Plasma Science and Technology, Vol.14, No.4, Apr. 2012

3D Numerical Analysis of the Arc Plasma Behavior in a SubmergedDC Electric Arc Furnace for the Production of Fused MgO

WANG Zhen (), WANG Ninghui (), LI Tie (), CAO Yong ()School of Electrical Engineering, Dalian University of Technology, Dalian 116024, China

Abstract A three dimensional steady-state magnetohydrodynamic model is developed for thearc plasma in a DC submerged electric arc furnace for the production of fused MgO. The arc is

generated in a small semi-enclosed space formed by the graphite electrode, the molten bath and

unmelted raw materials. The model is first used to solve a similar problem in a steel making

furnace, and the calculated results are found to be in good agreement with the published mea-

surements. The behavior of arcs with different arc lengths is also studied in the furnace for MgO

production. From the distribution of the arc pressure on the bath surface it is shown that the

arc plasma impingement is large enough to cause a crater-like depression on the surface of the

MgO bath. The circulation of the high temperature air under the electrode may enhance the arc

efficiency, especially for a shorter arc.

Keywords: numerical simulation, DC electric arc furnace, arc plasma, temperature field,flow field

PACS: 52.50.Nr

DOI: 10.1088/1009-0630/14/4/10

1 Introduction

Studies in the field of metallurgy, production of re-fractories, and other industrial sectors have been car-ried out for many years. The expertise gained in the useof a variety of metal treatment equipment leads to theconclusion that DC electric arc furnaces (EAF) can beeffectively used in many technological applications [1].A twin-electrode DC submerged electric arc furnace(SAF) was designed for MgO-crystal production. Thistechnique was also found as another effective method togrow high quality MgO single crystals [2]. Because theenvironment for crystal growth is significantly affectedby the arc behavior, a fundamental understanding ofthe heat and fluid flow in the arc plasma is necessary.Due to the hostile environment for observing the pro-cess occurring in the inner zone of the furnace, numeri-cal simulations for similar processes are used to obtaindetailed information in some studies. USHIO et al. [3]

and SZEKELY et al. [4] conducted numerical simula-tions of a DC EAF using the turbulent Navier-Stokesand Maxwell equations to predict the contributions ofthe different mechanisms of heat transfer from the arcto the bath. Recently, QIAN et al. [5] studied the arcplasma by a similar method, and obtained some resultsfor different arc currents and anode-cathode distances.WANG et al. [6] used the PHOENICS software packageto solve the governing equations and gave useful con-clusions in arcs heat transfer and bath circulation. Thesimulated results illustrated above [36] were all com-pared with experimental data by BOWMAN [7]. In hisstudy, BOWMAN measured plasma velocity distribu-

tions in free-burning DC arcs of up to 2160 A. Usinghigh-speed digital video cameras, the behaviors of high-current arcs were photographed by JONES et al. [8],and both shape and size of the depression caused bythe impingement of the arc were observed.In a steel making EAF, the arc was generated in a

large enough space above the bath surface, so it was rea-sonable to assume an unbounded bath surface for thearc plasma simulation in all previous studies. However,affected by continuously charged raw materials, the arcplasma may behave differently in the melting processof a SAF for MgO production. Additionally, the arcbehavior is often characterised by a shorter arc lengthwith a larger arc current during the realistic operationof the SAF. In this paper, a three-dimensional magne-tohydrodynamic model of arc plasma is presented. Theinfluence of the arc length on the arc characteristics isinvestigated to gain a better understanding of the heattransfer and fluid flow in the SAF.

2 Mathematical model

Since the highly purified MgO powder in the furnaceis an ideal thermal and electrical insulator and its melt-ing point is very high, the design of the bottom elec-trode in the DC furnace for steel making is abandoned.Instead, a twin-electrode submerged furnace is adoptedto produce high-purity MgO single crystals. This im-plies that one of the electrodes is the cathode and theother is the anode. In this study, main focuses are puton the effects of the arc plasma on the MgO bath. The

supported by the National High Technology Research and Development Program of China (No. 2008AA03A325)

Plasma Science and Technology, Vol.14, No.4, Apr. 2012

dimensions of the computational domain are shown inFig. 1. The arc is modeled in a three-dimensional cylin-drical coordinate system and the domain of ADD-FEEis used to solve the plasma flow field, as is shown inFig. 1. The MgO material is assumed to be chargedoutside the curve DDEE. The plasma arc can be de-scribed by mass, momentum and energy conservationequations. Before setting the governing equations forthe plasma arc, the following assumptions should bemade:

a. The arc is steady and radially symmetrical.

b. The arc is assumed to be in local thermodynamicequilibrium (LTE).

c. No interface deformation is considered. The in-terface between the plasma arc and the molten bath isa flat surface.

d. The plasma is optically thin, which implies thatre-absorption of radiation by the plasma is insignificant,compared to the total radiative loss.

