Lorentz force in the molten tip of an arc electrode

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1965 Br. J. Appl. Phys. 16 1169

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BRIT. J. APPL. PHYS., 1965, VOL. 16

Eorentz force in the molten tip of an arc electrode J. C. AMSON Churchill College, University of Cambridge MS. received 2nd February 1965, in revised form 31st March 1965

Abstract. A general expression for the Lorentz force in the molten tip of an arc electrode is derived from the surface integral of Maxwell stress. The Lorentz force in typical situations is calculated and its variation and eEect during the formation and detachment of the molten tip are discussed. Values much larger than those customarily associated with molten tips are seen to appear towards the end of the tip detachment cycle.

1. Introduction Consider an electric arc in a gas atmosphere, one of whose electrodes (of either polarity)

is a thin rod or wire. Heat from the arc process and the flow of current through the wire can cause the tip of the wire to melt. The molten tip thus formed is often detached as a droplet from the electrode and transferred through the arc by a combination of electro- magnetic, fluid-dynamic, and gravitational forces. If the electrode wire as it melts is advanced steadily to maintain a quasi-steady system in which the arc length and current etc. fluctuate only slightly during the cycle of melting and detachment of the electrode tip, the process of tip detachment can become regular. Such a well regulated system is the basis, for example, of the automatic consumable-electrode welding process (see Amson 1962 for a short bibliography of the process). (Typical rates of droplet detachment from the tip of an aluminium wire, & in. diameter, in an argon arc, vary between 1 per second at 30 A to over 1000 per second at 350 A, the electrode feed speeds being 1 a2 and 14 cm sec-l respectively.)

Gravity is not always a predominant force in tip detachment, since detachment can take place coaxially with the electrode wire, regardless of the systems orientation, providing the current is high enough. Forces other than gravity arise from the interaction of the electric current and its own magnetic field. The direct electromagnetic forces are the Lorentz forces within the body of the molten tip prior to detachment; the indirect ones are those due to the flow of arc plasma and the gas about the tip of the wire, induced by the Lorentz forces in the conducting arc plasma itself. These flows are often violent, and plasma velocities of the order of 1 km sec1 are not unusual, whilst detached droplets with a mass of about 10 mg, ir, free flight along the arc axis, have often been observed cinematic- ally to receive accelerations of over 100 g from such plasma jet streams (Amson 1962, Maecker 1955, Wells 1962, Amson and Salter 1963).

In this paper we study the contribution to the forces of tip detachment arising from the axial component of Lorentz force in the molten tip. For simplicity we only consider a symmetrical arrangement, i.e. one in which the current, the magnetic field, the droplet at the tip, and the arc are all disposed symmetrically with respect to the axis of the cylindrical wire electrode. This restriction also ensures that whatever torques appear within the molten tip shall have axial symmetry, and hence that the resulting force acting on the droplet is torque-free. When the system is not symmetric, the combination of thrust and torque in the detaching tip may well explain the very different motions of droplets observed in this situation (Needham 1963, British Welding Research Association Report A1/38/63). Whilst the radial components of Lorentz force produce an increase of pressure throughout a conductor in general, this contributes nothing to the axial force on the conductor in the absence of fluid motion in the latter. But when fluid motion is present, and especially for example in the compound conductor formed by the interior of the molten tip of an arc

1169

1170 J. C. Amson

electrode and the arc plasma about its exterior, the situation can alter radically and ad&. tional axial forces, the so-called magnetokinetic forces, can appear (Amson and Salter 1963, Serdjuk 1962). Both these two further aspects of Lorentz force in the molten tip of an arc electrode lie beyond the scope of the present study, and must be reserved for a later investigation.

2. General theory

We shall use a right-handed coordinate system and the notation shown in figure 1.

Unmelted elrctrode wire

\ Melt ing i n t e r f a c e '.

