T‐MACr29r2r0994w

Analysis of Electromagnedc Force for Shaping the Free Surface
of a Molten Metalin a Cold Crucible

T.MoHsue■磯jilnalL KumeS.FttimoJ

Rep血"d fromIEEE TRANSACTIONSON MAGNETICSVd。 29,No.29 mrch 1993

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1EEE TRANSACyrIONS ON MACNEICS,VOL.29,NO.2,MARCH 1993
Analysis of Electromagnetic Force for Shaping the Free Surface
of a Mo:ten Metal in a Coid Crucib:e
To Morisue, T, Ya」 ima, T. Kume, and S. Fu3■ mori
Department of Chemical Engineering, Nagoya University」apan
Furo― cho, Chikusarku, Nagoya 464‑01,Abstrα ct ― To control the shape of the freesurface ofa molten metal in a cold crucible is very important for

obtaining a g00d product.

In this paper; an inverse

problem in which the currents in the induction coils areto be calculated to obtain the given fFee Surface of amolten metal in a cold crucible is solved
by using theboundary int9gral equation method and the least square
method.

the acce■ eration of gravity, respectively. The

IIlagnet ic

PreSSule at the point P(r(z),z) is given as:pm(r(z),z)= [二 (r(Z),Z)]2/2■ 0(3)wherO ■ and llo are the magnetic f■ ux density
permeab■ ■ity of free sPaCe, respectively.

andtheI. INTRODUCT10N
The schematic v■ ew of cold crucibles is shown ■n Fig.
1. There are two types of the problem of the free sur―face of a molten metal in a co■ d crucible. The first is:
The direct problem, to ca■ cu■ ate the free surface
given the co■ l currents[1〕 ,[21,[31,and the second is:

The inverse prob■ em,to caldulate the coll
given the free surface.

currents

ChargeFig。 2A given free surface

旦 iS CalCulated fron the magnetic vector potential‖C。

li●(b)

lnverse prob■ em methodology is important in engineetingstateddesign[4]. In this Paper, the inverse i:Pibb■ em
above is solved by using electromagnetic and hydrodynam―lc equatiOns and the ■east square method.

AO ‐ O

underof(4) The fluid f■ ow of the molten metal is negligible.
The shape of the given free surface is as shown inFig。 2. Let the free surface be written as:

the'r全1)/an'。 1/(4TIェ ー 11)d「+ ∫Э▲(二ニΓ:'二 =[x,y,z]crb■ e。

where(7)Ω and r denOte free sPace and the free surfaceOf

the molten IIletal, respectively.

In equilibrium, the following relation isestab■

ish―ed:≦ Z ≦ Z2

The liquid Pressure at the point P(r(Z),Z) is given as:Z)

Manuscript received Septenber 4, ■992

fol■ owing

thedeta■ ls:)/(4π―二:￨)dΩ∫
0二 0(二l二Ω■―∫ (ェ 1)。 Э[1/(4TI二 ―二'￨)]/ani d「1/2 全(三 )=The molten metal is axisymmetric.
The magnetic field does not enter the mo■ ten metal,or the normal component of lllagnetic field intensity
is zero at the free surface of the molten metal.
(3) The surface tension of the molten metal is negligi―

where p and g are the density of the▲ iS Calculated byequation。 (For theboundary integral equation method, refer to [51.)

(1)
(2)

Ps(r(z),2)= ρg(Z2(6)

since the magnetic f■ ux does not ex■ st inside the mo■ tenmetal. In this Paper,

the fo■ 1。 wing assumptions:

Zl

(5)

at the free surface,boundary integra■

ヽIn this chapter, the inverse probleIIl is solvedPs(r(z),z)= pm(r(z),z),(2)moltenmetallwhere 二O is the cO■ l current density. The boundary con―dition for全 = [Ar, AO, Az]= [0, AO, Ol is given as:(a) continuous casting type, (b)batch typer = r(z),i■▽包 +■ 0三 0=0

Schematic view of cold crucibles

II. THE INVERSE PR03LEM(▽ ・ A=0)

The magnetic fie■ d equation ■n free sPace is written
terms of the lllagnetic vector potential as:。 g vater(a)Fig。 lB=▽ xA￨￨as:andorpg(Z2 Z)=[旦 (r(z),2)12/2口 0
Since the system■s linear and the boundary condit19niS ,Oll10geneous (see (6)), the idea of the transfer func―

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IEDE TRANSAC10NS ON MAGNEICS,VOL.29,NO.2,MARCH 1993
tion is convenient■ y introduced to the inverse problem。
Letュ (1,」 )be the transfer function between二 (r(Zi), Zl)
and the j― th coi■ unit current. Then, the co■ l currents:
11,12,― ――,IN produce the magnetic field:
N―,M
旦(r(Zi),Zi)= 】 ■(■ ,j)Ij, i=1,2,― ―Table 2Co■ ■ No.