Fig.1 Computational domain for the arc plasma

2.1 Hydrodynamic problem

Under the above assumptions, the governing equa-tions can be expressed as follows:

Mass conservation

1r

r(rvr) +

1r

(v) +

z(vz) = 0. (1)

Radial momentum conservation

1r

r

(rv2r

)+1r

r(vrv ) +

z(vrvz)

= Pr

+1r

r

(2re

vrr

)+1r

[e

(1r

vr

+vr

vr

)]+

z

[e

(vzr

+vrz

)]+2er

(vrr

1r

v

vrr

)+

v2r jzB.

(2)

Azimuthal momentum conservation

1r

r(rvvr) +

1r

(v2)+

z(vvz)

= 1r

P

+

r

[e

(vr

vr+1r

vr

)]+1r

[2e(1r

v

+vrr

)]+

z

[e

(vz

+1r

vz

)]+2er

(1r

vr

+vr

vr

) vrv

r.

(3)

Axial momentum conservation

1r

r(rvzvr) +

1r

(vzv) +

z

(v2z)

= Pz

+ jrB +1r

r

[er

(vzr

+vrz

)]

+1r

[e

(1r

vz

+vz

)]+

z

(2e

vzz

).

(4)

Energy conservation

1r

(rvrh)r

+1r

(vh)

+ (vzh)

z

= +1r

r

(rKecp

h

r

)+

1r2

(Kecp

h

)

+

z

(Kecp

h

z

)+jr2 + jz2 + j2

Sr

+5kB2e

(jrcp

h

r+jzcp

h

z+jcp

h

).

(5)

In Eqs. (1)(5), the variables are the pressure (P ),radial velocity (vr), azimuthal velocity (v), axial ve-locity (vz), current density (J) and plasma enthalpy(h), while the plasma properties are density (), heatcapacity (cp), radiative loss (Sr), viscous dissipationterm (), electrical conductivity (), azimuthal mag-netic field (B), Boltzmann constant (kB) and electroncharge (e).The viscosity (e) and thermal conductivity (Ke) in-

clude both laminar and turbulent components,

e = + t, Ke = K +Kt. (6)

The laminar components are derived from kinetic the-ory while the turbulent components are determined us-ing the k turbulence model.In the k turbulence model, incorporated into the

present calculations, the eddy viscosity t and eddythermal conductivity Kt are obtained from

t = Ck2

, Kt =

tcpPrt

, (7)

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WANG Zhen et al.: Arc Plasma Behavior in a Submerged DC EAF for the Production of Fused MgO

where k is the turbulence kinetic energy, is the dissi-pation rate of the turbulence kinetic energy, Prt is theturbulent prandtl number, and C is one of the con-stants of the model.Two turbulence variables k and are obtained by

solving

1r

(vrrk)r

+1r

(vk)

+ (vzk)

z

= t + 1r

r

[r

(l +

tk

)k

r

]

+1r2

[(l +

tk

)k

]+

z

[(l +

tk

)k

z

],

(8)and

1r

(vrr)r

+1r

(v)

+ (vz)

z

= C1t

k C2

2

k+1r

r

[r

(l +

t

)

r

]

+1r2

[(l +

t

)

]+

z

[(l +

t

)

z

],

(9)where

= 2

[(vzz

)2+(vrr

)2+(1r

v

+vrr

)2]

+(vzr

+vrz

)2+(vz

+vzr

)2

+(1r

vr

+vr

vr

)2.

(10)The constants in the turbulence model equations are

listed in Table 1.

