ISIJ International, Vol. 40 (2000), No. 11, pp. 1089–1097 Modeling of a DC Electric Arc Furnace—Heat Transfer from the Arc Jonas ALEXIS, Marco RAMIREZ1), Gerardo TRAPAGA1) and Pär JÖNSSON2) MEFOS, Box 812, SE-97525 Luleå, Sweden. Also, a graduate student at the Div. of Metallurgy, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. 1) Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, Cambridge MA, 02139, USA. 2) Div. of Metallurgy, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. (Received on March 27, 2000; accepted in final form on June 27, 2000 ) A mathematical model describing heat and fluid flow in an electric arc has been developed and used to predict heat transfer from the arc to the steel bath in a DC Electric Arc Furnace. The arc model takes the separate contributions to the heat transfer from each involved mechanism into account, i.e. radiation, convection, condensation and energy transported by electrons. The model predicts heat transfer for different currents and arc lengths. Model predictions show that arc efficiency is higher for lower power input. The model also predicts shear stresses and current density distribution at the steel surface. This information can be used as boundary condition input to simulate the effect of heating with electrodes in a DC EAF on the heat and fluid flow in the steel bath. KEY WORDS: modeling; arc; heat transfer. 1. ture, velocity and current in the arc and also the shear stress, velocity, temperature and heat flux at the border of the steel bath (anode) for four different arc lengths and three different currents are presented and discussed. Introduction Both in an electric arc furnace (EAF) and a ladle furnace (LF) heat can be supplied to the steel bath through graphite electrodes, making it possible to control the steel temperature. In order to improve the control of these metallurgical reactors, a fundamental understanding of the heat and fluid flow, mass transfer and electromagnetic phenomena is needed. In the past, several individual models of electric arcs, EAF steel baths, and stirred ladles have been developed.1–12) Coupling of these models is necessary in order to achieve accurate prediction of heating effects in EAFs and LFs. One approach to carrying out this coupling is presented in this paper. A model of a DC electric arc has been developed to predict heat transfer, current density and shear stresses at the boundary where the steel bath surface is located. This information is further used in the second part of this investigation to define boundary conditions and represent heating and mixing effects on the steel bath in a DC EAF.13) In an earlier publication7) a prediction of fluid and heat flow in the arc was made. Predicted velocities and temperatures were compared with experimental data from an investigation by Bowman14) and were found to agree well. These results showed that the model predictions were reliable. In the present model, further improvements have been made. The induced current has been included together with an improved representation of the boundary conditions necessary for modeling the steel bath (anode). In the first part of the paper a review of previous work is given and the mathematical representation of the DC arc is described. Thereafter results of the distribution of tempera- 2. Previous work Improvements in EAF efficiency require fundamental knowledge of the physics involved in the process, i.e. heat transfer, mass transfer, fluid flow and the electromagnetic phenomena. To achieve this goal the use of CFD models has become increasingly popular. In 1981 first Ushio and Szekely1) and then McKelliget and Szekely2) performed numerical simulations of a DC EAF using the Turbulent Navier Stokes, energy conservation and Maxwell equations in both the arc and the bath regions of the system. They were the first to predict the contribution of the different mechanisms of heat transfer from the arc to the bath. In their calculations a parabolic current density distribution was assumed through the arc region to simplify the magnetic problem. In 1985 McKelliget and Szekely3) used the magnetic diffusion equation to predict heat transfer and fluid flow in a welding arc. With more powerful computers it became possible to solve more complex problems. In 1992 Choo, Szekely and Westhoff,4) and in 1995 Qian, Farouk and Mutharasan,5) solved Laplace’s equation for the electric potential to determine boundary conditions for a model of the weld pool. In 1996 Larsen and Bakken6) used the magnetic transport equation to predict the current and magnetic field in an AC arc. The above-described work used different approaches to predict fluid flow in arcs, and in one case4) used the calculated heat flux as boundary conditions in a weld-pool 1089 © 2000 ISIJ ISIJ International, Vol. 40 (2000), No. 11 Fig. 1. Calculation domain for the DC arc model. model. More specifically these authors used three different methods to determine the Lorentz forces responsible for the fluid flow. These methods are i) The Laplace equation for electric potential ii) the magnetic diffusion equation, and iii) the complete magnetic transport equation. The first two methods, however, do not include the induced electric field term, v3B, in the solution. Assuming that the magnetic Reynolds number is much less than unity, these methods give a good approximation of the Lorentz forces. However, this is not always the case for high-current arcs, as in the case of EAFs. The third method does include this term, but a disadvantage with this method is that the system of equations becomes more difficult to solve. In the present work, the Poisson equation for electric potential is used to overcome these problems since the induced electric field is included and the current in the system can be correctly described. 