Mathematical Analysis and Numerical Simulation of High Frequency Electromagnetic Field in Soft Contact Continuous Casting Mold

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1 , pp Mathematical Analysis and Numeical Simulation of High Fequency Electomagnetic Field in Soft Contact Continuous Casting Mold Xianzhao NA, Xingzhong ZHANG and Yong GAN National Engineeing & Reseach Cente fo Continuous Casting Technology, Cental Ion & Steel Reseach Institute, No. 76 Xueyuan Nan Road, Beijing , China. (Received on Mach 12, 2002; accepted in final fom on May 24, 2002 ) The electomagnetic paametes in soft contact continuous casting billet mold wee analyzed in this pape by mathematical analytic method and numeical simulation; the optimized fequency ange was fixed on Hz. The distibution of magnetic flux density and electomagnetic body foce in split mold was obtained; the eseach esults achieved espectively by mathematical analytic method and numeical simulation agee with each othe. With incease of fequency of electomagnetic field, the electomagnetic body foces acted on the suface of stand incease, and attenuate apidly towads the cente of stand. When the meniscus was located at the middle of coil height, the electomagnetic body foce acted on it is biggest, and then deceased along the casting diection to almost zeo at the location of chage bottom. On the tansvese diection pependicula to the casting diection, the asymmetical distibution of magnetic field at the suface of stand pemeated though the slit of mold was enhanced slightly with the incease of fequency. The magnetic flux density at the slit aea is about 10% highe than that at othe aea at Hz, and the distibution of magnetic body foce is almost even on this diection. KEY WORDS: soft contact solidification; electomagnetic continuous casting; mold; mathematical analytic method; finite element method; numeical simulation. 1. Intoduction The initial solidification behavio of continuous casting stand is vey complicated owing to simultaneous existence of liquid steel, slag, coppe mold and mold oscillation. The initial solidification shell defoms peiodically along with the peiodical oscillation of mold, and finally the oscillation maks ae fomed at the suface of stand, it will become the cause of tansvese cack and dawing beakout in some exceptional condition. Theefoe, how to contol the initial solidification behavio of stand, educe and even eliminate the peiodical defomation of meniscus and initial solidification shell, is the fundamental of impoving suface quality of stand, inceasing casting speed and poduction efficiency. Enlightened by the aluminum casting without mold, contolling and impoving the initial solidification behavio of stand by imposing electomagnetic field on the meniscus aea is all though one of focuses was paid attention to by many metallugical eseaches. Due to the highe density, highe melting point, lowe conductivity of steel, and the chaacte limit of existing conducto, casting of steel without mold is eally difficult to ealize. In ode to imposing high fequency electomagnetic field on the meniscus of stand pemeating the coppe mold, and keeping the existing favoable heat tansfe and mechanical pefomances of coppe mold at the same time, the slit mold was bought fowad. The existing coppe mold tube is split into some segments along the casting diection, and insulation mateials ae filled in the slits between segments. High fequency electomagnetic field can be imposed aound the meniscus aea of stand pemeating the slit coppe mold. Electomagnetic body foce pependicula to the suface and pointing to the inne of stand is fomed in the stand, static pessue of liquid metal to shell and mold wall is counteacted patially accodingly. So the shape of meniscus can be contolled, and peiodical defomation of initial shell can be educed, this is so-called soft contact electomagnetic continuous casting technology. Consequently, the tagets of deceasing the depth of oscillation maks, impoving the quality of stand suface, loweing the beakout ates and incease of the casting speed can be ealized. Many eseaches had caied out long-tem eseach on soft contact solidification, they applied diffeent fequency in this technology, and obtained coesponding esults. In liteatue, 1) the fequency was fixed on 11 and 6 khz. Although the electomagnetic foce was calculated, but the deep analysis about it was not caied out. Application of low fequency electomagnetic field was eseached in liteatue, 2) the fequency is aimed at 60 Hz. In liteatue, 3) the authos eseached the effect of intemittent high fequency magnetic field on initial solidification, the fequency is 2002 ISIJ 974

