Modeling of magnetic field behavior in DC arc furnace bath for different designs of current lead of bottom electrode Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

MODELING OF MAGNETIC FIELD BEHAVIOR IN DC ARC FURNACE BATH FOR DIFFERENT DESIGNS OF CURRENT LEAD OF BOTTOM ELECTRODE

Yachikov I.M.,

Nosov Magnitogorsk State Technical University, Magnitogorsk, Russia

Doctor of technical sciences, Professor

Portnova I.V.

Nosov Magnitogorsk State Technical University, Magnitogorsk, Russia

ABSTRACT

The paper is concerned with the simulation of magnetic field behavior in the bath of a DC arc furnace for different designs of the bottom electrode busbar. The busbar can be designed as a circular loop, a flat Archimedean spiral, a cylindrical helix or as Archimedean spiral. The authors offer a mathematical model, which makes it possible to determine magnetic fields around the current leads of various forms for the bottom electrode in a DC arc furnace. Computer modeling was used to analyze the behavior of magnetic field intensity in the bath of the DC arc furnace. The authors give some recommendations on design parameters of the busbar. They calculated the relationships between the intensity of the magnetic field and the bath radius and the number of turns in busbars of different designs. It was found that the mofl promising designs of the current leads for conductive Sirring are the flat Archimedean spiral and the cylindrical helix.

Keywords: DC arc furnace, bottom electrode, intensity of magnetic field, current lead.

Introduction

Metallurgical melts are high-temperature current-carrying liquids. Their behavior can be described by magnetohydrodynamics laws dealing with the processes, which flart when electric current going through the melt interacts with its own magnetic fields and the external ones.

External magnetic can have both positive and negative influence in the technological processes developing in metal baths. The mofl common melting facility containing current-carrying melt is a DC arc furnace. On the one hand, magnetic field acts on molten metal generating electromagnetic forces causing conductive Sirring of the melt and intensifying heat and mass transfer processes. On the other hand, external magnetic fields can result in some negative consequences, in particular, they can cause intensive metal flows leaching the lining or they can cause deflection of the arc from vertical or arc movement along the melt surface.

One of the problems fleelworkers face in the process of metal production is the problem of melt Sirring. There is a number of ways for the liquid bath Sirring. The mofl advantageous ones are the noncontact electromagnetic methods of metal Sirring, such as, the conductive and electric vortex ones.

Okorokov N.V., Oflroumov G.A., Zubarev A.G., Zhilin V.G., Ivochkin Yu.P. and others [1-4] contributed greatly to the development of the theoretical basis of electromagnetic Sirring of liquid metal and to the invefligation of electric vortex flows.

Physical theory of electric vortex flows arising in the process of interaction between electric current and its self-magnetic field is well known [5]. Paper [6] for the firfl time showed that the longitudinal magnetic field causes azimuthal whirl of the flow and formation of the secondary toroidal vortex in the meridian plane of the bath in the vicinity of the furnace bottom; this secondary vortex turns in the direction opposite to the electric vortex flows. Vlasuk VH. and Sharamkin V.I. determined the criteria contributing to the development of the secondary flow in the meridian plane and the conditions, under which this flow can completely suppress the electric vortex flows [7].

In paper [5], the cylindrical model was used to carry out theoretical and experimental invefligation of the influence of the longitudinal magnetic field on the characteriflics of electric vortex flows. Analysis of the obtained results showed that when the external field increases, hydrodynamic pressure of the liquid metal jet on the surface of the large electrode decreases, while in

the vicinity of the electrode axis the vacuum area is formed and the direction of the heat convection changes.

Moshnyaga V.N. and Sharamkin V.I. experimentally proved that the increase in induction of the external longitudinal magnetic field results in the decrease of pressure on the bottom of the bath [8]. They also found that the change in the pressure can be controlled not only by the external magnetic field, but also by the deformation of the lines of force of the magnetic field in the operating bath when additional ferromagnetics are placed in it.