3. Fig. 2. Calculation grid for the DC arc model. • The anode (border to the steel bath) is assumed to be flat and the bath itself is neglected. • The MHD approximation20) is applicable, implying that a simplified form of Maxwell’s equations can be used to describe current and magnetic fields. Since the arc is axis symmetric, a two-dimensional calculation domain in cylindrical co-ordinates can be used. An outline of the calculation grid is shown in Fig. 2. Smaller calculation cells are used in the plasma region and close to the anode and cathode where greater calculation accuracy is required. 3.3. Transport Equations for the DC Arc According to the assumptions in Sec. 3.1 the governing transport and turbulence equations for the DC arc may be expressed in two-dimensional cylindrical coordinates as follows: Mathematical Model of a DC Electric Arc In the model presented in this report the arc is treated as a fluid1–12) with temperature-dependent thermodynamic properties.15,16) The coupled conservation equations of energy, mass and momentum, which define plasma temperature, pressure and velocity, are solved together with Maxwell’s equations. First the problem is stated and a description of the fluid-flow model is provided. Then source terms, boundary conditions and heat transfer mechanisms are discussed. Conservation of Mass: ∂( ρv ) 1 ∂( ρrw ) 1 5 0 .......................(1) ∂z r ∂r where r is the density, r is the radial distance, and z is the axial distance. The variables v and w are the velocity components in the radial direction and axial direction, respectively. 3.1. Statement of the Problem In Fig. 1 a schematic representation of the region of integration of the DC arc model is given. The system consists of the cathode (graphite electrode), the arc column, and the anode (steel bath). The calculation domain is defined so as to allow for entrainment and general interaction with the surrounding gases. Conservation of Axial Momentum: ∂( ρv 2 ) 1 ∂( ρrvw ) 1 ∂z r ∂r 52 3.2. Mathematical Formulation of the Flow The following assumptions are made in the statement of the mathematical model of a DC arc: • The arc is axis symmetric. • The operation of the arc is independent of time, i.e. steady state. • The arc is in local thermal equilibrium (LTE),17) i.e. the electron and heavy-particle temperatures are very similar. This assumption has been shown to be valid throughout most of a gas tungsten arc, except for in the fringes of the arc and near the anode and cathode surfaces.18,19) © 2000 ISIJ 1  ∂v   ∂P ∂  12  µ eff    ∂z ∂z   ∂z    ∂v 1 ∂  ∂w   1  1 J r Bθ .............(2)  rµ eff  r ∂r  ∂z    ∂r where m eff is the effective dynamic viscosity, P is the static pressure, Jr is the current density in the radial direction and Bq is the magnetic flux density in the azimuthal direction. The product Jr Bq is the axial component of the Lorentz force produced by the current and the induced magnetic flux density in the solution domain. 1090 ISIJ International, Vol. 40 (2000), No. 11 Conservation of Radial Momentum: Table 1. Constants used in the turbulence equations.23)  ∂w ∂( ρvw ) 1 ∂( ρrw 2 ) ∂P ∂  ∂v   1 52 1 1   µ eff  ∂z r ∂r ∂r ∂z  ∂r    ∂z 2 ∂ 1 r ∂r  2v ∂w   rµ eff ∂r  2 µ eff r 2 2 J z Bθ .....................(3)   G 5 µ eff where Jz is the current density in the axial direction. The product Jr Bq is the radial component of the Lorentz force. The effective dynamic viscosity used in Eq. (2)–(7) is defined as Conservation of Thermal Energy: ∂( ρvh) 1 ∂( ρrvh) 1 ∂z r ∂r 5 m eff5m l1m t ................................(8) ∂  µ eff ∂h  1 ∂ 1 ∂z  σ T ∂z  r ∂r  µ eff ∂h   r σ ∂r  1   T J z2 where m l is the molecular dynamic viscosity and m t is the turbulent dynamic viscosity. The turbulent dynamic viscosity can be calculated using the following relation: 1 J r2 σe 5 k b  J r ∂h J ∂T  ......................(4) 2 SR 2 1 z  2 e  C p ∂r C p ∂z  µ t 5CD ρ 3.4. Electromagnetic Transport Equations for the DC Arc The equations for conservation of axial and radial momentum (2)–(3) include source terms for the Lorentz forces. The Lorentz forces are given by: F5J3B ................................(10) In a two-dimensional axis-symmetric system the radial and axial components of the source terms in the momentum equations are given by: 3.3.1. Turbulent Transport Equations The following equations for the turbulent kinetic energy, K, and the dissipation rate of the turbulent kinetic energy, e , from the well-known K–e model has been used.23) Fr52Jz Bq ..............................(11) Fz5Jr Bq ............................