2 fixed on 20 khz. The authos of liteatue 4) investigated the shape of meniscus unde khz, consideed the fequency should be set above 20 khz. Fequency was aimed at 20 khz in liteatue, 5) and it was aimed at 25 khz in liteatue, 6) but the eason of selection of fequency was not given out. In this pape, the distibution of electomagnetic field, action of electomagnetic body foce pemeating slit mold on stand, the decisive effect of fequency on soft contact solidification and optimal fequency ange in billet slit mold ae discussed hee deeply. 2. Theoetical Foundation The most adical govening equation fo electomagnetic field is Maxwell equations. Electomagnetic hamonic wave of definite fequency is emitted by high fequency powe souce in soft contact technology. Within conducto, 0, J se whee,, density of fee chage (C/m 3 ); J, electic cuent density (A/m 2 ); s, electical conductivity (S/m); E, electic field intensity (V/m). The Maxwell equations of altenating electomagnetic field ae obtained as: E iwmh...(1) H iwee se...(2) E 0...(3) H 0...(4) Electomagnetic chaacte equations of medium ae stated as: D ee...(5) B mh...(6) The Ohm law within conducto is given as: J se...(7) whee, B, magnetic flux density (T); D, electic displacement vecto (C/m 2 ); H, magnetic field intensity (A/m); m, magnetic pemeability (H/m); e, dielectic constant (F/m). The electomagnetic body foce imposed on conducto is descibed as: F E J B...(8) 3. Analytical Model 3.1. Attenuation of Hamonic Electomagnetic Wave Within Conducto Fo electomagnetic wave of cetain fequency tansmitting pependiculaly into suface of conducto, Maxwell equations can be tansfomed as follows: 2 B iwmsb 0...(9) B 0...(10) 1 E B...(11) µσ Fig. 1. Fig. 2. Attenuation of electomagnetic wave of diffeent fequency in coppe mold wall of 6-mm thickness (220 C). Attenuation of electomagnetic wave of diffeent fequency in steel of 40-mm dimension (1 520 C). Fo plana electomagnetic wave, the solution fo tansmitting of electomagnetic wave within conducto can be obtained by solving the above equations. B B 0 e pfms...(12) whee, coodinates along the tansmitting diection of electomagnetic field (m); f, fequency of electomagnetic wave (Hz); B 0, magnetic flux density of stand suface (Tesla). B 0 h f mni h f mn(i 0 e iwt )...(13) whee h f, esidual pecentage of electomagnetic field passing though mold wall, elated to fequency; t, time (s). The attenuation cuves without dimension of electomagnetic wave of diffeent fequencies within 6 mm coppe wall, 40 mm adius steel stand and ound coppe mold filled by liquid steel ae shown espectively in Figs. 1, 2 and 3. Physical chaactes of diffeent elative mediums ae listed in Table. 1. We can conclude fom these figues that the electomagnetic wave deceases moe apidly with the incease of fequency. When fequency inceases fom to Hz, the attenuation cuve does not change obviously Electomagnetic Foce The electomagnetic body foces pependicula to the suface of stand and pointing to the inne of conducto is descibed as: F J B 1 B B 1 B B...(14) µ ( ) µ ISIJ

3 Fo electomagnetic solenoid, B z B 0 e pfms, B 0, B q 0 is taken into account, so following equations can be obtained, B ( B) e eθ ez 0 0 Bz 0 B z 0 Bz F 1 B B 1 B ( ) z µ µ e...(15)...(16) Substituting the esults of Eq. (12) into Eq. (16), Eq. (17) can be obtained as follows, 1 f F f B (17) µ π µσ π µσ ( ) e 3.3. Electomagnetic Pessue The centipetal pessue pependicula to suface of stand is obtained by taking F integal fom suface 0 to, DD l θfd p Fd...(18) DD l θ 0 0 B p e 2µ π fµσ Bz B z e...(19) The attenuation cuves without dimension of electomagnetic body foce and pessue imposed on the steel stand ae shown espectively in Figs. 4 and 5. Electomagnetic body foce and pessue changes a lot in stand accoding its fequency and distance fom the suface. With the incease of fequency, the electomagnetic foce suddenly deceases and pessue in the steel eaches apidly to a fixed level. The electomagnetic foce imposed to the inne of stand decease fast to almost zeo. When fequency inceases fom to Hz, the attenuation cuve of electomagnetic body foce and pessue does not change obviously. 4. Numeical Simulation The total distibution of electomagnetic field in coppe mold and steel stand is aleady compehended by analytical model in pat 3, the behavio of attenuation of electomagnetic field changes with its fequency. The slit stuctue of soft contact mold can not be easily simplified as 2-D stuctue, the electomagnetic field distibution and attenuation ule in the mold must be vey complicated in space. So, the distibution and attenuation of electomagnetic paametes in some special egions such as slits and cones in soft contact mold must be gasped though 3-D electomagnetic field numeical simulation. In this pape, a 2-D numeical model was established fistly to analyze the pemeation conditions of high fequency electomagnetic wave acoss coppe mold wall and insulated mold wall espectively simulating the segments and slits. And then a 3-D model was established to simulate the electomagnetic field distibution in soft contact mold Numeical Solution of Electomagnetic Field Magnetic vecto potential A (Wb/m) and electic scala Fig. 4. Distibution of magnetic flux density in liquid steel unde diffeent fequency (1 520 C). Fig. 3. Attenuation of electomagnetic wave of diffeent fequency in coppe mold (6 mm) (220 C) and liquid steel (40 mm) (1 520 C). Fig. 5. Distibution of electomagnetic pessue in liquid steel (1 520 C). Table 1. Physical chaactes of diffeent medium ISIJ 976