In the process of invefligation of the influence of electric vortex flows on liquid metal behavior, one can single out the fludy of reasons causing spontaneous azimuthal whirl of the axisymmetric electric vortex flow, which occurs in the process of electric current spreading from the point source in the volume of the electrically conductive liquid. Different specialifls [5, 9, 10] suggefl a number of possible reasons of this phenomenon. For example, in [10] analytical and numerical evaluations were used to show that the effect of magnetohydrodynamics -dynamo is possible for a certain kind of electric vortex flows with axial symmetry even under conditions of very small values of magnetic Reynolds number. In papers [3, 4, 9, 11] numerical invefligation of electric vortex flow was carried out for different values of current and external magnetic field. The authors showed that in real-life conditions there is always some force interaction between the electric current in the liquid metal and the external magnetic field including magnetic fields of relatively low intensity. Hydrodynamic flructure of the flows, which occur in the liquid bath, intensify the heat transfer from the hot region near the small electrode into the melt. As a results of additional vortexes arising, the direction of the dominating electric vortex flow changes. It acquires radial rather than axial direction.

So far, the issues of electric vortex flows behavior, when they are influenced by weak external magnetic fields, have not been sufficiently described in scientific literature and require further detailed invefligation.

At present conductive Sirring of liquid metal in a DC arc furnace bath is hardly ever used due to the lack of data on the influence of external magnetic fields on the current-carrying melt and the lack of simple and reliable mechanisms generating external magnetic fields with the preset or controlled configuration. However, proper application of conductive Sirring of molten metal makes it possible to improve

performance characteriflics of the fleel-making process as well as the quality of the manufactured metal and to minimize the negative influence.

The aim of this research work is to invefligate the behavior of magnetic field in the bath of a DC arc furnace making use of the bottom electrode busbar in the form of a cylindrical helix, a flat Archimedean spiral or Archimedean spiral.

The main task is to fludy the behavior of the external magnetic field flrength for different designs of current leads of the bottom electrode by means of computer simulation.

However, the role of magnetic fields and their application in electrometallurgy have not been sufficiently described even in periodical literature, and there are jufl a few experimental and theoretical papers on this topic.

Mathematical models of magnetic field intensity around current leads of different forms

One of the ways to control the intensity of conductive Sirring in different parts of the molten bath is connected with the change

in the intensity of magnetic field generated by outer current-carrying conductors in different regions of the current-carrying melt. To achieve this, it was suggefled to generate the external magnetic by the bottom electrode busbar designed in the form of a circular loop, a flat Archimedean spiral, a cylindrical helix or as Archimedean spiral (Fig. 1). A ferromagnetic core can be placed inside the cylindrical helix and moving the core in vertical and horizontal directions, one can control the magnitude of magnetic field intensity in different regions of the current-carrying melt in the bath [1, 2].

To determine the intensity of magnetic field generated by

current I going through the element dl (point B) in an arbitrary point A, Biot-Savart-Laplace equation in its differential form (Fig. 2) was used

I

dH = ■

4nr

dl x r '

(1)

Fig. 1. Different kinds of bottom electrode current leads: a - a circular loop; b - a flat Archimedean spiral; c - a cylindrical helix; d - Archimedean spiral; 1 - bottom electrode; 2 - rigid busbar; 3 - current direction

Let us consider the magnetic field generated around the conductor designed in the form of a cylindrical helix carrying current I (Fig. 2, c). We will use cylindrical coordinate syflem (r, 9, z) related with Cartesian one (x, y, z) by the following

, y((P) = Rn ■ cos() , x() = Rn ■ ,

z(() = b • (/(2 • n) where b is the helix lead determining the

relationships: r = Jx2 + y2 ,. Let us define the function of change of z coordinate when 9 increases by 2 -n , Rn is the

radius of helix. If the cylindrical helix has n turns, then its length

the conductor line in the parametric form

( = arctg (y I x )