(12) The turbulent kinetic energy is expressed as Thus, in order to determine these source terms the current density and the azimuthal magnetic flux density have to be calculated. Using the MHD approximation gives: ∂( ρvK ) 1 ∂( ρrwK ) 1 ∂z r ∂r ∂  µ eff ∂K  1 ∂ 5 1 ∂z  σ K ∂z  r ∂r K 2 ..............................(9) ε where CD is a constant. where h is the enthalpy, Cp is the specific heat at constant pressure, s T is the thermal conductivity, s e is the electrical conductivity, SR is the radiation loss term, kb is the Bolzmann constant and e is the electron charge. The thermal energy Eq. (4) consists of the two convection terms, the two diffusive terms, the Joule heating source term, the radiation source term and the transport source term of enthalpy due to electron drift (Thompson effect). In these calculations the net radiation losses are used to predict the radiative cooling of the arc.21,22) ∇ · J50 .................................(13)  µ eff ∂K  1 G 2 ρε ...(5) r σ ∂r   K For a moving conductive fluid, Ohm’s law takes the following form: where s K is a constant in the turbulence model. G is the volumetric generation of turbulent energy. The dissipation rate of turbulent kinetic energy is expressed as J5s e[E1U3B] .........................(14) where the first term inside the parentheses (i.e. the electric field E) defines the contribution to the applied current Japp and the second term defines the contribution to the induced current JI. 1 ∂ ∂ ( ρvε ) 1 ( ρrwε ) ∂z r ∂r ∂  µ eff ∂ε  1 ∂ 5 1 ∂z  σ ε ∂z  r ∂r 2 2 2 2    w    ∂v  ∂w  ∂w     ∂v   2   1   1  r   1  ∂r 1 ∂z      ∂z       ∂z     .........................(7) The electric potential is given by:  µ eff ∂ε  ε  r σ ∂r  1 K (C1G 2C2 ρε )   ε E52∇F ...............................(15) Combining Eqs. (13), (14) and (15) gives: .........................(6) ∇ s e · ∇F 5∇ · s e (U3B)5∇ · Ji .............(16) where s e , C1 and C2 are constants in the turbulence model. The values used in the turbulence model are listed in Table 1. The volumetric generation of turbulent kinetic energy is expressed as: Equation (16) can be solved numerically to obtain the electric potential, which is used to calculate the total current using Ohm’s law. Ampere’s law can be used to approximate the azimuthal 1091 © 2000 ISIJ ISIJ International, Vol. 40 (2000), No. 11 Table 2. Boundary conditions for the DC arc. is assumed to be a parabolic function of the radius r. Current density is given by   r  2  .......................(18) J 5 2 J C 12   RC     where the radius of the cathode spot is defined as RC 5 and where I is the total current in the system and JC is the average current density in the cathode spot. In the calculations presented in this paper JC is assumed to be 4.43107 A.25) The electric potential is assumed to be iso-potential (zero) at the anode (region f–g). This is based on the assumption that the conductivity in the metal is much higher than in the plasma, implying that the variation of the electric potential in the metal is much less than in the arc. The other boundaries are treated as symmetry lines in the electric-potential calculation. magnetic flux density in an axis-symmetric model: Bθ 5 µp r ∫ r J z rdr .........................(17) 0 3.6. Source Terms Used at the Cathode and Anode Regions 3.6.1. Cathode At the cathode boundary layer local thermal equilibrium conditions are not believed to exist due to a difference in temperature between electrons and heavy particles.26) For thermionic cathodes, McKelliget and Szekely3) suggested that a positive source term could be used to account for the energy used in the cathode boundary layer to ionize the plasma and thereby cause a drop in the electric potential. The cathode fall heat flux can then be expressed as where m p is he magnetic permeability in the media. In these calculations m p is assumed to be equal to m 0. 3.5. Boundary Conditions for the DC Arc In order to solve the transport equations for the calculation domain, boundary conditions need to be specified. A complete listing of boundary conditions for the DC arc is presented in Table 2. Details regarding the boundary conditions for each parameter that is solved are given below. When a reference is given to a region in the solution domain within parentheses it refers to the definitions presented in Fig. 2. QC5|JC |VC ..............................(20) 3.5.1. Velocities No-slip boundary conditions, i.e. zero velocities, are imposed at the anode (region f–g) and at the electrode surfaces (region a–d). At the other boundaries (regions d–e and f–g), a constant pressure is imposed to allow for the inflow and outflow of gases. and the cathode fall voltage, VC , can be described as 5 k bTelec ............................(21) 2 e where Telec is the temperature of the electrons. Telec is approximated by the following relationship10): VC 5 3.5.2. Turbulent Kinetic Energy and Energy Dissipation Rate Logarithmic wall functions are used to calculate the turbulent kinetic energy and energy dissipation rate at the anode (region f–g) and at the electrode surfaces (region a–d). Telec5Tc,g2Tcat ...........................