Table 1. Constants in the turbulence model

C1 C2 k C Prt

1.44 1.92 1.0 1.3 0.09 0.9

2.2 Electromagnetic problem

In order to get the magnetic flux density distributionB, the following electromagnetic equations are solved inthis paper.The Laplaces equation for the scalar electrical po-

tential , () = 0, (11)

with the electrical conductivity. Eq. (11) results fromthe Ampe`res law J = 0 and Ohms law J = .The equation for the magnetic vector potential A can

be written as2A = 0J, (12)

where 0 is the vacuum permeability. Eq. (12) resultsfrom the Coulomb gauge A = 0 and the Ampreslaw B = 0J .

2.3 Boundary conditions

Due to the symmetry of the computational domain,only a quarter of the flow domain is considered in thefinite element model. A complete set of the boundaryconditions for the arc is listed in Table 2. The cur-rent density is assumed to be of parabolic in the radius,given by

J = 2JC

[1

(r

RC

)2]. (13)

The radius of the cathode spot is defined as

RC =

I

piJC, (14)

where I is the arc current, and the average cathode cur-rent JC is given to be 3.5107 A m2 in the presentcalculations [9].

Table 2. Boundary conditions in arc plasma region

Surface v(ms1)/P (Pa) T (K) (V) k

ABB vr = v = vz = 0 4000 Formula(13) 0 (/z) = 0

BBCC vr = v = vz = 0 4000 (/z) = 0 0 (/z) = 0

CCDD P = 0 (T/z) = 0 ((/z) = 0 (k/z) = 0 (/z) = 0

DDEE vr = v = vz = 0 2000 (/r) = 0 (k/r) = 0 (/r) = 0

EEF vr = v = vz = 0 3100 0 0 (/z) = 0

FA vr = v = 0 (T/r) = 0 (/r) = 0 (k/z) = 0 (/z) = 0

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Plasma Science and Technology, Vol.14, No.4, Apr. 2012

2.4 Technique of solution

The governing equations with boundary conditionsare solved using a finite element method. The nonlinearsolution is based on the SIMPLE algorithm. Conver-gence is declared when the following condition is satis-fied,

Li=1

Nk=1

mi,k m1i,k max

< 0.01, (15)

where is a general dependent variable, L and N arethe total number of nodes in the r- and z-directions,respectively, and m is the number of iterations.

3 Results and discussion

To validate the algorithm of the model, a compari-son is made with BOWMANs measurements [7] in freeburning DC arcs. The temperature contours for anarc of 2160 A with a length of 0.07 m are plotted inFig. 2(a). In Fig. 2(b) a comparison of the calculatedresults from the model with the experimental data onthe axial temperature in the arc is shown. As is seen,the temperature calculated is higher than the measuredone. The possible reason is that the assumption of LTEis not valid near the cathode spot region. The flow fieldis presented in Fig. 3(a). In Fig. 3(b) a comparison inthe axial velocity of the calculated results to the exper-imental data is shown. It can be seen that both matchwell.

Fig.2 Calculated temperatures for a plasma arc of 2160 A

with a height of 0.07 m. (a) Temperature field (K), (b) Axis

temperature as a function of axial distance from the cathode

(color online)

Fig.3 Calculated velocities for a plasma arcs of 2160 A

with a height of 0.07 m. (a) Velocity field (ms1), (b) Axisvelocity as a function of axial distance from the cathode

(color online)

It is clear from the literature that arc parameters,such as arc length and arc current, affect significantlythe arc properties and arc-bath interactions in a steelmaking EAF. However, as one considers the environ-mental enclosure in SAF and the high melting point ofMgO, the arc behaviors may be different. In order tostudy the differences between EAF and SAF, two caseswith and without limited space above the anode surfaceare presented.The calculated temperature field of a plasma arc of

10 kA with a height of 0.09 m for two cases is shown inFig. 4(a) and (b), respectively. Due to the higher cur-rent density near the cathode spot, the self-magneticcompression accelerates the plasma away from the spot,towards the lower current-density region. This phe-nomenon dominates the whole structure of the high-current free burning arc if it is assumed to be in asteady state. As is expected, the overall temperatureof the cylindrical region is higher in a limited space,shown in Fig. 4(b), while the arc radius is almost notaffected by the higher ambient temperature since thecathode spot radius in both cases is assumed to be thesame. The corresponding flow fields for these two casesare shown in Fig. 5. Because there is no lateral wall,the high temperature air flows away along the anodesurface, as shown in Fig. 5(a). In Fig. 5(b) it is shownthat the high temperature air may flow out of the cylin-drical space along the lateral boundary or take part inthe circulation with the inlet flow.The effect of the arc length is shown in Fig. 6. Due

to a relatively small space, the overall temperature ap-pears to be slightly higher for a shorter arc length, asis seen in Fig. 6(a), and with a larger velocity, shown

324

WANG Zhen et al.: Arc Plasma Behavior in a Submerged DC EAF for the Production of Fused MgO

in Fig. 6(b). These characteristics of a shorter arc maylead to a higher efficiency of heat transfer. The arc effi-ciency here is defined as the ratio of the power absorbedby the surrounding materials to the total power in thearc.

