Electromagnetic force computation

8/13/2019 Electromagnetic force computation

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ISI J International. Vol. 29 (1989), No. 6, pp. 462~468Three-dimensional Velocity Fields forMelts Produced by a Rotating Magnetic

NewtonianField

and NonNewtonian

O.J. ILEGBUSIand J. SZEKELYDepartmentof Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA02139. U.S.A.

(Received on June27. 1988, accepted in the final form on November18. 1988)

Amodel is developed to calculate fluid flow in Newtonianand non-Newtoniansystemssubjected to rotational electro-magnetic stirring. In the former case, transport equations for K, the turbulence energy, and L, its rate of dissipation, wereused to deducethe effective viscosity, while in the latter case, constitutive relations wereused to relate the shear stress tothe rate of strain. Thebody forces due to the rotating magnetic field are deducedfrom the established analyiical resultsobtained from the solution of Maxwell's equations.KEYWORDS:elocity; Newtonian; non-Newtonian; rotating; magnetic field.

l . IntroductionThe electromagnetic stirring of continuous castingsystems has gained widespread acceptance in recent

years. Anexcellent review of this subject is availablein Refs. l) to lO). Furthermore, in a recent paperSpitzer et al.11) have shownvery good agreement be-tween the theoretical predictions and experimentalmeasurements conducted using a mercury modelfluid.Most of this work has been concerned with. rela-tively modest fields, Iargely in the millitesla range,with corresponding melt velocities rarely exceedingl m/s and often being significantly below this figure.Indeed, the electromagnetic stirring literature sug-gests that melt velocities in the range of about 50cmls are likely to provide good operating conditions.The purpose of this work is to explore the effectof significantly increasing the magnetic field; underthese conditions, one can expect two things to hap-pen:(1) The melt velocities are likely to increase sig-nificantly.(2) For alloy systems, a melt-solid slurry willform, giving rise to the conditions usually termed rheocasting . Indeed, the works of Flemings andMehrabianl2) and Danzigi3,14) suggest that theserheocasting conditions would be attained by shear

rates in the range of I OOO~1Bo_th points (1) and (2) are essentially qualitative.Before an experimental program maybe rationallydesigned, one needs information on the actual fieldstrengths that are needed to provide these criticalshear rates and to examine how a non-Newtonianfluid containing a fairly high fraction of suspendedsolids would respond to these fields. Or, in otherwords, what fleld strengths would be needed in orderto produce acceptable velocities in melt-solid suspen-sions ?In this paper weshall examine the response of the

system to a somewhatidealized field confi~uration,but these results will, ofcourse, bc helpful in designinga real rheocasting system.2. Formulation

Let us consider a cylindrically-shaped metallic sys-tem consisting of magnesiumn which boron carbideparticles are suspended. This melt is stirred by arotating magnetic field ofstrength Bo, such as sketchedin Fig. 1.Thequestion is, then, to calculate the melt velocityfields for both turbulent and non-Newtonian condi-tions.This system maybe represented by writing downthe equation of continuity:

a laaz (u) +T~7(rv) =O ... .. . .. .(1)and the three componentsof the equation of motion :Axial momentumau au au avp I a(v ~ +7 ar rpeff- )]- -+u-- ::= -+-r az ar aza au

J ~~2D~ uoff7~ +FozTEuLr)

/RotatingMagnetBoron CarMagneium

E oro Pure MagnL~l h-5, 7cm-+

Fig.

,(2)

Carbide/MagnesiumSlurryMagnesium

1 Schematic of the electromagnetically stirred system.462 C1989 ISI J


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ISIJ International, Vol. 29 (1989), No. 6Radial momentum

- ap I a avP(v~~+u av 2rpeft ar- +- -r az ar r ar)] r2~r2+1[P au av 2peffv +Fr+ Pw--- +~z eff ar az

Azimuthal momentum3 )](v ar +u a~ 7 ar peffr ~(wr-:=:6lw o~~w I a

e ea aw ~raz p ff az Pvw +F..(3)

.(4)

2. 1. Turbulent FlowThe effective viscosity is defined in terms of themolecular and turbulent componentsthus:

Peff =p+pt pt is obtained from two characteristic properties ofturbulence, namely the energy lr and its rate of dis-sipation e, from the relation:

pt=CFPK2/e ........................(6)Kand e are, in turn, calculated from the followingtransport equations :

and

aK aK aK( ~i(v r-- +u =z arr aK_ __Laaz(~K az +GK-Pe

p(vae as I a(Pt ~i)+ _~(Pt aa)-+u --r az Ta'~~r a' (~r az (T= ~~.(7)

+CK(GK+S)- C2p,r - '(8)Theterm GKn Eqs. (7) and (8) is the rate ofproduc-tion ofK, and is dcfined as:

2 2GK= ;L av 2 _) azau 2 aw, [(() +2 ++ *~zar' )J} [r~(~u av 2+ --+- .(9)ar az

The term Sin Eq. (8) is used to account for the in-fluence of streamline curvature on the turbulence.Following Spitzer et al.,15) whoassumeda suggestiondue to Bradshaw,13) this term is defined thus:7(~r(~~)S= Cspt .(lO)

in which C* is an empirical constant whosevalue isl.6.2. 2. JVlon-JVle~)tonian Flow

At high solid loading, the flow becomesnon-New-tonian. For this situation, it is convenient to writethe equation of motion in vectorial form in terms ofthe stress tensor r thus:

'_u'Vp~ = -VP V r+FE .........(11)in which ~E is the electromagnetic force per unitvolume.Weshall assumethat the semi-slurry system obeysa power-law relationship thus:

~==-m V~(A=A) n-1 }A .(12)where, Ais the rate of deformation tensor and mandn are empirical constants defined by the relations :

m= e(9.783f.+1'4345) ..........................................(13)O.1055+0.4lfs for 0.15:~fs ........(14)n= _O.308+1.78f., for 0.03~f.s~0'60 ........(15)in whichf, is the solid fraction.

It should be noted that the precise nature of thisrelationship has been deduced for a lead-tin systemby Joly.16) It is not immediately obvious whetherthis relationship will hold for a magnesiumjboroncarbide system; incleed, it is likely that somemodifica-tions will be needed. Work is currently in progressin that respect. It is felt, however, that the generalnature of these systems, particularly the shear-thin-ning beh_avior, would be well represented by a powerlaw type equation.Wenote here that the terms Fz' Fr and Fe in themomentumquations (2) to (4) represent the appro-priate electromagnetic body force field components.Thesequantities maybe evaluated through the mani-pulation of Maxwell's equations. Details of the esti-mation of these fbrce componentsfor a rotating mag-netic field, assumedsinusoidal in time and in ang_ularcoordinates, have been given inll) and will not berepeated here. Weshall assumethat the system ofequations used in Ref. I l) for calculating the electro-magnetic force field will be applicable to thc presentcase. The non-Newtonian behavior will, of course,not affect these considerations and the fact that afinite, rather than an infinitely long cylinder is beingused is thought to be a good flrst approximation forobtaining information on the general nature of thesvstem.

Thus the force componentsfor a single-pole mag-netic fleld with amplitude Bo are:Fz =O ... .........(16)B~(w-T)2 3 ...l z~'Fr = ~~ '(17):2 porF I we B~(w . . . . . .( 18)- ,~r~r

2. 3. Boundary ConditionsThe boundary conditions needed to specify thefollowing physical constraints are :-symmetryabout the centerline ;-no slip at the solid surfaces introduced through wallfunctions; and

-zero shear at the free surface.In these calculations wedid not allow for the de-formation of the free surface, which was certainly an463


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ISII International. Vol. 29 (1989), No. 6oversimplification; thus, the computed results to bepresented would correspond to a system where freesurface deformation was artificially prevented, e.g.,through the use of a frictionless lid . This assump-tion is not thought to introduce a serious error as faras the general nature of the system is concerned.3. Methodof Solution

The governing equations were put in a finite dif-ference form and the.n solved using a modificationof the PHOENICSomputational package.17) Theprincipal novel features were:-the allowance for the electromagnetic force fie]d ;-modification of the program to represent non-New-tonian behavior; and-modification of the turbulence model to allow forthe extra source term due to swirl.A spatially non-uniform 28x 18x 36 grid wasused,and a typical computational run required about 3hon a MicroVAX11 digital computer for acceptab]econvergence.4. Results

