J. Etay, A.J. Mestel and H.K. Moffatt- Deflection of a Stream of Liquid Metal by Means of an...

8/3/2019 J. Etay, A.J. Mestel and H.K. Moffatt- Deflection of a Stream of Liquid Metal by Means of an Alternating Magnetic Field

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BY M E A N S O F A N

J . ETAYMADYLAM D.A.M.T.PInstitut de Mdchanique de GrenobleB.P. 95 Silver Street38400 St Martin d'H&res CCdex

France United Kingdom

A. J . MESTEL & H. K. MOFFATT

Cambridge University

Cambridge CB 3 9EW

of liquid metal due to externalcy currents is investigated. A simple theore tical model assuming un i-directional flow is

of deflection of th e strea m and the pow er supplied toa free-falling mercury

s deflected by two anti-parallel line currents. Th e agreement between theory and experimentreasonable, despite a tendency towards three-dimensionality in the lat ter .

. I n t r o d u c t i o n

or many purposes involving th e processing of liquid metals, it m ay be desirable to be able t o

flect a str ea m throug h a given angle. In thi s paper we describe how such deflection can behieved by the action of th e m agnetic pressure p~ associated with a high-frequency magnetic field

a simple one-dimensional analysis

estel & Moffatt 111.

We consider a two-dim ensional configuration, in which a sheet of metal, of initial thickness do andvelocity UO,moves under the influence of altern atin g line currents. Such sheets are used in

as the manu facture of metallic ribbon. In order t o analyse th e effect, we

e certain simplifying assumptions, as follows:(a) We assume that the field frequency, w / 2 a is sufficiently high for it to be reasonable to treat

6 = ( 2 / p0uu ) ' / ~where U is the electrical conductivity of the liquidpo its permeability) be small compared w ith the und isturbed thickness of the strea m , do .

(b) We assum e th at do is small compared with the scale L on which the magnetic pressure variesas determined by the current source distribu tion and the surface curvature.

(c) Finally, we assume th at grav ity may be n eglected a t least over th e scale L on which theuo, he length-scale on which gravity acts is I, = ui/g.

These three assumptions imply a hierarchy of length-scales as follows:

265

Lielpeferis and R . Moreau (eds.) ,Liquid Metal Magnetohydrodynamics, 265-271.1989 by Kluwer Academic Publishers.

www.moffatt.tc



266

0

The factor of 1/2 is included in (1) because th e m agnetic pressure is qu adr atic in t he magnetic field

and thus decays twice as quickly. Assumptions (a), (b) and (c) are obviously restrictive, but theypermit significant progress to be made. The limitations of the analysis are considered in [l].

W X

2. Quasi-one-dimensional analysis

Th e assumption (b ) above allows the use of a quasi-one-dimensional analysis, in which the liquid st rea m is in effectlocated by the position of its left-hand boundary. This is

a curve C parameterised by arc length s from some fixedpoint 0, as in figure 1. The stream thickness, d ( s ) , is(weakly) non-uniform when deflection occurs. We supposethat, upstream of the region of magnetic influence, thestr ea m has uniform velocity ug and thickness do 80 tha t the

volume flux is Q = z lodo . As viscous effects a re negligible,the flow is irrotational by virtue of the assumption (a)which ensures that the sole effect of the magnetic field is

to provide a magn etic pressure d istrib utio n over the liquidsurface.

Let (s , n) be taken as coordinates tangent and normalt o C, as in figure 1, and let u(s,n) be the velocity withinthe stream, effectively parallel to C. To leading order inthe small parameter do/L, th e velocity is uniform, U w uo,

and the n-component of the equation of motion is Figure 1: The coordinate system.