4 Table 2. Physical chaactes of diffeent medium. Table 3. Paametes of 3-D numeical simulation. Fig. 6. Fig. 7. Magnetic field in the mold with insulated mateials aound the meniscus. Shielding of coppe mold to high fequency electomagnetic field. Fig. 8. The eal billet soft contact mold model (1/4). potential f (V) wee inducted fo solution of electomagnetic field. B A...(20) A E φ...(21) t Inducting Loentz limitative condition φ A µε...(22) t into Maxwell equations, magnetic field and electic field wee solved espectively to obtain the solution of electomagnetic paametes. Physical chaactes of elated mediums appeaed in numeical simulation model ae shown in Table. 2. Related electomagnetic paametes set in numeical simulation model ae shown in Table The 2-D Simulation Results The distibution of high fequency ( Hz) magnetic field in simplified 2-D mold segment model and slit model ae espectively shown in Figs. 6 and 7. In Fig. 6, the mateials of mold wall wee consisted of Cu and insulation mateials. The mateials in the middle of mold wall aound stand meniscus is insulation mateials, so the electomagnetic field can pemeate entiely the mold wall and imposed on the metal stand completely. Due to the skin-concentated effect of high fequency electomagnetic field, it almost imposes on the suface of metal stand. The thickness of skin-concentated laye lies on the fequency of electomagnetic field. In Fig. 7, the high fequency magnetic field is shielded completely by coppe mold wall. So the high fequency electomagnetic wave can not pemeate the metal mold without slit to impose on metal stand. Fo futhe eseach on the details of high fequency electomagnetic field penetating acoss slit coppe mold and imposing on metal stand, it is eally necessay to simulate the 3-D distibution of electomagnetic field, electomagnetic body foce and eddy cuent in soft contact mold. So we established a 3-D model in the next step The 3-D Numeical Simulation The Reseach Object The eseach object is designing a soft contact continuous casting slit billet mold, the stuctue of soft contact mold is shown in Fig. 8, and its concete size is shown in Fig. 9. Hee the stuctue and size of simulation model is same as the actual mold. Due to the symmety of the billet mold, so the quate of mold is eseached. The inne size of the billet coppe mold tube is mm, and its length is 200 mm, the thickness of coppe wall is 6 mm. Thee ae 4 slits wee split along the casting diection on evey side of coppe mold, and insulation mateials ae filled into the slits ISIJ