Lb = nJ 4n2 r2 + b 2

Fig. 2. Calculation of intensity of magnetic field generated by current leads in the form of: a - a circular loop; b - a flat Archimedean spiral; c - a cylindrical helix; d - Archimedean spiral

Thus, coordinate of vectors R, c, r [3] take the following form

OB = R = (Rfi • cosp, Rfi • sinp, z)

Rn = (Rn -cosp Rn •sin() ,

AB = r' = (Rn-cosp-xo, Rn-sinp-yo, z-zo) ,

dl = dp-[Rn-sinp, - Rn *cosp, ———

/ ljn - (Rn sin V-ro sin Vo ) ■ b/(2n)- Rh cos yl — - z0

Hx(r0,V0, zo) = —■ J -n-372

4n o ( f

I Rn + ro - 2Rnr ■ cos(y- Vo )+( - zo

dy

(2)

/ Inn (Rn cos V - ro cos Vo )■ bl(2n)-Rn sin vl 2v - zo

Hy (ro,Vo, zo) = 4-- i "- " ,3/2

4n o ( (b

R/5 + ro2 - 2 Rnrocos(V-Vo )+| bT-zo

(3)

/■R~ 2m Hz(ro,Vo,zo) = -R- j

I 2n

Rn - roCos(v-vo )

dy

4n o ( ( b \2

I R/5 + ro2 - 2Rnro cos(V-Vo )+lb,V-Zo

ydy

Hz (r0, zo):

(5)

Hr (r0, zo) =

(6)

/■R 2n

c r

4n

Rc - ro ■ cos(v)

o R2 + ro2 -2RC - ro ■cos(v)+zo2 Jn

-dy

/■Rc-zp 4n

2n

o (Rc2 + ro2 - 2Rc ■ ro ■ cos(v) + zo2 f/2

2no

Là = J

spiral can be calculated as o

2

a

V2+1)+

b

2

2

dy

. The coordinates of vectors

R

(Fig. 2, d): R = (a •(• cosp, a •(• sinp, z),

r'= |aB| = (a• (•cosp-x0, a• (•sinp-y0, z-z0)

Similar to the cylindrical helix case, we calculate the vector

L

length dl

di

2+u 2+d2

unit vector direction "e and vector

2- n

Cartesian projections of magnetic field intensity in an arbitrary point A(r0, 90 , z0), around the current-carrying conductor designed as cylindrical helix containing n turns take the following form

Projections of magnetic field intensity in the point set by cylindrical coordinates A(r0, 90 , z0) around the conductor designed as Archimedean spiral containing n turns take the following form

sinp - a -p•sinp)•bj(2 -n))

H x (ro.Vo, zo) =

2n n m.v) ■ (y ■ cos(v) ■ (zo - z)-(ro

■ J -/-

o „2. ,„2^2

—dy

2 ■a ■ V r ■c

H (r V z ) = _L 2'in A(V) ■b(2n)-y^osy-ro ■cos Vo)-V ^sin'

HyVro.yo.zo>- . ' J

a2 V2 + ro2 - 2 ■a ■ V ■ ro ■ c°s(v- Vo)+(z- zo)2

■(z - zo)) J

/ 2n n

Hz (ro.Vo. zo) ^^--J -,-

4 n o 2 2 o a ■ V

■(a V2 - r0 ■ V ■ cos(V- Vo))

V2 + ro2 - a V■ro ■cos(V-Vo )+(z - zo)2 J3

-dV

(7) , (8) (9)

4^a2 n2 + 4 a 2 2 2 2 n y + b

4 ■n2 ■v2 + b 2

¿p) = ■

where

In a special case, when b=0, one can obtain a mathematical model of the magnetic field around the flat Archimedean spiral (Fig. 2, b) [4]:

H x (ro.Vo. zo)

/ ■ a ■ zo 4n

J-

1+V cosv

(io)

(4)

In a special case, when b=0, n=1 we obtain a mathematical model of an axially symmetric magnetic field generated by the circular loop (Fig. 2, a) [3, 4]:

/ ■ a ■ zo 2nn Hy (ro, Vo, zo) = ——— J

4n

(11)