Fig.4 A comparison of the temperature fields for two

cases, with a current of 10 kA and an arc length of 9 cm.

(a) Without limited space, (b) With limited space (color

online)

Fig.5 A comparison of the velocity fields for two cases,

with a current of 10 kA and an arc length of 0.09 m. (a)

Without limited space, (b) With limited space (color online)

Fig.6 Axial distribution of temperature and velocity for

different arc lengths in a range of 5 cm to 9 cm. (a) Axis

temperature as a function of axial distance from the cath-

ode; (b) Axis velocity as a function of axial distance from

the cathode

The effect of the arc length on the pressure distri-bution at the anode surface is shown in Fig. 7. Amaximum pressure appears at the center of the an-ode. Blocked by the lateral wall, the pressure becomesslightly higher near the fridge of the anode. It is ap-parent from this figure that a shorter arc length re-sults in a higher pressure. Such a high pressure meansthat the force generated by the impingement of the arcjet on the bath surface can be very significant. Thearc plasma photographed by JONES et al. [8] exhibitedto be a high velocity turbulent self-constricted jet. Acrater-like depression caused by the arc in the surfaceof the bath is clearly shown. The temperature, flow andcurrent density fields of the bath are then believed tobe significantly affected by the force. The interactionbetween the arc plasma and the molten bath is worthyof further study.

Fig.7 Radial distribution of pressure on the anode surface

for different arc lengths in a range of 5 cm to 9 cm

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Plasma Science and Technology, Vol.14, No.4, Apr. 2012

Since the heat flux transferred from the arc to themolten bath plays an important role in the operationof the furnace, it is necessary to estimate its value andefficiency. There are four different mechanisms of heattransfer considered, namely convection, the Thompsoneffect, condensation of electrons and radiation [4]. Anintegration of these four mechanisms is considered tobe the total effective arc power. It should be pointedout that the bottom area of the electrode is much largerthan that of the cathode spot. Although the arc insta-bilities in a realistic operation of DC arc furnaces maycause the cathode spot to move around irregularly onthe electrode surface [10], the total effective arc poweris still assumed to be valid here. The effect of the arclength on both arc power and arc efficiency is shown inFig. 8. The Joule heating power of the arc is calculatedby the following equation,

PJ =V

Qdv, (16)

where V is the conductive volume of the arc, and Jouleheat power per unit volume Q is calculated by

Q = E J, (17)

with E the electric field. It is found that the arc powerdoesnt increase significantly with the increase in arclength with a fixed arc current. This is because thatmost of the arc power is dissipated near the arc root.For a short arc the environmental enclosure leads to acirculation of the air in SAF, so the arc efficiency ismuch higher than that in a DC EAF for steel mak-ing [11]. Additionally, a shorter arc length leads to ahigher arc efficiency in the operation of SAF.

Fig.8 Effect of arc length on the arc power and arc effi-

ciency

4 Conclusions

a. A model for a DC arc was developed. The pre-dicted arc temperatures and velocities agree well withthe experimental data by BOWMAN.b. The behavior of arcs in SAF for MgO production

is predicted by the model, including the temperaturefield, the velocity field, and the pressure distributionon the anode surface.c. The calculated results show that the maximum

pressure at the anode surface is in an order of mag-nitude of 104 Pa, and the pressure distribution is af-fected by the arc length significantly. Modeling of thebath considering impingement of such a jet is worthyof further study.d. The circulation of the high temperature air under

the electrode bottom in SAF can lead to a higher arcefficiency, especially for a shorter arc.

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(Manuscript received 13 January 2011)(Manuscript accepted 29 April 2011)E-mail address of corresponding authorWANG Ninghui: ninghuiw@263.net

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