.

(ii) When a drop necked off from the electrode, the arcbrightness was typically confined to the portion of the dropbelow the narrowest point of the neck (the waist). Figure 10shows two drops during necking. It is believed that thebright spots above the neck waists were arc light beingreflected by the liquid metal surfaces rather than evidenceof asymmetrical current emission.

(a) (b)

Figure 11. The plasma flow around a free drop in anAr–2% O2 plasma at 220 A. A single drop (a) is shown3 ms later in (b).

(iii) It appears that little arc current, maybe no currentat all, flowed through the metal drops while they were inflight in the arc. In figure 11, the current which heats theplasma appears to be flowing around the drop (this is evenmore evident when the drop is viewed in motion). Belowthe free drop, there was no arc brightness such as wouldbe expected if significant current were flowing throughthe drop. Furthermore, there was no visual evidencethat the free drops were subjected to magnetic stresseswhich would have resulted from current flowing throughthem†. The conductivity of the drop was of the orderof 200 times that of the plasma, but the apparent lack ofcurrent flowing through the drop may have been because theconditions necessary for cold-cathode electron emission didnot exist on the surface of the drop. Therefore, the currentexperienced the drop as an open circuit and flowed aroundit.

For the present model of a drop in an Ar–2% O2 plasma,the welding current was modelled as emitting uniformlyfrom the entire surface of the truncated-ellipsoid shape(figure 6) and uniformly from the surface of the truncated-ellipsoid/polynomial-volume shape below the waist of thepolynomial volume (figure 8). The effect of modellinga more constricted current path, appropriate for a CO2plasma, is considered at the end of section 8.

7. The magnetic force on model drop shapes

The z-directed component of the magnetic force must bedetermined for truncated-ellipsoid and polynomial-volume

† If a drop has current flowing through it and the drop has a higherconductivity than the surrounding fluid, it would tend to elongate (becomeprolate). If its conductivity were lower than that of the surrounding fluid,then it would tend to flatten (become oblate) [10]. Drops traversing aGMAW arc exhibit neither tendency.

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L A Jones et al

rc1

rcs

r

n

s = 0

s = S

t

I I Is d p= -

IdSurface Area Sp

Figure 12. The frustum approximation of a polynomialvolume below its waist.

I

Idrcs

1

5

2

4

3

Is1a

F = z

c

Figure 13. Magnetic stress tensor surfaces ①–⑤ fortruncated-ellipsoid shapes. The shading indicates the areaof surface-current emission for the Ar–2% O2 model.

surface profiles. In particular, the traction vector must beintegrated over these shapes as in the second term of thesum in (19)

µ0π

∫ S0

(∫ s0j (s ′)r(s ′) ds ′

)2 drds

ds

r(s). (26)

It must be remembered that this term has meaning onlywhen it is combined with the contributions from all of thesurfaces forming a closed surface. For uniformj (s) and aspherical surface profile, both the inner squared integral andthe outer integral may be evaluated in closed form (see (23)and (24)). However, there are very few other suitable dropshapes for which both integrals may be evaluated in closedform. It is shown below that, for an ellipsoidal surface, theinner squared integral may be evaluated in closed form, butthe outer integral cannot. For a polynomial-volume surface,neither integral can be evaluated in closed form.

I

Id

rcs

1

5

2

4

3

6

aIs1

rc1

F

c

Figure 14. Magnetic stress tensor surfaces ①–⑥ fortruncated-ellipsoid/polynomial-volume shapes. The shadingindicates the area of surface-current emission for theAr–2% O2 model.

7.1. The magnetic force on ellipsoids

For uniform surface-current emission, the inner squaredintegral in (26) becomes

Ms =∫ s

0j (s ′)r(s ′) ds ′ = J

∫ s0r(s ′) ds ′. (27)

The task of evaluating this integral for a prolate ellipsoidalsurface is eased by using prolate spheroidal coordinates(η, θ, ψ) [11]. Substituting the differential element ofsurface area, expressed in metrics for prolate spheroidalcoordinates, into (27), together withr = f sinhη sinθ ,wheref = (c2− a2)1/2, yields

Msp= Jf 2 sinh2 η∫ θ

0sinθ ′

(1− 1

sinh2 ηsin2 θ ′

)1/2dθ ′.