Molten droplet 41 \

\i?r"%. I= 0 A Is.0

I

Figure 1. Typical form of a pendent droplet at the molten tip of an arc electrode, showing notation used in the text. a maximum droplet radius, b wire radius, T stress vector, n normal

vector.

The Lorentz force L on a conductor in a magnetic field can always be represented by the surace integral of the Maxwell stress tensor 9:

L = . d S . The absolute magnitude of the stress tensor Y is & p p o H 2 (Abraham and Becker 1944,

Landau and Lifshitz 1960), independent of the stress direction, where p is the relative permeability of the conductor ( p = 1 in the molten tip since its temperature is then above the Curie point), p o = 477 x IO-' is the permeability constant, H = I(r)/2nr is the mag- nitude of our magnetic field strength vector H and I(r) is the total current enclosed by the circle of radius Y about the z axis. The direction of the stress at the surface of the con- ductor is such that the magnetic field H there always bisects the angle between the stress T and the outwards surface normal vector n. It is easy to see that T is always directed in- wards and perpendicular to the surfaces of our conductor.

In general the axisymmetric surfaces of a molten tip are described by two pairs of rela- tionships, namely Y = r(s), z = z(s) for the outer surface, and r = ?(U), y = y ( ~ ) for the

Lorentz force in the molten tip of an arc electrode 1171

interfacial surface between the molten tip and the still unmelted wire. The total Lorentz force in the molten tip is then

L = SJ F p . dSp f iJ^ FQ . dSB from which we can calculate the magnitude of the axial component in a downwards direc- tion as

(AB) (BB)

wherej( ) is the current density distribution function with respect to s or U. The following assertion shows that the non-conducting surfaces do not affect this quantity. The Lorentz force in an axisymmetric conductor is independent of the suvfaceprojile of any

zone where the surface current density is zero. For supposej(s) is defined over the interval [0, S] such thatj(s) = 0 for s1 d s < sp and

is arbitrary elsewhere, and sl, se are such that 0 < s1 < s2 < S but otherwise arbitrary, then the contribution to the Lorentz force from the outer surface of the conductor is

s s, S

- p O r 1 [( j ( s ) r(s) ds -+ j (s ) r(s) ds - - s ] 2 r:)* S? 0 S?

Here the first and third members of Lo are obviously independent of the profile function r(s) in the zone [s,, s2]. The second member reduces to

0 SI

where b,, b, are the radii of the points on the surface at sl, sp respectively, and I , is the current enclosed by the circle of radius b, (or b2 for that matter, since no current emerges from the zone [s,, s,]). Thus the second member is also independent of the profile function r(s) in Isl, s,] since I,, b,, b2 are each themselves independent of it there. Thus Lo is inde- pendent of the shape of the conductor in [s,, sz] . Since s1 and s, are arbitrary in [0, S ] the assertion follows.

The contribution from the non-conducting zone [sl, s2] is the same as if the actual con- ductors surface was replaced by a coaxial cylinder and an annular disk with radii b, and b2, since the contribution from the cylinder is zero by symmetry, and from the disk is just 3 p o l l 2 log (b,/b,). Moreover, if b, > b, the contribution hinders detachment of a molten tip, and if b2 < b, the contribution assists detachment. The contribution is nil when s1 = 0, whatever s, may be, since

b lim 112 log 2 = O s,+o bl

that is, a non-conducting cap at the apex of a molten tip contributes nothing to the Lorentz force Lo.

We may assumej(u) = j , constant, in the first member of L above. Further, the inter- face BB can usually be approximated by a cone, and then this first member reduces to

1172 J . C. Amson

(p,,/l6~) I d 2 (where I d (< I ) is the current actually passing through the molten tip). This value is independent of the cone angle, as can easily be shown. Accordingly, the contri- bution Li from the interface BB is the same as if the interface was a flat disk. For the remainder of this work we shall adopt these assumptions and in particular that the first member of L is (p0/16r) 16. When the current density is uniformly distributed over the

S

outer surface, we have j ( s ) = j = I /A = constant, where A = 277 ! Y(S) ds is the area of the outer surface. The contribution Lo from the outer surface is then

0

where S S

p ( s ) = [I r(s) ds] [J Y(S) dx1-I 0 0

and P(S) is a function depending only on the profile of the tip. Thus Lo consists essentially of the product of the square of the current passing through the tip, and a coefficient whose value is determined solely by the shape of the molten tip. Similar expressions can readily be calculated for linear or Gaussian distributions of the current density.