(9)

j=1

where N is the tota■ number of the co■ ls, and M is thetotal number of the points where the magnetic field in―tensities are evaluated. Combining (9)with (8) and us―
ing the least square method gives the coi■ currents: 11,12,― ――,IN・The computed resu■ ts for the case of N = 8 and M = 80

are shown in Table l and Fig.3, and the cOmputed results
for the case of N = 15 and M = 80 are shOwn in Tab■ e 2and Fig.4. The physica■ properties used are as f。 1lows:ρ = 2.3 x 103

kg/m:

rO = 2.5 x 10 2
Table lCoi■ No.

m,

口0 = 47r X 10 7 H/nl,z,―zl=5x10 2m

Computed coil currents fOr
the configuration with 8 co■ ls
COi■ current [A]1620.2

2‑79。

39o.8

4215.05420.96366.77293.081659.5

Computed coil currents forthe configuration with 15 coils
Coi■current

11591.6

2‑1344.5

3471.2

4‑3.5

5‑23.46117.67112.18133.49229.310191.011194.412233.0

1314322.6‑265.0154002.5[A〕

3,Fig。 4

Computed co■ ■ currents forthe configuration with 15 cOils

Ps(r(z),z)=pg(z五 ― z)+p*(2)1

Ps(r(2),2)― pm(r(2),Z)= τ[1/Ri(■ (2),2)+ 1/R2(r(2),2)]
Fig.3

where Rl and R2 are the principa■ radii of curvature of
the free surface and are exPressed as follows:

Computed coi■ currents forthe configuration w■ th 8 co■ ■sAs ■s seen from the computed results, it maybe

1/Rl + 1/R2 =d/dz[(dr/dz)//(1 + (dr/dz)2)]■m―:;it:in[° ti:allil.:liall:ilwi[ie8 11[:::i:l° ::il:. i:ithe other hand, it may be possible to realize iting the cOnfiguration with 15 induction coi■ s.
'°(8):

by us―

III. MODEL REFINEMENT
In the preceding chapter, the surface tension of themolten lneta■ is neg■ ected fOr simp■ icity. However,
thepurface tensiOn P■ ays an important role for shaping thefree surface Of a mo■ ten meta■ . In this chapter,thesurface tension is taken intO accOunt. Eqs。 (2) and (8)are changed to the fo■ lowing:+ (1/r)//(1 + (dr/dz)2),(10)

and t is the surface tension. P■ in (2)' is a constantand is determined by (8)i since Rl and R2 are known in
the inverse prob■ em. (As is shown in the■ast sectionof this chapter, the determination of Pお is complicatedin the direct prob■ em。 )Since the surface tension has a self― regulation func―tion, the refined mOde■ using the surface tension equa―tions ■mproves the computed resu■ ts obtained bythes■ mple lllode■ described in the preceding chapter.
Fig.5(a) and (b) denOte respect■ vely the computed results by
the simple model and the refined model for the cOnfigu―
ration with 8 induction co■ ■s. Fig.6 (a) and (b) denOte

IEEE TRANSAC■ ONS ON MACNEICS,VOL.29,NO.2,MARCH 1993

1564

respectively the computed resu■ ts by the simplemodeland the refinё d model for the configuratiο n with 15 in―duction co■ ls.
The colnputation procedure us■ ng the refined model isas fo■ lows:Obtained(1)The induction COi■ Currents: 11,12,― ――,INby the silllPle lnodel are used without any change.
(2) Calcu■ ate the shape of the free surface

using eqs.
(2)', (8)' and (10).
〈3) Calcu■ ate the magnetic PresSure at thё uPdated freesurface using eqs。 (3),(4),(6) and (7).

(4) Iterate the processes (2) and (3)uitil the shapeёfthe free surface converges.

r(.)
Fig.6 (a)Computed resu■ t by the si面 p■ e mode■

for the configuratiOn with 15

coilsr(■ )Fig.5 (a)Computed resu■ t by the simp■ e mode■

for the configuration with 8 co■ lsZ(m)

r (m)

‑0.