(28)

Substituting the definition of constantη in prolatespheroidal coordinates,η = sinh−1[a/(c2− a2)1/2], yields

Msp= Ja2∫ θ

0sinθ ′(1+ e2p sin2 θ ′)1/2 dθ ′ (29)

whereep = (c2 − a2)1/2/a. This integral has the closed-form solution [12, No 437.3]

Msp= Ja2[

1

2− cosθ

2(1+ e2p sin2 θ)1/2

−1+ e2p

2epsin−1

(ep cosθ

(1+ e2p)1/2)

+1+ e2p

2epsin−1

(ep

(1+ e2p)1/2)]. (30)

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Magnetic forces acting on molten drops

11.26250 11.26275 11.26300 11.26325 11.26350 11.26375

11.26400 11.26425 11.26450 11.26475 11.26500 11.26525

11.26550 11.26575 11.26600 11.26625 11.26650 11.26675

11.26700 11.26725 11.26750 11.26775 11.26800 11.26825

11.26850 11.26875 11.26900 11.26925 11.26950 11.26975

Figure 15. Drop detachment with a 330 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.

A similar calculation using oblate spheroidal coordinatesyields

Msb= Ja2(

1

2− cosθ

2(1− e2b sin2 θ)1/2

−1− e2b

2ebln

[eb cosθ + (1− e2b sin2 θ)1/2

]+1− e

2b

2ebln(eb+ 1)

)(31)

whereeb = (a2− c2)1/2/a.Sincer = a sinθ and dr/dθ = a cosθ , the integral of

the traction vector over an ellipsoidal surface becomes

fzs= µ0π∫ 2

0M2s (θ) cotθ dθ (32)

where2 = tan−1

( ca

tan8). (33)

The angle8 specifies the portion of the truncated ellipsoidfrom which current is emitted, as shown in figure 5 for asphere (a = c). No closed-form solution of (32) exists andso it is necessary to evaluate it numerically. Care mustbe taken in performing the integration since the integral isimproper, but convergent, atθ = 0.

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 16. The same conditions as those in figure 15,except with a 290 A pulse.

7.2. The magnetic force on polynomial volumes

Neither of the integrals in (26) can be evaluated inclosed form for a polynomial volume. In the dynamicmodel presented in the companion paper [5], (26) mustbe evaluated a large number of times and computingboth integrals numerically would be very time consumingsince they represent, in a discrete sense, the sum of asquared sum. Therefore, to speed up the simulations inthe companion paper, the polynomial volumes below theirwaist were approximated by frustums – line segments inprofile as shown in figure 12. Not surprisingly, the integralscan be evaluated in closed form for line segments; however,it is easier to start with the form of the second summed termon the right-hand side in (16). For uniform surface-currentemission density,

I (r) = Ip(r2− r2csr2c1− r2cs

)+ Is. (34)

Therefore, the integral of thez-directed component of thetraction over a linear surface profile emitting a uniformcurrent density is

fzp = µ04π

∫ rc1rcs

[Ip

(r2− r2csr2c1− r2cs

)+ Is

]2dr

r. (35)

Note that ifIp = 0, this expression reduces to (20), whereasif rcs= 0 andIs = 0, this expression reduces to the negative

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L A Jones et al

of (21), as expected. Performing the integration yields

fzp = µ04π

{I 2p

4(r2c1− r2cs)2[(r4c1− r4cs)− 4r2cs(r2c1− r2cs)

+4r4cs ln(rc1

rcs

)]+ IpIsr2c1− r2cs

[r2c1− r2cs

−2r2cs ln(rc1

rcs

)]+ I 2s ln

(rc1

rcs

)}. (36)

The model of uniform surface-current emission density,both from the polynomial volume and from the truncatedellipsoid (as in (27)), allowsIs andIp to be easily computeda priori on the basis of fractions of the total surface areaof emission.

7.3. Model calculations

In the surface-current emission model for an Ar–2% O2plasma, current is emitted uniformly from the entire surfaceof the drop, as shown in figure 13. The currentI flowingdown through the solid electrode is modelled as equal tothe currentId emitted by the liquid drop. In other words, nocurrent is emitted directly from the solid electrode. Currentemission from the electrode could be modelled by allowingId < I .