Two factors influence the value of the integrated stress over the outer surface of the tip more than any others; these are the shape of the surface arid the way in which the current is distributed over the surface.

The shape of the surface is the result of a number of difTerent forces. The mass of the tip is influenced by gravity; the atmosphere about the drop is in motion, giving rise to aero- dynamic and plasmadynamic forces ; thermal and electromagnetic convection inside the tip produces additional surface forces. Though the surface energy of the tip reacts against these many forces. its own nature is modified to an unknown extent by the arc root mechanisms. (At the anode of our arcs, current densities range between 1 and 60 k A cm-l, power densities between 6 and 1000 kw cm-I; at a cathode, values up to 100 times larger than these can occur (Ecker 1961). Again, nearly all these factors depend on the polarity of the tip, and on which metals, gases, gas pressures, impurity levels and operating modes are being used in any given situation, Moreover, the magnitude and disposition of the forces often depend on the shape of the tip itself. Consequently it is very difficult to determine theoretical tip profiles. The best we can do at present is to observe experi- mental profiles, to assign approximate formulae to them, and then use these formulae in our expressions for the Lorentz force.

In a similar way the current density distribution depends on so many arc-physical cir- cumstances, that only the simples: specifications of its distribution function are reasonable. For the anode of a non-consumable-electrode arc, the current distribution may be either almost Gaussian or else concentrated in a fairly wide axially central zone at almost uniform density (Nestor 1962). However, almost nothing is known about the comparable situation in a consumable-electrode arc because of the practical difficulties that have so far pro- hibited its investigation. For the cathode, the current is always concentrated at such high but uncertain densities in the small cathode spot, and nowhere else, that the distribution may again be represented either as a disk of constant current density or by a Gaussian function of very tight concentration. The overall shape of the molten cathodic tip is then often irrelevant to the calculation of Lorentz force. With high melting point, high density metals of good thermionic emittance, such as tungsten, molybdenum and zirconium, tip detachment from the cathode often occurs not by whole droplets of roughly the same diameter as the wire, but as a rapid succession of micro-droplets which first appear as small protuberances somewhere near the apex of the molten tip (G. R. Salter private com- munication, Cooksey and Milner 1962). If we regard such protuberances as cathode tips

1173 Lorentz force in the molten tip of an arc electrode

themselves, then the distribution of current will be with respect to the surface of the pro- tuberance alone and not the whole of the molten tip, and the remarks made about the anode tips will apply here as well.

3. Lorentz force in molten tips with empirically determined profiles We shall take advantage of the many observations, made by high-speed cine photography

during recent years, of the shapes adopted by the molten tips of consumable arc-electrodes (G. R. Salter private communication, Cooksey and Milner 1962, Needham et al. 1960, Ishizaki 1963, Greene 1960),and calculate the Lorentz force in tips whose shapes are simple geometrical approximations to those observed.

/ I

1 1 I I I I I I I

,'Arc mantle \ I i I I I

I I

I

I !E: --_______e- --='

rd

1174 J. C. Amson

where the shape coefficient P is

-0.5-

a sin CD 1 1 2 2 P = log--- - - - b 4 1 - cos CD + (1 - cos C D ) Z log 1 -k cos @ '

p (0) = l o g ' - I 026 6 4'

- 1 . 0 " ' " I '

This value for L was calculated by Greene (1960) using a more direct method only applicable with ease when bothj(q5) andj(r) are constants.