5 Fig. 9. The dimension sketch of mold system. Fig. 10. The simulation aea of magnetic field. between segments of mold, the size of evey slit is mm. In view of simulation of gap between stand and mold wall and the divisional ationality of elements, the thickness of gap is set as 1 mm. On the bases of the eseach esults of efeence liteatues, 4 6) the electomagnetic foce imposed on meniscus egion is consideed maximal when stand meniscus is located at the middle position of induction coil height. So, in the eseach wok, the liquid metal level was supposed as plane, and its position located at the middle of coil height Establishment of Finite Element Simulation Model The finite element model was established accoding to the eal stuctue of mold, the simulation aea is shown in Fig. 10. Mold, stand, coil and sufficient fee space wee included. Basic assumption: 1) The liquid metal level is supposed as plane, and the effect of meniscus shape on magnetic field was neglected. 2) The liquid metal level is located at middle of coil height. 3) Solidification pocess of stand is not consideed. 4) Oscillation of mold is ignoed. 5) Gap is not conductive. Thee ae elements and nodes totally in the finite element model. On account of the symmety of model, paallel flux conditions ae set as bounday condition of symmetical faces. The magnetic vecto potential at the distance of 6 times Fig. 11. The magnetic flux density in the simulation aea. mold height is set as zeo. The fequency paametes ae set as 50, 1 000, , , , Hz, and the ampee-tun of coil is Simulation Results Magnetic flux density B is vecto sum of B x, B y, and B z, its unit is Tesla. Electomagnetic foce F is vecto sum of F x, F y, F z, its unit is Newton, epesents the integal value of volume density of electomagnetic foce in element. In fact, thee ae so many elements of diffeent size in a simulation model, the absolute value of electomagnetic foce acted on diffeent size element can not show the actual effect. So electomagnetic foce must be tansfomed into electomagnetic body foce, viz. volume density of electomagnetic foce. Electomagnetic body foce is the key paamete to explain the essence of function of high fequency electomagnetic field, it can explain and compae clealy the magnitude of electomagnetic foce acted on any position in mold. So in the latte analysis, we tansfom electomagnetic foce into electomagnetic body foce, its unit is Newton/m 3. The unit of density of eddy cuent JT is A/m 2. The vecto simulation esults of 3-D electomagnetic field at Hz fequency ae shown in Figs. 11 to 14. The simulation esults of magnetic flux density in whole egion ae shown in Fig. 11, its distibution is accoded vey well with ight hand spial ule. At the same time, owing to 2002 ISIJ 978

6 Fig. 15. Analysis path of simulation esults. Fig. 12. Fig. 13. Magnetic flux density on the suface of stand. The induction eddy cuent in stand, mold, and coil. Fig. 14. Electomagnetic foce in the stand. the shielding effect of metal on high fequency magnetic field, electomagnetic field mainly acts on the suface laye of metal stand. Magnetic flux density has its maximal value, Tesla, in the gap nea meniscus of stand in whole simulation aea. The simulation esults of magnetic flux density in metal stand ae shown in Fig. 12. Unde the slit of mold, the magnetic flux density on suface of stand is highe than in othe egion of the stand, and it mainly acts on the suface laye. Magnetic flux density has its maximum value 0.12 Tesla at the suface of chage nea meniscus, and decease monotonously into the chage. The simulation esults of induction eddy cuent in metal stand, slit mold and coil ae shown in Fig. 13. Unde the action of high fequency magnetic field, eddy cuent mostly fixed on the suface of conducto, and fom espectively a close loop in each sepaated conducto. The distibution of electomagnetic foce in metal stand is shown in Fig. 14. All of the electomagnetic foces ae pependicula to the suface and pointing to the inne of stand, and thei values attenuate apidly towads cente of stand Analysis of Numeical Simulation Results Fo futhe undestanding the simulation esults, 3 typical paths on the suface of metal stand ae selected to analyze the change ule of electomagnetic paametes, and shown in Fig. 15. The esults of numeical simulation wee compaed with the esults of analytical model. Path 3 is on the suface of stand, locates at the middle of segments between 2 slits and paallels to the axis of mold. Distibution of magnetic flux density and electomagnetic body foce and thei concete values on path 3 ae shown espectively in Figs 16 and 17. We can see fom these two figues the changes of diffeent fequency electomagnetic field along the casting diection. Along with the incease of fequency, the electomagnetic body foce on path 3 inceases geatly, especially at the location of liquid metal level. At the exit of mold, the value of electomagnetic body foce deceases close to zeo due to the spaseness of magnetic flux. Attenuation of magnetic flux density and electomagnetic body foce and thei concete values on path 1 ae shown espectively in Figs. 18 and 19. Path 1 locates at the intesectant position of symmety face and liquid level. The attenuation ate of magnetic flux density and body foce incease apidly, and the actual value of electomagnetic body foce on the suface of metal stand is impoved geatly with the incease of fequency. The simulation esults of magnetic flux density and electomagnetic body foce on path 2 ae shown in Figs. 20 and ISIJ