(a2 ■ y2 + ro2 - 2a■ yro ■cos(y-Vo )+ zo2

-J 1 + y2 sin y

(a2 ■ y2 + ro2 - 2a■ y ro ■ cos(y- yo )+ zo2 f

+ y2(a■ y ro ■cos(y-Vo))

■dy

■ dy

Let us consider the magnetic field generated around a conductor in the form of Archimedean spiral carrying current I (Fig. 2, d). The form of such a conductor in its parametrical

form is set as y(p) = a cos(p) , x(p) = a sin(p)

z(p) = b p/(2n) , a = Rc/(2n^n) . , , where is a parameter

determining radial displacement of the line and characterizing the

number of turns n, which are necessary for the spiral to achieve

r = R

the external radius c . The total length of the Archimedean

4n

r can be written as

j m a 2nn

Hz ^^ z0) = — \ Y vj/2

0 (a p + r0 -2a-p-rQ-cos(p-pQ)+ Z0 )

(12)

On the basis of this mathematical model, a computer program «Calculation of magnetic field around the current leads of different shapes» was developed, this program makes it possible to model magnetic fields in tabular and graph forms [5].

Results of computer modeling

Computer modeling was used to invefligate the behavior of magnetic field intensity in the molten fleel bath for bottom electrode current leads of different design. The calculations were carried out for the bath of a fleel-making DC arc furnace - 5 (5 ton metal capacity) of flandard design with one bottom electrode and the following parameters: arc current Id=6 kA; radius of the liquid fleel bath Rv =1.245 m, bath height H=0.34 m.

In the process of modeling, it was accepted that the bath with the melt is located in the positive half-subspace (z>0) with respect to xOy, while the busbar is located in the negative half-subspace (z<0). Only the influence of the magnetic field generated by the current leads was taken into account.

Computer modeling was carried out for a flat and Archimedean

spirals with the following parameters: a Rc ^ 2n n) =0.048 m - radial displacement of spiral turns; n=4 - the number of turns necessary for the spiral to achieve the external radius R =1.2 m;

/2

4 n

1

b= -0.3 m - helix lead; ® 15.3 m - the length of the helix. For current leads designed as a circular loop and a cylindrical helix, computer modeling was carried out for the same values of

R , b, n, while the length of the cylindrical helix was

L

b

30.2

m. All the current leads were located at a diflance of z0=0.3 m from the upper turn surface to the bath hearth, which is in direct contact with the molten metal.

The research group carried out analysis of the relation between the axial Hz and the radial Hr projections of intensity

of magnetic field generated by the cylindrical helix and the r coordinate on the hearth, in the middle of the bath height and on its surface (Fig. 3). It was found that the axial and radial projections of magnetic field intensity are values of the same

r ^ R

order, closer to the edges of the turns ( c), the value of

H decreases, while the value of H increases. If the diflance z

z 7 r

from the current lead increases, the intensity of magnetic field decreases.

Fig. 3. Dependence of axial (a) and radial (b) projections of intensity of magnetic field generated by cylindrical helix on the bath radius for 9=n/3: 1 - on the hearth (z0=0.3 m); 2 - in the middle of the bath height (z=z0+Hv/2); 3 - on the bath surface (z=z0+Hv)

Dependence of the axial Hz and the radial Hr projections of intensity of magnetic field for the current leads designed in the form of Archimedean spiral on the r coordinate (Fig. 4) was analyzed. It was found that the value of the axial projection Hz of magnetic field intensity is by an order of magnitude greater than

that of the radial one H. When the diflance z„ from the current

r0

lead increases, the value of magnetic field intensity decreases for the both projections. Radial component of magnetic field for Archimedean spiral along the r coordinate has an extremum.