(2)

where p is th e liquid density, p ( s ,n) the pressure, an d K ( s ) s the curvature of C at position s. Tothe same appro xim ation, the appro priate boundary conditions are

(3 )P ( S , do ) = P O - on the right-hand boundary,

p ( s ,0) = p o + y K +PM on the left-hand boundary,

wherepo

is atmospheric pressure,7 s

the surface tension andp ~ ( s )

s the magnetic presaure.In

writing (3) , we are assuming that all the current sources occur on the left-hand side of the metalstream. Integrating (2) using (3) we get

p ( s ,n)= PO -7 K + pu;K(do - n) (4 )

and hence the required magnetic pressure P M ( S ) is given by

T he perturbation to the velocity u(s,n) and width d ( s ) may now be found from Bernoulli’s theorem,

p + 4pu2 = po + 1P 212 , (6)

and mass conservation. From (4 ) and (6), e have

7



267

also Q = uOdO, th e stream thickness, d , is given by

d ( s ) = do + 4 K d ; X . (9)

It is clear from the form of (5) that the inertia of the uniform stream acts somewhat like a

As the magnetic pressure is positive, the direction in which the stream is(as determined by the sign of I < ) may be seen from (5) to depend on the sign of A. We

A is positive, i.e. when the magneticX < 0, the magnetic pressure

strongly resisted by surface tension, w ith t he m om en tum flux being less im po rtan t. In both thesed 2 do everywhere. It is of interest to note th a t when both X and p~ are

C gives rise to an admissable steady state, to lowest order in the jet thickness.

Equation (5) defines a non-linear relation between the curve C and the external currents. 111

eory, any desired s trea m p at h C (for which K > 0) can be obtained by choosing the distributioncurrent sources to give the required mag netic pressure in (5). On e such d istribution (and there willmany others) is provided by placing coils so as (in effect) to provide a current sheet J ( s ) os wt

ar to the deflected stream . T hi s curre nt sheet produces a magnetic field B ( s )cos w t in the gaptween t he coils and the stre am , where B ( s ) = p o J ( s ) , and an associated time-averaged magnetic

p~ = B2/4po = i p o J a . Hence the required magnetic pressure (5) is achieved provided

zontal, then by integ rating (10) we ob tain th e resultIf we use the expression K = d$/ds, where li, is the angle tha t the tangent to C makes with the

a is the tota l angle of deflection of the s trea m. We may express a n terms of the power

pplied to the coils (per unit length in th e %-direction). T hi s power,W ,

s balanced by the Joule

terms of the skin-depth ap prox imatio n. Tog ether (11) and (12) give a simple relation ship between

6, we see that the power needed to obtain a given deflection tl behaves4, rovided w is sufficiently large for the skin-depth approximation to apply. Thus, in practice

at which the deflection effect is still pronounced, but for whiche power dissipated in th e metal is relatively low. T hi s optim um value will occur when 6 / d o is



268

3. Deflecting action of weak line currents.

It is difficult to find th e stream path C analytically for a

given current distribution, as the magnetic field is stronglycoupled with the shape of C. A problem of this type wassolved numerically by Shercliff [2] in his work on the shap-ing of liquid metal columns, when the balance is betweensurface tension and magnetic pressure. If the deflection isweak, however, then progress is possible by perturbationanalysis. We illustrate this with reference to the action ofa concentrated line current I coswt placed at a distance Lfrom a stream whose undisturbed position is 0 < z < do.

When I # 0, we suppose that the stream is symmetricallyperturbed as indicated in figure 2. To leading o rder , how-ever, the magnetic field distribu tion may still be calculated

as if the stream were in the undisturbed position. Thereis then an image current -I coswt at x = L, = 0, andthe net magnetic field on x = 0, is B = (0, B,, 0) where

Icosw1

1 1

Figure 2: Deflection by a weak

current source.

Icosw1x

1 1

Figure 2: Deflection by a weak

current source.