A selection of the computedresults is given in thefollowing. Theprincipal input parameters employedin the computation are summarizedin Table l.Figs. 2 and 3((a) to (c)) show a set of the com-puted azimuthal velocity flelds.In all these plots (a) refers to a position close to thebottom of the container (z/H=0.2), (b) to a positionabout midway(z/H=0.5), and (c) to a plane close tothe top surface (z/H=0.8). It is seen that there isrelatively little variation with vertical position.Fig. 2depicts a system with a field strength of 1.25kG, while Fig. 3 shows the behavior of the systemwith a field strength of 0.5 kG. Both these refer toturbulent conditions, i.e., a solid fraction of zero.Figs. 4(a) and 4(b) showthe radial and axial variationof the maximumalues of the azimuthal velocity,respectively, for turbulent flow (f.=0). The strongspatial dependenceof this variable is readily noted inFig. 4(a). As expected, the azimuthal velocity showsa very markedradial dependenceand decrease quitesharply as the central part of the system is being ap-proached.

Figs. 5(a) and 5(b) show plots of the radial andaxial variations of the maximumate of turbulentene-rgy dissipation, respectively. Onceagain, there isstrong spatial depenclence with maximumalues oc-curring in the top core region. However, the im-

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\ \--\+-:;~~~~i:+';__,>i~i+': ::..

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_--'~'+*~ '* ~~;~ ~~~

(b)

~~:~.;~~( ~--~ ._=_~~~:'~:~::::__

Table 1. Proposed input parameters.Melt MegnesiumSolid Boron carbideParticle density 2 52x 103kg ' m~3Melt densitv

. I . 81 x 103 kg.m~8Kincmatic viscositY 6. 9x l0-7 m2/sAmplitude of magnetic field O-I .25 kgAngular velocity of field 3OOOrpm- 7 M/S(a) z/H=0.2 forB0=1.25kG and.f*=0

(b) z/H=0,5 for B0=1.25 kGandf* =0(c) ~/H=0.8 for B0=I ,25 kGandf* =OFig. 2. Velocity field.

464


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ISIJ International, Vol. 29 (1989), No. 6~_~~:\1'~~:~~~:~;~~~4'-~-~t-1~__~'~~ *

~'____;~~\~~\~~t\\~\\\~~~:~;~~(~~:r//::::/:7:r/::/,:///iE:~ ~ii:.\\\\~::::\\~~:,\\~~\\

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/~;////~i///// ~: ~~, \ll

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~~c) ~~~- 6'O M/S(a) z/H=0'2 fOrBo;=:1'25kG andfs=0(b) z/H=0'5 for Bo:=0'5kG andfs=0(c) z/H=0'8for Bo:=0'5kGandfs=0Fg' 3' velocity field'

15 O12.510 O

\ 75~~ 5.025

f. I~ O

B = 125KG

B. = O50KG

OO 1O2 O4 O6 0.8r/RFig. 4(a). Radial variation of maxirnumazimuthal velocity

forf* =Oas a function of magnetic field.15 O1~.510.0~,)\~ 75~ 5025Oo

is = O

B. = I e5KG

B = 050KG

O 0~ 04 06 08 1z/HFig. 4(b). Axial variation of maximumzimuthal velocityforf*=0, as a function of magnetic field.

portant point to note here is the quite high value of a.Fig, 6depicts a system with a solid fraction of 0.4,exhibiting strongly non-Newtonian behavior. Thevariation of maximumzimuthal velocity with mag-netic fleld as a function of solid fraction is showninFig. 7. The effects of solid loading on the spatialvariations of the maximumazimuthal velocity areshownin Figs. 8(a) and 8(b).Examination of these plots showsthe following:(1) Comparisonof Figs. 2and 7, and observationof Fig. 6, showan almost linear relationship betweenthe maxirnumvelocity and the applied field strength;this is the expected behavior, which maybe foundfrom order-of-magnitude approximations as well asfrom the prior computedresults of Spitzer et al.11)(2) Perhaps of the greatest practical interest isthe behavior of the non-Newtonian melt with the40 o/o Solid fraction, where the velocities are seen toexhibit a maximumomedistance from the wall; thisis due to the nature of the constitutional relationship.More specifically, the local shear rate will be at itsmaximumt someposition near the vertical solid sur-face, which in turn will cause a local minimumn theapparent viscosity. Thenet effect of this behavior is

465


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,l

ISIJ International, Vol. 29 (1989), No. 6

co

\~~

2000

1500

1000

500

o

f. = O

B. = 125KG B = O50KG

Fig. 5(a).

cl,\\2~

O 02 04 06 08 1r/RRadial variation of maximumurbulent energydissipation rate forfs=0, as a function o_f mag-netic field.