7 Fig. 16. The magnetic flux density on the path 3 unde diffeent fequency of electomagnetic field. Fig. 19. Electomagnetic body foces on the path 1 unde diffeent fequency of electomagnetic field. Fig. 17. Electomagnetic body foces on the path 3 unde diffeent fequency of electomagnetic field. Fig. 20. Magnetic flux density on the path 2 unde diffeent fequency of electomagnetic field. Fig. 18. Magnetic flux density on the path 1 unde diffeent fequency of electomagnetic field. Fig. 21. Electomagnetic body foces on the path 2 unde diffeent fequency of electomagnetic field. 21. Path 2 locates at the intesectant position of suface and liquid level of stand, cosses the egions unde slits and segments of mold. We can see fom these two figues, the uneven distibution of magnetic field is enhanced slightly with the incease of fequency. When the fequency is set as Hz, the value of magnetic flux density at the slit egion is 10 % highe than the segments egion on path 2, and that the distibution of electomagnetic body foce on path 2 unde diffeent fequency ae basically unifom. In Fig. 20, magnetic flux density B is vecto sum of B x, B y, and B z. We can see fom Fig. 22, vecto B unde silts of mold has angle with the suface of stand (diection Z), but vecto B unde segments uns paallel with the suface of stand. In fact, centipetal foces shown in Fig. 21 ae fomed mainly by B z, we can conside that vecto B z on path 2 is unifom, this is the eason of unifomity of electomagnetic body foce shown in Fig. 21. So we can conclude that electomagnetic field can be imposed homogeneously compaatively on the metal stand pemeating slit coppe mold. In the egion nea the cone of mold, the value of electomagnetic body foce imposed 2002 ISIJ 980

8 Fig. 22. The diection of vecto-magnetic flux density unde silts of mold on the path 2. on the cone of metal stand deceases, the eason is that the cone of coppe mold has no slit. 5. Compaison of Results Obtained fom Analytical Model and Numeical Simulation Compaing Fig. 2 and Fig. 18, Fig. 2 shows attenuation of magnetic flux density without dimension fom the suface to cente of stand, and Fig. 18 shows the concete attenuation value of magnetic flux density fom the suface to the cente of stand. The change tend of cuves in these two figues agees with each othe. The attenuation cuves of electomagnetic body foce without dimension ae shown in Fig. 4, and the coesponding simulation value is shown in Fig. 19. Fom these two figues, we can conclude that the explanatoy issues deduced fom the two methods coincide well. 6. Conclusion (1) The attenuation of magnetic flux density, electomagnetic body foces and magnetic pessue along the adius diection in coppe mold filled with liquid steel can be gasped wholly by analytical model. Fom these analytic esults, we can conclude that the optimal fequency adopted in soft contact technology should be set in the ange of khz. (2) The fequency of powe supply is the key paamete fo soft contact technology. (3) In the ange of khz, the value of electomagnetic body foce imposed in metal stand, especially in the egion aound meniscus, incease geatly with the incease of fequency. But thei uneven distibution on the tansvese diection will be enhanced with the incease of fequency, especially when fequency changed fom 40 to 100 khz. (4) Fom the simulation esults on path 2, magnetic fequency anges fom khz is optimum in ode to maintain meniscus homogeneously. (5) When the fequency inceases, the attenuation speeds of magnetic flux density and electomagnetic body foce fom the suface to the inne of metal stand incease, and the value of electomagnetic body foces at the suface of stand, especially at the liquid level, ae enhanced geatly. (6) Along with the casting diection, magnetic flux density and electomagnetic body foce in stand decease gadually to almost zeo at the exit of slit mold. (7) Electomagnetic field can pemeate slit coppe mold and act unifomly compaatively on the suface of stand. REFERENCES 1) P. R. Cha, Y. S. Hwang, H. S. Nam, S. H. Chung and J. K. Yoon: ISIJ Int., 38 (1998), ) T. Toh, E. Takeuchi, M. Hojo, H. Kawai and S. Matsumua: ISIJ Int., 37 (1997), ) T. J. Li, K. Sassa and S. Asai: ISIJ Int., 36 (1996), ) A. Y. Deng, G. W. Yu, G. L. Jia and J. C. He: J. Ion Steel Res., 13 (2001), 10. 5) H. Nakata, M. Kokita, M. Moisita, K. Ayata: Int. Symp. on Electomagnetic Pocessing of Mateials, ISIJ, Tokyo, (1994), ) S. Fuuhashi, M. Yoshida and T. Tanaka: Tetsu-to-Hagané, 84 (1998), ISIJ

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