(14)poL2I2 1

so t ha t pw =-x2 ( LZ+ y 2 ) 2 .os w t ,

B,=--POI L

?F L Z + y ’

The deflection of the surface, from ( 5 ) with I< FZ d 2 x / d y 2 ,is given by

N = - Po12wherea x N L3- = -dy2 4r2 ( L 2+ y2)2 pu8doLA

We int egra te th is with ‘initial’ conditions x (0) = $ 0) = 0 with the result

When l y l> L , we have c - & y l , so that the net angle of deflection is given by

N87

a = -

Equation (17) is strictly valid only provided the deflection is weak, i.e. CY < 1;however the resulb

of section 2 suggest th at an a rbitra rily large deflection of th e st rea m may be achieved if the dimen-

sionless parameter N is increased to a sufficiently large value. Thi s pa rame ter may be regarded Y

the magnetic interaction parameter, giving a measure of the transverse flux of momentum generated

rby th e m agnetic forces relative t o the flux of mom entum pu@o in the incoming stream.

Ihe magnetic field lines of a line current and its image consist of a family of co-axial circle.

Thus, the analysis of this section and the result (17) in particular may be extended to the more,

realistic case when the line current is replaced by a circular wire, of radius r, whose axis lies on

x =4, y = 0. At high frequency, the current in the wire flows near the surface and is equivalenl’to a line current at x = -L, y = 0, where La= b2 - 2 . For this value of L, he deflection obtainedis given by (17) with (15). It should be noted that no such exact result holds for low frequenciea (or ’

D.C .) for which the surface of the w ire is not a magnetic field line.In the experiments that follow we use for electrical convenience a device which may be modelled



269

which can then be squared and integrated to calculate the deflection angle a. When the twoa horizontal plane a distance a ap ar t, the result corresponding to (17) is

(19), (Y varies between a value due to deflection by both branches of the inductor in turn, ande where only th e upperm ost branch is imp ort ant , according to th e relative size of a / L and N .

So far, we have supposed that the current sources are all placed to one side of the stream.are

of C do not interact . Th e st ream pa th , C, will still be determinedthe differential equation (5) with the appropriate value of p~ being the difference between the

C. In [l], t is described how tw o line curren ts, o ne on each sidethe metal stream , can act as a sensitive control over th e u ltim ate direction of the s tream.

Experiments

inents were performed in orde r t o measure t he deflection of a nearly two-dimensional streaniline currents for various values of the interaction parameter N . The experimental set-up used is

& Garnier [3]. Essentially, th e facility consists of a mercury-filled hydraulica freely falling column . T he ini tia l cross-section of th e column is determined

a gently converging nozzle ending in a slit l m m . by 39m m. T he alterna ting magnetic fieldto adjustable capacitors, and powered by a

kWatt gene rator . Th e indu ctors are ma de from hollow copper tu bin g of 3-4m m. diameter througltA single line current would give rise to a large self-inductance with

sultant inconvenience for th e t un ing cap acitor. Ins tea d, hairpin-shaped inductors were bu ilt, withe current flowing along one branch of the hairpin and returning along the olher. I n the middlethe inductor the field is approximately th at due t o two line currents. T h e distance between

e branch axes is 15mm. The current flowing in the inductor varies from 0 to 1520 Amps. at a

6, is therefore 0.8mm. and the m agnetic field near the sheetfrom 0 to 2000 Gauss. A higher current could not be obtained with this kind of copper tube

ductor. In Riga we were ta ug ht t he phrase “T he expe rim ent w as a success: the apparatus burnedIndeed, one of the inductors used in our experiments proved very “successful!”

A truly two-dimensional geometry cannot be achieved experimentally, even in the absence ofectromagnetic effects. Surface tension causes rolls to form at both ends of the initally flat sheet

grow as the m etal falls, as in figure 3a. Of course, in any industri al appl icatio n, th e sheet would

longer and th us end effects less im po rta nt th an in these experim ents. A further departure froni-dimensionality occurs due t o the slight twisting of the she et when it leaves th e nozzle. Whcii

alternating, high frequency current flows in the inductor the departures from two-dimensionality

three-dimensionality of the magn etic field there . T h e resul ting curved cross-section aggravatestendency of the sheet to cont ract and its thickness increases. T h e me asure me nt of the deflectioii



270

(a ) No current, I = 0. (b) Wit h current, I = 1520Amps.Figure 3: Evolution of a two-dimensional mercury sheet.