30002500200015001OOO500

Fig. 5(b).

o

fs = O

B. = I ~5KC;

B = 050KG

O 0~ 04 C8 16z/HAxial variation of maximumurbulent energydissipation ratc ibrfs=0, as a fhlnction of mag-netic field.

a rather sharper local maximumn the melt velocitythan would be expected for turbulent flow conditions.5. Discussion

Amathematical formulation has beendeveloped torepresent the fluid flow in a cylindrical body of puremagnesiumand a magnesium-boroncarbide slurry,when agitated by a rotating electromagnetic forcefleld. The appropriate electromagnetic force fieldequations were taken from the prior work of Spitzeret al.11); in the statement of the fluid flow equations,allowance wasmadeor either turbulent behavior inthe case of pure magnesium, or ibr non-Newtonianbehavior, in the case of a magnesium-boroncarbideslurr y.Virtually all of the prior published work on theelectromagnetic stirring of molten metal systems hasbeen confined to relatively low flelds, such that thecorresponding melt velocities tended to be belowabout I m/s or less. Furthermore, up to the present,no published data have been available concerning the

(a)

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(c ) -- 5.0 l(/S(a) z/H=0.2for B0=1.25kGandf*=0.4.(b) z/H=0.5 forB0=1.25kG andf*=0.4.(c) z/H=0.8 forB0=1.25kGandf*=0.4.

Fig. 6. Velocity field.

1

466


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ISIJ International, Vol. 29 (1989), No. 6

\~~

Fig. 7.

~,,\~~

Fig.

~,,\s~

Fig.

15 OIZ.510 O7.55,0

2.5

o.o

fs * Ofs = 08f. = 04

O O~5 O5 O75 1 1~5BO(KG)Maximumzimuthal velocity with magnetic fieldwith solid fraction as a parameter.

15 O12.510 O7.5

5,02.5

0,0

8a).

Bo = 1.25KG

fs = 08

fD = 04

O O~ O4 0.6 O8 1r/RRadial variation of maximumzimuthal velocitywith solid fraction as a parameter.

15.0l~.510.07.55,0

250.0

8(b).

Bo = 125KGfs = OS

fs = O4

O 0~ 04 08 16z/HAxial variation of maximumzimuthal velocitywith solid fraction as a parameter.

velocity fields in electroma~netically stirred melt-solidsuspensions. In this earlier work, again, relativelylow fields were considered, in conjunction with avertical motion in the crucible, and again, quite lowmelt velocities. Indeed, the preliminary conclusions

that were drawn from this earlier paper were theinherent difliculties associated with electromagneticrheocasting. While electromagnetic patents do ex-ist,13,14) thcse do not provide either measurementorcalculations concerning the melt velocities.Themost important flnding of the present paper isthat, for a rotating system, provided high enoughfields are employed, it appears to be quite possible toagitate melt-solid suspensions at high enough shearrates to obtain good fluidity and hence quite highmelt velocities. The calculations presented in thispaper indicate that fields of the order of I kG orhigher should perform adequately.While the calculations have been performed for amagnesium-boroncarbide slurry, qualitatively verysimilar considerations should bc expected for a broadrange of metallic systems. Indeed, these calculationsshould be indicative of the behavior of both rheo-casting and compocasting systems.The preliminary nature of this work must bestressed at this stage. Areas where further researchis needed, and is indeed pursued by the present re-search team, include the following:

(1 ) Thecalculations were presented for a specific,predetermined field. Theactual coil design to bringabout such a field in the absence of excessive Jouleheat generation is a critical practical issue. Never-theless, the computed results presented here shouldprovide adequate incentive for developing such coildesigns.(2) The calculations were presented using one

speciflc constitutive relationship relating shear stressto strain in the melt-solid slurry. It is uncertainwhether the actual relationship used is universallyapplicable; nonetheless, the general shear thinningbehavior of these systems was thought to be wellrepresented, at least in a semi-quantitative sense.(3) In the calculations, the electromagnetic forcefield was calculated fbr an infinitely long cylinder.The behavior of finite systems will be somewhatdif-ferent, although the general nature of the findings isunlikely to be affected by such a refinement.(4) In the calculations weassumeda non-deform-able free surface. Clearly, unless somemechanicalrestraint is placed on the top free surface, significantdeformation will occur due to melt rotation. Indeed,this deformation, if not prevented, could cause sig-nificant practical operating problems. However, inexamining the computedresults, there were no dra-matic differences in the behavior betweenthe top andthe botto_m part of the vessel. T~his would indicatethat the results would be representative of a systemwith a solid cover.Notwithstanding these caveats, the results seemtoindicate that electromagnetic stirring, with properlydesigned stirrers, could play an important role in bothrheocasting and compocasting.Nomenclature

Bo: Amplitude of magnetic fieldf*: Solid fractionFE: Electromagnetic force per unit volume467.


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ISIJ International, Vol. 29 (1989), No. 6F., Fa, F. : ComponentsofF. in r, 6and z directions,respectivelyGK Rate of generation of turbulence energyH: Height of' meltK: Turbulent kinetic energym, n: Empirical coefficients in power-law relation-

shipP: Static pressurer: Radial coordinateR: Radius of containerS: Termin source of e due to swirlu: Velocity componentin z-directionu~: Maximumv: Velocity componentin r-directionw: Velocity componentin e-directionw~: Maximum~'z: Axial coordinateA: Rate of deformation tensore: Rate of dissipation of turbulence energyMaximum~ .6: Azimuthal coordinate

,i:: Electrical conductivity;L : Molecular viscosity

pt : Turbulent viscosityP*ff : Effective viscosity

po : Permittivity of free space(rK, (T. : Prandtl numbersfor Kand e, respectively

t: Shear stress tensora' : Angular velocity of electric current

AcknowledgementThis work was sponsored by the National Aero-nautics and Space Administration under ResearchGrant #NAG~3-808.

l)

2)

3)4)5)6)

7)8)

9)10)ll)12)13)14)

15)16)

17)

REFERENCESR. Alberny and J. P. Birat: in Continuous Casting ofSteel, Biarritz, (1976), I16.E. D. Tarapore and J. W. Evans: Metall. Trans. B, 7B(1976), 343.H. S. Marr: IronSteellnt.,51 (1978), 87.K. H. Tackeand K. Schwerdtfeger: Stahl Eisen, 99 (1~79),7.S. Kolberg: ASEA., 53 (1980), 89.K. H. Spitzer, K. H. Tacke and K. Schwerdtfeger: Proc.Symp.Metallurgical Applications of Magnetohydrodynam-ics, Trinity College, Cambridge, (1982).H. K. Moffat: ZAMM,8 (1978), T65-T71.K. Fujiwara et al.: Electromagnetic Stirrer for Use in aContinuous Steel Casting Apparatus , U.S. Patent No.4,406,321, Sept. 27, 1983.J. P. Birat andJ. Chone Ironmaking Steelmaking, 10 (1983),269.R. D. Matthewson, L.J. Longand D. T. Hackworth: IronSteel Eng., 63 (1986), No. 9, 36.K. H. Spitzer, M. Dubkeand K. Schwerdtfeger: Metall.Trans. B, 17B (1986), 119.M. C. Flemings and R. Mehrabian: Trans. Am. Found.Soc., 81 (1973), 81.J. A. Danzig: Process and Apparatus for Casting Metalsand Alloys , UKPatent No. 8005620, (1980).J. A. Danzig: Process and Apparatus Having ImprovedEfficiency for Produc,ing a Semi-Solid Slurry , EuropeanPatent No. 82105446.7, (1982).

Effects of Streamline Curvature on Tur-. Bradshaw:bulent Flow , AGARDographo. 169, (1973).P. A. Joly: Ph.D. Thesi_s to Departmentof Materials Sci-ence, Massachusetts Institute of Technology, (1979).D. B. Spalding: Mathcmatics and Computers in Simula-tion, XIII, North Holland, Holland, (1981), 267.

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Electromagnetic force computation