Fig. 4. Dependence of the axial (a) and the radial (b) projections of intensity of magnetic field generated by Archimedean spiral (when 9=n/3) on the bath radius: 1 - on the hearth (z0=0.3 m); 2 - in the middle of the bath height (z=z0+H/2); 3 - on the bath surface (z=z +H )

0v

For the current lead designed as a cylindrical helix, the projection of magnetic field intensity Hz does not change significantly along the r coordinate until r/Rc<0.5, then it decreases rapidly (Fig. 3, a). For the current lead designed as Archimedean spiral, the value of axial intensity of magnetic field decreases smoothly (Fig. 4, a). In both cases the values of Hz are of the same order of magnitude. When the value of the r coordinate increases, the value of H increases in case of the cylindrical helix (Fig. 3, b), while in case of Archimedean spiral magnetic field intensity achieves its maximum value (Fig. 4, b).

Dependence of the axial and the radial projections of intensity of magnetic field generated by the current lead designed as a flat Archimedean spiral on the r coordinate was analyzed (Fig. 5). Values of Hz and Hr projections are of the same order of magnitude. The value of Hz has its maximum values on the axis. For Hr, as well as for Archimedean spiral, the maximum value of the field intensity is achieved in the process of moving from the bath axis to its periphery.

Fig. 5. Dependence of axial (a) and radial (b) projections of intensity of magnetic field generated by current lead designed as a flat Archimedean spiral on the r coordinate (b): 1 - on the hearth (z0=0,3 m); 2 - in the middle of the bath height (z=z0+Hv/2); 3 - on the bath surface (z=z +H )

0v

When comparing Archimedean spiral and the flat Archimedean spiral, one can see that the profiles of the obtained curves look similar, but the values of intensity are different. For the flat Archimedean spiral, the value of axial projection of magnetic field intensity is 1.3 times higher on the bath axis and it has almofl zero value when r=1.2 m (Fig. 5, a). The value of radial projection of magnetic field intensity is by an order of magnitude greater (Fig. 5, b), compared with Hr generated by Archimedean spiral (Fig. 4, b).

The research group obtained the relation between the axial projection of magnetic field intensity and the number of turns for current leads of different design in the middle of the bath (r=0, z=0.47 m) (Fig. 6). It was found that the increase in the number of turns in the current lead designed as cylindrical helix results in the smooth increase of Hz and magnetic field intensity becomes saturated at 5 6 turns. For the current lead designed as Archimedean spiral, Hz has the maximum value when the number of turns is n=2-4. In this case the value of H is twice

z

as low as for the current lead in the form of a cylindrical helix.

Fig. 6. Dependence of axial intensity of magnetic field at the point of molten metal with the coordinates r=0, z=z0+Hv/2 on the number of turns for current leads of different forms: 1 - cylindrical helix; 2 - Archimedean spiral

Of all the invefligated forms of current leads, magnetic field is characterized by axial symmetry only around the circular loop (when b=0, n=1). The research group obtained the relation between the radial projection of magnetic field Hr and the azimuthal coordinate 9 for different number of turns in the

middle of the bath height (Fig. 7). It was found that if the number of turns n increases, diflribution of magnetic field becomes close to axially symmetric, that is why, in order to obtain such a profile of magnetic field, current lead with several turns should be used.

Fig. 7. Dependence of radial projection of magnetic field intensity Hr on the azimuthal coordinate 9 for different number of turns

n for Archimedean spiral (r =R) in the middle of the bath height (z=z0+H /2):

- n=3;

- n=4;

- n=5

Conclusion

1. The research group offered a mathematical model, which makes it possible to invefligate the behavior of projections of magnetic field intensity in the molten metal bath of an arc furnace for different designs of current leads to the bottom electrode.

2. Computer modeling was used to find out that the mofl promising design of the current lead for conductive Sirring is the current lead in the form of the Archimedean spiral with 2-4 turns

and the current lead in the form of cylindrical helix containing 5-6 turns. Other things being equal, the current lead designed as a flat Archimedean spiral is more compact and provides higher values of magnetic field intensity in the processed melt.

The results of this research work can be used by engineering companies specializing in design and revamping of fleel-making units with the current-carrying molten metal bath.

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