Numerical analysis of electromagnetic force field through the use of two examples

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Baosteel Technical ResearchVolume 7，Number 3，September2013，Page50Numerical analysis of electromagnetic force field through the use of two examplesBAI YunfengSpecial Steel R &D Center，Research Institute，Baoshan Iron& Steel Co．，Ltd，Shanghm 200940，ChinaAbstract：In modem processing of conducting materials．such as stee1．the time—varying electromagnetic field plays a keyrole in obtaining the desired microstmcture or eliminating solidification defects such as porosity and segregation in castbillets．Up to now．few studies on the induced electromagnetic force f also called the Lorentz force)field in 1iquid metalhave been reported．Compared with the magnetic field．the induced force field is the real and only direct cause for flowcontro1．The electromagnetic force is comprised of two components．One is time—independent and the other is time—dependent．Th e time—dependent component varies with time in both amplitude and direction．W hen it reaches the extremevalue，it can be one dozen times 1arger than the time—independent component．In this paper．a new method to quan titativelydescribe the induced electromagnetic force in liquid metal under a harmonic electromagnetic field．including both its time—independent an d dependent components，was proposed based on the formula derivation from the data of amplitude andphase an gle．Through this method，the features of the time—dependent component were discussed．including the directions ofrotation and the long axis．As a result，the force pattem was described．W ith two example calculations．the method wasexplained in detail．Th e results of both examples show that the force field in liquid metal can be divided into severalregions with diferent force features．Exam ple 1 shows the effect of coil position on the evolution of the force field patternin liquid meta1．Exam ple 2 is a kind Of sti州 ng bV the travelling magnetic field．whose results present the sub—structures inmeta1 and show that most of them have almost the same rotating direction.

Key words：electromagnetic force field；numerical an alysis；time—dependent component；quan titative description；sub—structure in force fielddoi：10．3969／j．issn．1674—3458．2013．02．0101 IntroductionA1temative electromagnetic fields are widely used inthe processing of conducting materials．This is one ofthe hot topics in electromagnetic processing ofmaterials (EPM ) and many studies have beenpublished recently ,2 A
． In the liquid metal processing，electromagnetic fields are USUally imposed to obtain acertain kind of electromagnetic force field，also calledthe Lorentz force，which affects the flow field directlyand then affects thermal and concentration tran sferindirectly．Thus the solidifcation process can becontrolled to some extent ’ ．The typical applicationsof electromagnetic force fields are electromagneticstirring by travelling or rotating harmonic magneticfieldsl J and electromagnetic confinement shaping E6,7]
． To the author’s knowledge．the induced force fieldin liquid metal under a harmonic electromagnetic fieldhas so far not been described mathematicaly．In thisstudy，the author has developed a method to describethe force field pattern ．based on numerical calculationresults and form ula deftvations of the phase angles andamplitudes of sinusoidal quantities in the eddy currentzone under an electromagnetic field.

Corresponding author：BAI Yunfeng；E-mail：b~yunfeng2013###sina．cn2 Mathematical fundamentalsThe folowing assumptions have been made in thisstudy：the problem discussed is a 2一dimensional one：the exciting current density，referred to as JxExct，and theinduced eddy current，refered to as JGx(Y，z)，areparallel to the X axis；the induced magnetic fieldsBy(Y，z)and Bz(Y，z)are in plane YOZ，and theirfrequency and angular frequency are refered to as fand c【J．respectively．The source or the exciting curentin coils can be written as follows：JxE t=JXExctmsin(mt)= 艇xcf sin(2~-ft) (1)Then the sinusoidal quantities of the inducedelectromagnetic field at the given point(Y，z)can bewritten as follows：By( ，z)=B (Y，z)sin[ogt+ l(Y，z)]Bz( ，z)=Bzm(Y，z)sin[ogt+ 2(Y，z)]Gx(y，z)= G (Y，z)sin[mt+ 3(Y，z)In the above equations，t is time，suffix m(2)standsf0r the amplitudes of sinusoidal quantities，and ．(i= 1．2，3)stands for their initial phase angle(IPAfor short) relative to the exciting source．Both theamplitudes and IPAs can be obtained by thecalculation of transient quantities and the method canbe found in literature .