Fig.6 (b)COmputed result by the refined lllode■for the configuration with 15 coi■ swhere w = (dr/dz)//11 + (dr/dz)21. The bouǹary
tions for eqs。 (11) and (12) are given as:=(2)lz=0=rl(not knOwn in advaice),
W(2)lz=o=0.
(14)

r (In)

Fig.5 (b) Computed resu■ t by the refined mode■
for the configuration with 8 c。 1ls
ln comparison with the inverse Problem, consider thedirect problen defined in chapter I.
The mathematica■
model fOr the direct problem is derived fromeqs.(2)',(8)i and (10)as fol■ ows:

dr/dz=w//(1 ‑w2),dw/dz = [/(1 ‑ w2)]/r(11)― [pg(22― pm(■ (z),z)]/τ

condi―(13)

rl in (13) is iterative■ y determined such that w(2) Sat―lsfies w(zl) = ‑1. Pお in (12) is iteratively determinedsuch that the mass of the mo■ ten meta■ coincides
withthe given mass. The lnagnetic Pressure pm in (12)depends
upon the shape of the free surface. The shapeofthefree surface is iteratively ullated until the cOnverged
shape is obta■ ned. Therefore, the direct problem
hasthree iterative■ y determ■ ned Parameters, whi■ e the ■n―verse prob■ em has only one iteratively determined param―eter. An examp■ e of the computed resu■ t fOr the directprob■ etll is shown in Fig.フ . The tno■ ten meta■

used is ironand the assulned initial shape of the free surface

isasphere。

Z)+ P*(12)Note. There may exist a fluid f■ ow insidethemeta■ , and it IIlay be expected that the ve■ocity

fluid flow is very small asmo■ ten■ong as a stab■ e freeOfthesur―

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IEEE TRANSACrIONS ON MAGNEIICS,VOL.29,NO.2,MARCH 1993
face is maintainddo Assume that the fluid flow is asteady flow. The fluid flow is expressed by the Navier―Stokes equation[6]as follows:(ェ・▽)y = [ニ ー ▽p + η▽2ェ ]/ρ

[6]LoPrandt■ and OoG.Tletjens, Fυ r7dα θηどσノs ο′ わノdrο ―σnd αerOmecわ σr7′ cs′ Dover Publications, INC., 1957,ch.15, pp.251‑265.

(15)

where x, 二, P, and n are the vё locity, the volume force, the dynamic Pressure, and the viscosity, respective―
ly. The ca■ culation of the fluid flow has not been car―ried out yet.

Biography
ToMorisue is a Professor Of the DePartlllent of ChemicalEngineering, Nagoya University, 」apan.
T.Yalima iS a doctoral course graduate student of theDepartment of Chemical Engineering, Nagoya University,」apan.

T.Kume is a masteris course graduate student of the De―partment of Chemical Engineering,
Nagoya University,」apan.

S.Fulimori iS a masteris course graduate student of theDePartment of Chemical Engineering, Nagoya University,」apan.

molten metal(iron)

Fig.7

Computed result for the direct prob■ em

Nbte: The operating frequency of a cO■ d l crucible̲lies
between 10 kHz and 100 kHz, and the corresPonding skindepth ■■e, between l ltun and O。 3 nln. Therefore, the as―sulllpt iOn (2) in chapter II may be va■ id.

IV. CONCLUS10NS
In this Paper, an inverse problelll of the free sur■face of a mo■ ten metal in a dold crucible is solved byusing the boundary integral equation methodand the■east square method.
To foFmu■ ate the problem of the free surface as aninverse problem may be prom■ sing since ithe ‐inverse
prob16111 has ■ess itOrative■ y― determined― Parameters thanthe direct method. The inverse approach IIlay be i cOnSid―
ered to be comp■ ementary to the direct approach.
By

combining the direct prob■ em with thO inverse, pr6b■ em
gives an efficient and accurate approach to thefreesurface problem of a molten metal of a c。 ld crucib■ e.
REFERENCES[1l A.Cagnoud and 」。P.Brancher, ''Model■ ■,3 。f COup■ ed′Fff
Phenomena in e■ ectromagnetic levitation,1:rransacI′ οη5 0r7 ″αgnerics′ v。 ■.21, No.6,
pp.2424‑2427, 1985。

[2]A.Gagnoud and I.Lec■ ercq, "Free boundary problems
in electromagnetic levitation melting and continu―
ous casting," ′ffF rrα nsσ ctFons Oη ″σgη ef′ cs′ v01.
24, No.1, PP。 256‑258, 1988.
[3]RoMoreau, 〃αgnerOカ ノd″ Ody"σ m′ cs′ Kluwer Acadelllic
Publishers, 1990, ch.3, pp.87‑94.
」。
L.Coulo血 b,[41 SoR.H.Hoole, S.Subrananiam, RoSaldanha,and 」。C.Sabonnadiere, ::Inverse problem
methodolo―gy and finite elements in the identification
ofcracks, sources, materials, and their geometry ininaccessible locations," ′fFF rrα rtsσ ct′ Or7S O″ 〃ag―
nef′ cs,Vo■ .27, N。 .3, pp.3433‑3443, 1991.
ysls σnd des′ g″ οf[5]S.R.H.Hoo■ e, Co″ ρυter¨ αFded oη α′erecrrο

gnetFc deソ ′ces′ E■ sevier, 1989,"α ch.4, 22.122‑135。
29‑71, andch。 2,Pp.