The greatest error in calculating the experimental value of N derives from the estimate of L.

defined in terms of the distance between the centre of the coil and the liquid stream. The value 01L was measured on a photograph taken with a 45O mirror placed below the inductor. For maximaleffect, L should be small, when the accuracy of this measurement is low. The deflection angle, a .

was measured on photographs taken by a camera mounted adjustably in the plane perpendiculgto the metal sheet. The three-dimensionality of the deflected jet ensures that the measurement 01

a s not easy. It was decided to define Q as the average of the maximum and minimum defleclronsattained over the sheet. A curious feature of the observed deflection was that the stream did 1101

appear to bend until below the position of maximum pressure, whereas theoretically this shoultl

occur very slightly above that position. This was probably an observational efTect.T w o sets of experiments were performed, one w ith th e inductor horizontal (so th at only one b r a n d

of the inductor has a large effect on the stream) and the other with it vertical. The deflection R ~ S

noticably greater when the inductor was vertical, T he measured values of a in degrees) are plotletl

in figure 4 as functions of N , and N h . The theoretical line (18) and curve (19) are drawn on llir

figure for comparison. T h e agreement is satisfacto ry fo r a small, but not surprisingly, the theor!over-estimates the obtained deflection angle fo r N (and a ) arge. The manner in which this occur<

appears to be fairly systematic. There are a number of reasons why this might be expected. First,

equations (18) and (19) apply only when a is small, whcreas the experiinents cover a wide ratigt’

of a. The “weak deflection” approximation, whereby the magnetic field may be easily calculated,thus breaks down. Secondly, assumptions (a) and (b) are only weakly satisfied in the experimentsAs a result, the unidirectional irrotational flow assumed in the theory may not be wholly accurate

Th irdly , it is clear th at the three-dimensionality of the experiments will lead to a reduced deflectioiiIt also hinders observation of a as we have already discussed. Finally, we should recall the problenirinherent in the measurement of L . It may well be that there is a systematic under-estimation of I,

when a is appreciably large. Bear ing in m ind t he p ract ical difficulties, i t was felt th at the agreementbetween theory and experiment was satisfactory.



27 1

N”

Inductor horizontal Inductor verticalFigure 4: Deflection angle a n degrees plotted against Nh and N u .

. Concluding remarks

he use of electromagnetic fields as controlling devices in the metallurgical industry is growing.to mechanical means.

ally the fluid flow th at results is complex in nat ure , an d is often turb ule nt. In this paper,con tra st, we have investigated a process for which the flow is particularly simple, in which the

of a s tream of liquid metal is controlled. The analysis is fairly general and mayextended to cover particular geometries of industrial interest. Althou gh the fluic! dynamicalhniques we have used are elementary, they do seem t o give a reasonable description of the real

as witnessed by experiment.he main limitations of the process derive from its reliance on two-dimensionality and a veryfrequency. In [l] a second configuration is considered which avoids these difficulties, relying

If a thin, cylindrical metal

to carry a current, then it can be deflected by other suitably positioned currents. The

a j e t is calculated, and the question of it s stabili ty is briefly discussed.In conclusion, the au tho rs would like to offer their congratulations to the organisers of this veryccessful a nd enjoyable symp osium , and to express their g rat itud e for t,he lavish hospitality shown

in Riga.

References

[ l ] ETAY J . , M ESTEL A . J . , 8~ M O F F A T T II.I<., 1988 ‘Deflection of a s tream of liquid metal by

[2] SI1ERCLIFF J . A . , 1981 ‘Magnetic shaping of molten metal columns.’ I’roc. I t . Soc. 1,ond. A

[3] ETAY J . , 9r. GAR NIE R M . , 1082 ‘Le contrgle electromagndtiqiie des surfaces mbtalliqiiec:

of an alternating magnetic field.’ J . Fluid Mech Vol. 194, 309.

375, 455.

ses applications.’ J . Mdch. The or. Appl. Vol. 1, 911.

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