When the amplitudes B ，B and JG ，and IPAsare all prepared wel，the body force F，in N／m ，canBAI Yunfeng．Numerical analysis of electromagnetic force field through the USe of two examplesbe calculated with the folowing equation：一 1 一~，o BzmCOS( 3一 2)J。 {— _ +JGxmBzmcOs(2o9t+ 3+ 2)J②JGxmB v,．COS( 3一 1)k③+JGxmBymcos(2wt+ 3+ 1)k、— — ④= [① +③]+[② +④]=F 。 +Fp (3)where J and k are the unit vectors along the Y and Zaxes，respectively．The body force F is comprised of4 terms，represented as①，②，③ and④ respectively.

According to their relationships with time，F can bedivided into two parts，F and Fp ls t g
：1F 。 一I — 1 J
G [一BzmCOS( 3一 2)J+B cos( 3一 1)k]1p l 。 g = ÷-， [Bancos(2wt+ 3+ 2)J—Brmcos(2o9t+ 3+ 1)k] (4)F is composed of① and③ and is independent oftime．It is a constant quantity that only varies withposition．F Du】 is composed of② and④．It isdependent on time and also varies with position．Bothits magnitude and direction chan ge with time with acycle of0．5T(T=l／f).

The end of the composition vector Fpu ￡is anelipse with time[． Its rotation direction can beclockwise or anticlockwise．Its long axis，which is inthe direction of the maximum forces，can be located indifferent quadrants in the 2一D case．Both its rotationand long axis directions can be determinedmathematically according to the following method：The phase angle in term ② in Equation(4)is OL=(2wt+ 3+ 2)，and then term ④ can be writen as：1④ =一÷ G cos(2rot+ 3+ 1)= —1
．，G 曰 c。s[ +( 1— 2+7r)]： -，。 BymCOS( + ) (5) =— o where／3=( l一 2+7r)．The folowing statements aretrue：the rotation of Fp ln is clockwise when sin／3>0and anticlockwise when sin <0，and when sin =0，the ellipse of Fp l ti g degenerates to a line segment；the1ong axis of the ellipse 1ocates in quadrants 1 and 3when COS 8>0 and in quadrants 2 and 4 whenCOS 8<0 .

3．1 GeometryFig．1 shows the calculation mode1 and its dimensionsare in milimeter．Th e model is symmetrical to the Z5laxis．1]he properties of the materials are all constant．The1iquid metal’s size is 40 1Tlnl (width) × 100 IL1TI(height)，its resistivity is =1．71×10一 D_m，thefrequency fis 60 Hz，and the current in every coil is Jq= 3．6×104sin(2盯疗)A／mz．1]he three coils are equallyspaced with the center at point D．Th e distance from Dto the Y axis is represented as L．In the calculations inthis Paper．L varies from 40—7(】mm downwards.

yFig．1 Illustration of Calculation Example 1 and thecoordinate systemIn the data processing，the area concerned is{(Y，z)l0．5≤Y≤39．5．一99．5≤z≤ 一0．5}．which is fl 1ittlesmaller than the one in Fig．1．Th e benefit is that theerror caused by the data management method ofelectromagnetic field calculation software can beminimized.

3．2 Distributions of amplitude and IPAWhen L = 一40 1TITI，the amplitudes，B ，曰 andJaXm as well as their IPAs， l， 2 and 3 were ploted，as shown in Fig．2．At the point of z= 一40 mlTl(thevertical center of coils)， ‰ reaches its minimumvalue，as shown in Fig．2(a)，while and reachtheir maximum values．as shown in Figs．2(c)and2(e)．Al the three amplitudes， ，曰 ，and ，attenuate along the width direction from the outersurface to the center(the Z axis)．The IPA of B1eaps at the point of z= 一40 inin from about 0．5(1．570 8)to about 1．5竹 (4．712 4)，as shown inFig．2(b)． ， and are about 1．51丁 and 2-rrespectively at the outer 1ayer of 1iquid metal f f．e.