The shape coefficient P can be conveniently written as the sum of two independent functions, P = P(a) + P(CD), where

a 1 b 4 P(a) = log - - -

1 , 2 2 1 - cos @ (1 - cos CD)2 log 1 + cos @ P(@) = log sin Q, -

where P(a) depends only on the ratio between the maximum tip radius a and the wire radius b, and P ( 0 ) depends only on the extent CD to which the conduction zone covers the droplet surface (see figure 3). Whilst P(a) is a simple increasing function of its argument,

P(Q,) reveals the way in which the tip profile intervenes in the Lorentz force values: the latter grow rapidly from negative to positive as the conduction zone spreads out over the tip's lower surface, and then begin to shrink again as the conduction zone spreads even further up the tip on to the diminishing area of the tip's shoulders.

Case (ii): a = b

This is a special case of case (i), where @ = $T, fixed, and a = b. When, as above, j(+) andj(r) are constants and related, then their relation depends on the height c of the cylindrical collar CB' :

I 1 j(q5) = __ _ _ - ~ 27b2 1 + cia'

In this case

L = I*.o PP(c) 4a

where

P(c) = - (i - log 4)(1 f f2)-' (see figure 4).

Lorentz force in the molten t@ of an arc electrode 1175

Figure 4. Variation of the Lorentz force coefficient P(c) with the relative droplet length c/a.

Case (iii): a < b This is a further specialization, insofar as the current I d actually passing through the

molten tip proper is now only part of the total current I . Here L = (p&) Id2P(C), the coefficient P(c) being the same as in case (ii).

It is useful, as shown by Amson and Salter (1963), to combine the separate equations in the three cases above into the single equation

where the shape coefficient q is a function solely of the ratio a/b. To do this we make two assumptions, both of which appear plausible in the light of

observation, but difficult to test directly. First, when a > b, the area A of the conduction zone on the tip's outer surface remains constant and equal to A" (the area it has when a = b); second, when a S b, then I d = CI and c is a function solely of alb. Then

A* cos@ = 1 - __ 27i-a'

and we can substitute for cos 0 in the coefficient P(aj (case (i)) thus converting it into P{O(a/b)}, a function not of @ but of aib. Also, for fixed c, the coefficient c2P(c) (case (iii)) is a function of a/b. Thus for fixed c, we have

P ( 4 P{@(a/b)} (a > b)

a'P(c) (a d b) 4 =

q being a function solely of aib as required. This form of the function is shown in figure 5 , using the specific values c = +a and a = (a/b)?.

alb

Figure 5. Variation of the Lorentz force coefficient q with the relative size of the droplet a/b.

1176 J . C. Amson

The two assumptions behind q are in fact assumptions about the relationships be:ween the current I, current density j , and tip size a. For if current I passes through just the entire surface A* of the tip whose radius is a = b (case (ii))at a densityj* = I*/A*, then since I < I when a > b, but A = A* by assumption, we have

which does not appear to contradict the experimental evidence. Similarly, since I > I* when a < b, and A = (a/b)2 A X (by geometrical similarity), so

and > j * whenever a(b/a)* 2 1, for example when a = ( ~ / b ) ~ as above, or when in general a = a2/bb, with bib, 2 1 (where bo is the critical electrode radius as defined by Amson (1962)). Here again j (u j * appears to be supported experimentally. Thus we see that our two assumptions imply the wider assumption that current density and total current are related, in that low or high values of each occur together.

4. Variation of Lorentz force during tip detachment We now study the changes that take place in the korentz force during the growth-cycle

of the molten tip. This is an important aspect of the detachment of droplets from the molten tip of an arc electrode that does not appear to have received any attention before. Let us remind ourselves once again that we can expect no generality in our treatment at this stage of our knowledge. We can however construct a sequence of droplet profiles which approximate an actual droplets profile-cycle, one which has all the essential features of the growth, necking-down, and detachment of a symmetric droplet from a molten tip of an electrode. The korentz force can now be calculated for each profile in the sequence.