The magnetic forces acting on the parts of the dropmodel above and below its equator may be computedseparately. The axial magnetic forcefml acting on thelower part of the drop may be computed by integrating thez-directed component of the traction vector over surfaces① (the area of current emission below the drop equator)and ②. Only a portion of the total drop currentId passesthrough surface②. The model of uniform surface-currentemission density may be used to obtain the currentIs1passing through surface②,

Is1= Id(

Sl

Sl + Su

)(37)

whereSl is the surface area of the ellipsoid below its equator(0 < ζ ′ < π/2) andSu is the surface area of the ellipsoidabove its equator(π/2 < ζ ′ < 8). The currentIs1 is thenused in (21) rather than the currentId flowing into the dropfrom the solid electrode.

The axial magnetic forcefmu acting on the upper partof the drop may be computed by integrating thez-directedcomponent of the traction vector over surfaces③–⑤. Thecontributions from surfaces③ and ⑤ do not cancel sincethe currents passing through these surfaces are differentand hence the magnitudes of the traction vectors on thesesurfaces are not equal. The contribution from surface④(the area of current emission above the drop equator) iscomputed using (32) with the integral limitsπ/2 and2.

Computation of the magnetic forces acting on themodel of a drop with a neck is slightly more complicated.Figure 14 shows current emerging from the entire surfaceof the truncated ellipsoid and from the polynomial-volumesurface below its waist. The model of uniform surface-current emission may be used to calculate the portion ofthe drop currentId that is emitted through the polynomial

2re

2rw

2a

c

c + +g d

Figure 17. Necking drop measurements. The drop imageis from figure 16, time 13.096 00 s.

volume, Ip, and the portion of the drop current that isemitted through the truncated ellipsoid,Is,

Ip = Id(

Sp

Ss+ Sp

)(38)

Is = Id− Ip (39)where Sp is the surface area of the polynomial volumethat is emitting current andSs is the surface area of thetruncated ellipsoid. These currents are used in (36) tocompute the integral of thez-directed component of tractionover surface⑤. The z-directed magnetic forcefmu actingon the upper part of the necking drop in figure 14 may becomputed by integrating thez-directed component of thetraction vector over surfaces③–⑥ using equations derivedfrom (21) (usingIs1 and multiplying by−1), (32) (with theintegral limitsπ/2 and2), (36), and (21), respectively.

8. Magnetic forces on measured drop shapes

The reactions of welding drops to sharp pulses of currentwere captured and measured using laser backlighting [13]and high-speed videography. Model drop shapes (truncatedellipsoids and polynomial volumes) were fitted to themeasurements and the equations derived in the previoussection were used to calculate thez-directed magneticforces acting on these shapes. Examples from the set ofdrops studied are shown in figures 15 and 16. The imagesin these figures are at 250µs intervals. The electrode is1/16 inch (1.6 mm) diameter ER70S-3 wire shielded withAr–2% O2 gas flowing at 50 cfh (1.4 m3 h−1) through a3/4 inch (19.1 mm) diameter gas cup. In all of the casesstudied, the voltage set point was 18 V, the base currentwas 40 A and the pulsing frequency was 5 Hz with a 2%duty cycle (4 ms pulse width).

It was noted in section 6 that the luminosity of thearc is only a marginal guide for determining the current

102

Magnetic forces acting on molten drops

Cur

rent

(A

)

0

50

100

150

200

250

300

350

Image Number(250 Ps/image)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

z -di

rect

ed M

agne

tic F

orce

(× 10

-3 N

)

-7

-6

-5

-4

-3

-2

-1

0

Figure 18. The measured current and the computed total z -directed magnetic force acting on model shapes in an Ar–2% O2plasma during current pulses.

path in the drop. The images in figures 15 and 16 providestriking evidence of this fact. Measurements of the weldingcurrent synchronized relative to the images revealed that thewelding current was at its maximum value by the fourthframe in the first row. However, the arc light did not attainits maximum brightness until 1.25–1.75 ms (5–7 images)later. In contrast, the arc lost its brightness as fast as thecurrent returned to its base level of 40 A in the secondimage in the fourth row. The rise and fall times of thecurrent pulses were virtually identical, but the arc lightsuggests long rise times and short fall times.

As shown in figure 17, various parameters weremeasured from these images, including the electrodediameter (2re), the overall drop length from the tip ofthe drop to the liquid/solid boundary (c + γ + δ), thewaist diameter when a neck waist could be seen (2rw),the maximum diameter of the drop (2a) and the ellipsoidparameter c. If a neck waist had not yet formed,then a truncated-ellipsoid drop shape was computed from

the measurements. Once a neck had formed, then atruncated ellipsoid with a connecting polynomial-volumeneck was computed from the measurements. Such shapeswere computed from the measurements via a nonlinearsolution for the unknown parameterζ of the truncated-ellipsoid portion of the shape†. The model shapes cannotaccount for any axial asymmetry and, when performingthe measurements shown in figure 17 on asymmetricaldrops (such as the drop in figure 15), the measurementswere skewed slightly in order to obtain the best fit of thesymmetrical model shapes to the asymmetrical drops.