Y=40 ITnTI)，as shown in Figs．2(d)and (f)．A11 ofthe three IPAs also attenuate along the width directionfrom the outer surface to the center.

3．3 DiseussiOilS on force vectorsFig．3 shows the distributions of electromagneticforce vectors．Fig．3(a)shows the distribution of， ．Figs．3(b)and 3(c)show the distribution ofF=F 。 +Fp l。 ti when t=0 and t=盯／(6w)respectively．W e can see that ean is relatively smalcompared with F (when t=0 and =~r／6oJ)．Th ere aretwo reasons for that．Firstly，F is a superimposition ofF 。 and Fp l ti ，which increases the probability of itbeing larger than Tm。 ．Secondly，according to
BA1 Yunfeng．Numerical analysis of electromagnetic force field through the use of two examples3．4 Discussions on FpulsatingIn this case of L = 一40 mm．within the region ofz> 一40 rain，there is approximately：一 B、 COS(2wt+3．57r) 一BB： COS(209t+3．57『) B：(6)Within the region of z< 一40 mn-i，we can also get：一 B、 cos(2rot+2．57r) Bv ?B cos(2wt+3．5万) BThe above results show that the directions of longaxes of the F。 1 ating ellipse are about the same in regionsof z> 一40 mlTl and z< 一40 mm.

Fig．4 shows three ellipses at three points in liquidmeta1，A (39．5，一99．5)，B (39．5，一40．5) andC(39．5，一0．5)．Fig．4(a)shows the elipse ofFpat point A，with its long axis located in quadrants 1 and3．At this point，the elipse is anticlockwise．Fig．4(b)is the ellipse at point B，where the ellipse degeneratesto a 1ine segment．which means that the short axis isalmost zero and the composition vector has only twoopposite directions，fight or left．Fig．4(c)shows theelipse of F= 1In。 +Fp l i )at point C．The elipsethere is clockwise．with its 1ong axis located inquadrants 2 and 4．In the case of this calculation，thethree ellipses at the three points have the same featurein that the ellipses are all very thin．That is to say．thelong axis is very 1arge compared with the short axis.

The red arrows in Fig．4 represent F 。 ．It can be seenthat F? there is rather smal1.

Fig．5 shows the directions of rotation and the longaxes of the force elipses when L= 一40 mm．Fig．5f a)shows the rotation direction of the force elipses：“1”stands for clockwise an d“一1”for anticlockwise：“0”stands for the case of the force ellipse degenerating to aline segment．We can see that the force vectors areclockwise in most regions in the upper part of liquidmeta1．while in the 1ower part of liquid meta1．the forceellipses rotate anticlockwise.

(a)A(39．5，-99．5)， (b)B(39．5，__4O．5)， (c)C(39．5，一O．5)anticlockwise left or right clockwiseFig．4 Elipses of F at three points in liquid metalFig．5(b)shows the direction of the long axes：“2”stands for the long axis direction located in quadrants 1and 3；“一2’ for the long axis direction located inquadrants 2 and 4．We can see that in the region of0> 一40 mm ，the long axes are in quadrants 2 an d 4，(a)Direction of rotationwhile in the region of z< 一40 mm．the long axes arein quadrants 1 and 3．From Fig．5，we can see that thevertical center of the coils plays an important role indeciding the direction of the force ellipse long axis inliquid meta1.

O(b)Direction of long axesFig．5 Direction of rotation and polarization long axesThe line segment area and z= 一40 lnln divide liquidmetal into three parts，labeled as A，B and C，as shownin Fig．6(a)．In region A，the rotation direction isclockwise，as shown by the circle arrow，and the longaxis is in quadrants 2 and 4．as shown by “＼”．RegionC has the exactly opposite features as A，where the∞【x 。0。口。【] 554force rotates anticlockwise and the 1ong axis is inquadrants 1 and 3．as shown bv “／”．Region B is ahybild zone of A and C，where the force ellipse rotatesin the same direction as the one in region A．but thelong axis is in the same direction with the one inregion C.