Such a profile is shown generally, in figure 6, and the corresponding sequence of such profiles in figure 7. The geometrical construction of the profiles is deliberately naive and yet remains quite representative at least of those associated with medium diameter wires in commonly used metals; droplets from small diameter wires tend to have rounded shoulders near the wires melting zone, their cycle usually starts further along the sequence shown, and their neck during detachment is often more attenuated and filamentary-but

1

Figure 6. Geometrical construction of the Figure 7. Sequence of approximate pendent approximate pendent droplet profile, showing droplet profiles during one complete cycle of

the definition of the timing parameter B. droplet growth and detachment.

Lorentz force in the molten t@ of an arc electrode 1177

these are details, and their inclusion here would not substantially change the form of the conclusions reached in this section. The 'timing' of this droplet cycle is not on a uniform natural time scale of course, but is referred instead to a simple geometrical parameter (in this case the angle j3). Some relation between j3 and natural time can readily be obtained from a suitable inspection of an actual growth-cycle (cf. the illustrations of Needham et al. 1960, Ishizaki 1963); the relation is non-linear, especially near the value j3 = 90", i.e. at the onset of neck-down.

The results of the Lorentz force calculations under the assumption of a uniform dis- tribution of current density over the entire surface of the droplet, are shown in figures 8, 9, IO. The gross Lorentz force at constant current (i.e. the force that exists in the total volume of the molten tip; see figure 8) is at first negative, acting to retain the incipient droplet. After becoming positive, the force reaches a maximum value of about 0.18 (p,,/47) I 2 newtons, the value adopted for the stylized droplets employed by Amson (1962). Thereafter the force diminishes as the neck forms rapidly, and it becomes once

!42.5"

Figure 8. Variation of the gross Lorentz force coefficient l(p) in a detachhg droplet (assmir,g uniform current density distribution over all the droplet surface, and constant droplet current).

I Net Lorentz force L,,=$ I ' m n I 90" 105" 120" 135" 142.5' 150"

B Figure 9. Variation of the net Lorentz force coefficients ma(& and mb(p) in a detaching droplet (assuming uniform current density over all the droplet surface, and comtant

droplet current).

h

54

h

QL

Net Lorentz tension T=$ 1 2 n (PI

c

XIo

Figure 10. Variation of the Lorentz tension coefficient n(p) at the equator of the neck of a detaching droplet (assuming uniform current density distribution over all the droplet

surface, and constant droplet current).

1178 J . C. Amson

again negative (retentive). This last reversal of sign shows that the gross Lorentz force can obviously not be wholly responsible for detachment. We must look further at the division of Lorentz force between the parts of the droplet above and below the equator of the droplet's neck, as soon as the latter appears (figure 9).

Let La and Lb stand for the values of the Eorentz force in the parts of the molten tip above and below the droplet equator, respectively. Since there is no neck before ,B = go', we have La = 0 and the gross Lorentz force L = the net Lorentz force Lb, for 0 < ,f3 < 90'. Between ,8 = 90" and ,R N l l l " , La and Lb are both positive and both are aiding detach- ment. But at about ,B = l l l " , the net Lorentz force La in the shoulders of the droplet has once again vanished and a detaching force now exists in the lower droplet alone, whose value is again about 0.18 (po/47r)P newtons, the same as the gross value for the whole droplet at that point. Thus at his critical value ,B = 11 1 " in the droplet cycle, the positive (detaching) Lorentz force ceases to occupy the whole droplet and becomes concentrated in the lower droplet. Thereafter the upper part of the droplet is the seat of only a retaining force. We might say that up to /3 = 11 1" the droplet is merely growing ready for detach- ment, but that detachment properly commences when ,B = 111" and L = 0.18 (p0/4ir) 1s newtons, and continues throughout the shrinking of the droplet's neck. The upper part of the droplet is then being forced upwards (La negative) whilst the lower part is being forced downwards (Lb positive), and the combined result is the appearance of a 'Lorentz tension' at the neck's equator (figure 10) with values T = Lb - La which increase very rapidly indeed as the necking-down proceeds. At the same time the ability of the neck to withstand this tension is shrinking as rapidly as the neck's radius. This combination of rising tension and falling retention will readily account for the marked and puzzling accelera- tion of a tethered droplet approaching detachment so often observed in practice, quite apart from the additional acceleration to be derived from the increasing surface area of the droplet exposed to the plasma streams about it.