The model shapes computed from the image measure-ments were then used to compute the axial magnetic forcesacting on the upper and lower parts of the shapes. Fortimes before a neck had formed, the upper and lower parts

† A truncated ellipsoid with a connecting polynomial-volume neck isuniquely specified bya, c, ζ , re and ze, the latter two measurementsbeing the electrode radius and position, respectively. Onlyζ could not bemeasured directly from the images.

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L A Jones et al

Image Number(250 Ps/image)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

z -di

rect

ed M

agne

tic F

orce

(× 10

-3 N

)

-7

-6

-5

-4

-3

-2

-1

0

Figure 19. The upper (M) and lower (O) components of the z -directed magnetic force, fmu and fml, respectively, and theirsum (◦) for a 330 A current pulse (figure 15).

Image Number(250 Ps/image)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

z -di

rect

ed M

agne

tic F

orce

(× 10

-3 N

)

-5

-4

-3

-2

-1

0

1

2

Figure 20. The upper (M) and lower (O) components of the z -directed magnetic force, fmu and fml, respectively, and theirsum (◦) for a 330 A current pulse and the CO2 current emission model.were the volume of the ellipsoid below its equator and thetruncated volume of the ellipsoid above its equator. Whena neck had formed, the lower part of the shape was thevolume below the ellipsoid’s equator and the upper part ofthe shape was the volume between the ellipsoid’s equatorand the waist of the polynomial volume.

The current pulses applied to the drops are shown inthe upper graph of figure 18. The totalz-directed magneticforce acting on the model shapes, that is, the sum of thez-directed magnetic forces acting on the upper and lowerparts of the model shapes, is shown in the lower graph. The

data have been plotted such that all of the current pulsesbegin at the same time. The symbols indicate samples ofthe welding current, and they may be correlated with thedrop images in figures 15 and 16. Data collection occurredat 2 kHz and the images were collected at 4 kHz. Therefore,each data point corresponds to every other image (500µsintervals) beginning with the second image in the first rowof figures 15 and 16.

The results in the lower graph of figure 18 show that, formost currents, when a drop initially elongates in responseto the magnetic forces, the magnitude of the axial force

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Magnetic forces acting on molten drops

decreased slightly. This was because the current divergedless in an elongated (prolate) drop. Once a neck had begunto form – typically about halfway through the current pulse– the axial magnetic force increased rapidly. This wasbecause the narrowing of the neck resulted in a greaterdivergence of the current. This observation is not surprisingor new since it has long been known that magnetic forcesact to pinch drops off the end of an electrode. Theresults shown in figure 18, however, represent the firsttime the axial magnetic force has been computed fromexperimentally measured shapes.

Consider the 330 A pulse data in figure 18 (marked withcircles (◦)) that correspond to the drop shown in figure 15.The components of thez-directed magnetic force acting onthe upper and lower parts of the model,fmu and fml, areshown in figure 19. The magnitude of the force on the upperpart of the drop is always much greater than the magnitudeof the force on the lower part of the drop. This is a resultof the current emission model for an Ar–2% O2 plasma,whereby current is emitted uniformly from the entire dropbelow the waist of its neck. When the drop currentId flowsdown through the drop, it begins emitting into the plasmaat the neck waist; much of the current is already flowingin the plasma at the level of the drop equator where itdrives plasma flow and thus has little effect on the drop[5, section 7]. The reduced currentIs1 remaining in thedrop and passing through the drop’s equator results in lessmagnetic force on the lower part of the drop.