Another 3 cases were calculated an d the force fieldpatterns were plotted，when the coil position moveddownwards with L = 一 45 rnn1． 一 50 1Tlnl and一 70 111．respectively．The force patterns are shown inFigs．6(b)，6(c)and 6(d).

Fig．6 f b)shows the case when the coils moved5mm down further．It can be seen from the figure thatthe force field features in regions A，B and C are． ＼，一 ／ ／， ／cBaosteel Technical Research，Vo1．7，No．3，Bep．2013similar to those in the case of L= 一40 nui1．but a smal1region D emerges，whose features are exactly oppositetO those of region B．When the coils continued to movedown to L= 一50 mm ．both the eddy current zone andcoils were symmetric to z= 一50 1TUTI．It was a specialcase and the force pattern is shown in Fig．6(c)withsymmetric features．We can see from the figure thatregions D and B have the same size and so do regionsA and C.

W hel the coils moved down further to L =一 70 mm．the force patern is shown in Fig．6(d)．Itcan be seen from the figure that region B disappears，region C decreases and regions A and D expand，whichis like a mi ror image ofFig．6(a).

§、／ ／刃 c～
、
t .

D ＼ ‘，9 、
(a) =—4Ofilm (b)￡=—45mlTl (c)三 一50mm (d)，． 一70millFig．6 Electromagnetic force patterns versus the position of coils4 Calculation Example 2， results anddiscussiOilS4．1 GeometryFig．7 shows the dimensions of Example 2．Theresistivity of the billet is 2．42 × 10 Qm．theresistivity of the coil material is 1．673×10～ llm ．andthe relative permeabilities of the two materials are both1．The resistivity and relative permeability of the ironcore are 1．25 x 10～ fZm and 2000 respectively．Thewhole model is symmetrical along z=0．The initialphase angles for coils A ．B an d C are 0，120。 and一 120。respectively．The frequency f is 60 Hz．and theamplitude of exciting current is J = 1 0 ×sin(2~ )A／ ． ．
．fi m In the folowing discussions the areaconcerned is{(Y，z)I l0．5≤Y≤69．5。一60≤z≤60}.

4．2 Distributions of amplitude and IPAThe distilbutions of induced magnetic fields．eddycurrent field and their IPAs are shown in Fig．8.

Figs．8(a)and 8(b)are B and its IPA；Figs．8(c)and 8(d)are日 and its IPA；Figs．8(e)and 8(f)arethe eddy curent density J and its IPA.

It can be seen from Figs．8(a)．8(c)and 8(e)thatB Ym reaches the mi nimum value，while B and Jreach the maximum values at the coil centers along theZ axis．We can see that although the model issymmetrical in geometry．the induced electromagneticfields are not．The distilbutions of IPAs are shown inFigs．8(b)，8(d)and 8(f)．The three figures havealmost the same shape，like a piece of cloth being tomat diferent positions，for IPA of B at z= 一10 mm.

for IPA of B at z= 一16 ran3 and for IPA 0f G，Y『"atz= 一20 1TITI.

Fig．7 Dimensions of calculation Example 2 and thecoordinate systemThe rotating direction of F。 l t． ellipse is shownin Fig．9．It can be seen that the rotating direction inmost regions is clockwise． Fig．1 0 shows thedirection of the polarization 1ong axis．It can beseen from the figure that the force field is dividedinto several sub—regions，but no obvious principlecan be drawn.

The distribution of F is shown in Fig．1 1，whichindicates that the force vectors have almost the sameleftward direction in the region near the coils．Thiskind of distilbution can help to drive the liquid to flow
56gg＼ 吉≥80604020- 60 —40 —-20 0 20 40 60Height／mm[4]Z暑苫 [5]l
§Fig．11 Distribution of time-averaged electromagnetic forcevectorWith 2 calculation examples，the evolution of theforce patern with coil movement was plotted and theforce field in a bilet stirred by a travelling magneticfield was analyzed．Both examples show that the forcefield has sub—structures，which need to be studiedfurther.

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