The infinitely large values apparently attainable by La and Lb as /? tends to 150" (and the neck radius tends to zero) will not of course be attained in reality, since these values are calculated on the assumption that the current density remains uniform over the whole droplet surface, which in turn implies that a large proportion of the current will continue to flow into the detaching droplet through the vanishing neck. Clearly this will not happen; there will come a point when the current must dispose itself afresh throughout the entire conductor comprising the electrode tip, droplet, and arc, in order to secure for itself the paths of least overall resistance, and then the arc root will vacate the almost insulated lower droplet and concentrate itself anew on the fresh shoulders of the wire. In fact this point will be delayed for some time even though the resistance of the filamenting neck is increasing rapidly in response to its shrinking diameter and rising temperature. For apart from any arc heat proper, the resistive heating of the neck and its vicinity will augment the supply of metal vapour from this region and this will produce a compensating reduction in the local arc resistance. Unfortunately the shift of current from the droplet to the shoulders is not yet identifiable under observation, since this copious production of metal vapour is nearly always seen to persist even after the droplet has become detached and is in flight across the arc. The actual values of T, La, L b , as the droplet approaches very close to detachment will reach maxima which may still be very large; after this, T and Lb will cease to grow, will diminish, and then vanish as the neck severs, La will then become L again at a value equal to that of an incipient droplet at some earlier point in the cycle; and the whole cycle will repeat itself.

5. Conclusions

(1) The Lorentz force in the molten tip of an arc electrode can be found by integrating the Maxwell stress over the total surface of the tip; when current density is uniformly distributed over the surfaces the result can be expressed as the product of a constant, the square of the total current passing through the tip, and a coefficient which depends on the shape of the tip.

Lorentz force in the molten tip of an arc electrode 1179

(2) The Lorentz force in an axially symmetric tip is independent of the shape and size of the melting interface if this is conical or flat.

(3) The Lorentz force in an axisymmetric conductor is independent of the surface profile of any zone where the surface current density is zero; the integrated Maxwell stress there has a particularly simple form. (4) The Lorentz force in molten tips with empirically determined profiles can be readily

evaluated when these profiles lend themselves to simple geometric approximation; in certain instances the Lorentz force can then be expressed as the product of a constant, the square of the total current passing through the tip, and a coefficient depending solely on the ratio of the diameters of the tip and the electrode wire.

(5) Calculation of the Lorentz force in a sequence of tip profiles, representative of the growth and detachment cycle of a droplet on the molten tip, reveals that Lorentz force at first hinders detachment, then assists it, and finally promotes it as soon as the neck of the droplet has been substantially formed.

(6) A new concept-'Lorentz tension'-can be defined and seen to play an important part in the severance of a detaching droplet from the parent tip of an arc electrode; an evaluation of the components of the Lorentz tension in the shrinking neck of a droplet reveals the rapid rise to large values of the forces aiding detachment in the last phase of the detachment cycle ; this behaviour could then account for the frequently observed in- rush of acceleration to the droplet.

Acknowledgments This paper is condensed from Report A1/35/63 prepared for the A1 Committee of the

British Welding Research Association, whose permission to publish is gratefully recorded.

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ECKER, G., 1961, Erg. exact. Natnr. Wiss., 33, 1-104. GREENE, W. J., 1960, Trans. Amer. Inst. Elect. Engrs, Part 2, 194203. ISHIZAKI, K., 1963, Japan Weld. J . , 32, 1047-53. LANDAU, L. D., and LIFSHITZ, E. M., 1960, Electrodynamics of Continuous Media (Oxford:

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