In a carbon dioxide plasma, current is emitted froma tight anode spot on the bottom of the drop [9, 14, 15].A model for this observation is that current is emitteduniformly from a small anode spot on the bottom of thedrop below the point on the drop surface where the surface-tangent angle isξ = π/4 (see figure 5). Thez-directedmagnetic forces acting on the upper and lower parts of adrop were recalculated for a 330 A current pulse (figure 15)using such a current emission model and the results areshown in figure 20. Although the shapes in figure 15 wereobserved in an Ar–2% O2 plasma, the results in figure 20show the axial magnetic forces that would act on a drop ina carbon dioxide plasma if the drop ever actually attainedsuch shapes. Only if a narrow neck could form would thez-directed magnetic force aid detachment (be negative) andeven then only about half as much as it would in an Ar–2% O2 plasma. It is known that drop detachment in a CO2plasma is violently erratic [16] and drops rarely achievethe relatively symmetrical necking shapes seen in figure 15.Before a narrow neck can form, the axial magnetic forcespush the drop onto the electrode, preventing the formationof a neck. The results in figure 20 predict that currentpulses will aid the detachment of drops in a CO2 plasmaonly if a neck can somehow be formed.

9. Summary

In a GMAW drop, magnetic diffusion is virtuallyinstantaneous relative to the time scale of mechanicalmotion of the drop. At each moment in time the magneticforce acting on a welding drop is dependent only on theshape of the current path and is independent of any resulting

fluid flow generated in the drop – a point that was assumedboth in [3] and in [4], but not discussed or mentioned. Themagnetic force on a drop is determined by the shape of thedrop’s surface and the current emission density along thesurface. If these two items of information are known, thenthe magnetic force acting on the drop at any time may becomputed.

Our axial magnetic force calculations, performedusing magnetic stress tensors integrated over shapes thatapproximate experimental observations of detaching drops,are the first calculations in which the general resultsin [4] have been applied to geometries measured fromexperimental images. The shape approximations (truncatedellipsoids and polynomial volumes) were fitted to dropsdetached by current pulses (where magnetic forces are thedominant forces acting on the drops) and the associatedaxial magnetic forces were computed. Because theapproximate shapes closely match actual drop profiles (atleast until the neck is close to severance), the results area more accurate representation of the instantaneous axialmagnetic forces acting on welding drops during neckingthan are those in previous studies. This technique forcomputing magnetic forces is applied in a dynamic modelof drop detachment in the companion paper [5].

Acknowledgments

Support for this work was provided by the United StatesDepartment of Energy, Office of Basic Energy Sciences.This paper was extracted from chapter 5 of the firstauthor’s doctoral thesis [17] at the Department of ElectricalEngineering and Computer Science at the MassachusettsInstitute of Technology.

References

[1] Dattner A 1962 Current-induced instabilities of a mercuryjet Ark. Fys.21 (7) 71–80

[2] Maecker H 1955 Plasmaströmungen in Lichtb̈ogen infolgeeigenmagnetischer KompressionZ. Phys.141 198–216

[3] Greene W J 1960 An analysis of transfer in gas-shieldedwelding arcsTrans. AIEE Part II79 194–203

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

[5] Jones L A, Eagar T W and Lang J H 1998 A dynamicmodel of drops detaching from a gas metal arc weldingelectrodeJ. Phys. D: Appl. Phys.31 107–23

[6] Haus H A and Melcher J R 1989Electromagnetic Fieldsand Energy(Englewood Cliffs, NJ: Prentice-Hall)

[7] Becker R 1964Electromagnetic Fields and Interactions(New York: Blaisdell)

[8] Nestor O H 1962 Heat intensity and current densitydistributions at the anode of high-current, inert gas arcsJ. Appl. Phys.33 1638–48

[9] Mechev V Set al 1983 Calculation of the welding arccharacteristics in consumable electrode weldingWeld.Production30(7) 32–5

[10] Lancaster J F 1986The Physics of Welding2nd edn (NewYork: Pergamon)

[11] Moon P and Spencer D E 1971Field Theory Handbook2nd edn (New York: Springer)

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[12] Dwight H B 1961Tables of Integrals and OtherMathematical Data4th edn (New York: MacMillan)

[13] Allemand C D, Schoeder R, Ries D E and Eagar T W 1985A method of filming metal transfer in welding arcsWeld. J.64 (1) 45–7

[14] Rhee S and Kannatey-Asibu E Jr 1992 Observation ofmetal transfer during gas metal arc weldingWeld. J.71(10) 381–6s

[15] Nemchinsky V A 1996 The effect of the type of plasmagas on current constriction at the molten tip of an arcelectrodeJ. Phys. D: Appl. Phys.29 1202–8

[16] Kim Y-S and Eagar T W 1993 Analysis of metal transferin gas metal arc weldingWeld. J.72 (6) 269–78s

[17] Jones L A 1996 Dynamic electrode forces in gas metal arcwelding PhD ThesisMassachusetts Institute ofTechnology

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