(22) where Tc,g is the temperature of the gas in the cell closest to the cathode and Tcat is the temperature in the cathode. In the model the distance from the anode surface to the center of the first cell is 0.5 mm. This is comparable to the maximum observed thickness of the anode fall region, 0.1 mm.27) 3.5.3. Temperature Inside the cathode spot (region a–b), the surface temperature is assumed to be 4 000 K, which is below the boiling point of carbon.20) Outside the cathode spot (region b–c and c–d) and at the anode (region f–g) the temperature is assumed to be 1 800 K, which is a common temperature of the steel in the EAF. In reality the steel temperature can vary, but this variation will not affect the studied arc characteristics. The other boundaries are treated as symmetry lines, so the temperature gradient is zero normal to the symmetry axis. 3.6.2. Anode The energy lost by the arc in the vicinity of the anode is electrical and thermal energy. The electrical part is transferred to the plasma by making atoms vibrate faster (joule heating). This is taken into account in Eq. (4), which is the transport equation for conservation of thermal energy. The enthalpy of the electrons is taken into account in the form of the Thompson effect. The thermal energy loss is due to conduction, convection, radiation and vaporization. However, the energy loss due to vaporization is considered to be negligible and has not been dealt with in this study. In summary, the following four different mechanisms for the heat transfer from the arc to the anode are considered: 3.5.4. Electric Potential In the cathode spot, RC (region a–b), the current density © 2000 ISIJ I ..............................(19) πJ C 1092 ISIJ International, Vol. 40 (2000), No. 11 • • • • Convective heat transfer Radiative heat transfer Thompson effect Condensation of electrons (work function and anode fall) which means that source terms need to be added to include each of these mechanisms. 3.6.3. Convective Heat Transfer The convective heat transfer from the plasma to the molten steel, QCon, can be described by the following equation28): QCon 5 0.915  ρ b µ lb  σ Tw  ρ w µ lw  0.43  ρ w µ lwν b    r   0.5 (hb 2 hw ) ....(23) where the subscript w denotes the values of the bath surface and b denotes the values in the boundary layer and h is the enthalpy. Fig. 3. Radial distribution of current density (z-component) in the arc column, at different cathode distances. Arc current is 36 kA and arc length 25 cm. 3.6.4. Radiative heat transfer The sum of radiative heat transfer, Qr, from each volume element in the calculation mesh to the liquid steel surface is calculated by means of approximate view factors as follows: Qr 5 ∑ S R, j 4πdi2, j cluded in the rest of the calculation domain and is included in the discussion of the source terms at the anode to point out its importance in the vicinity of the anode. 3.7. Method of Solution The solution of the transport, turbulent and electromagnetic equations and the equations for boundary conditions and source terms was obtained using the commercial code PHOENICS. During a calculation the equations are solved by iteration until the residuals are less then 1%. With a twodimensional grid containing 2 025 cells a typical simulation takes about 4 hours of CPU time on a Sun Ultra Enterprise 4000. The temperature dependent thermodynamic properties for air, which is the gas used in all the calculations, was calculated by Murphy.15,16) A detailed discussion regarding the effect of using other gaseous media on the arc characteristics can be found in an earlier publication.7) cos Ψi DV j ..................(24) where SR, j is the radiation loss from volume element DVj , in the arc column to the steel surface i, di, j is the vector joining the surface i and the volume j and y is the angle between d and the normal vector of the surface. In these calculations the net heat losses are used to predict radiative heat transfer to the anode.21,22) 3.6.5. Thompson Effect The transport of thermal energy by the electrons, Qe, due to the Thompson effect can be described by the following equation10): Qe 5 5J A k b (TA,g 2TAnode ) ..................(25) 2e 4. Results and discussion The heat transfer to the anode is described by Eq. (23) to (26). These equations imply that current density, temperature and velocity at the anode, and temperature in the arc are required to predict the rate of heat transfer between the arc region and the steel bath. To determine these parameters an accurate description of the arc is needed. Results from model predictions presented below are from cases with three different currents (36, 40 and 44 kA) and four different arc lengths (ranging from 15–30 cm). where JA is the current density at the anode, TA,g is the temperature at 0.1 mm from the anode27) and TAnode is the temperature at the anode. In this study TA,g is taken as the temperature at the cell closest to the anode. The distance from this cell to the anode surface is 0.5 mm. 3.6.6. Condensation of Electrons The contribution from the condensation heat flow being generated by the electrons crossing the anode fall and entering the liquid steel, QA, can be described as follows: 4.1. Calculation of Magnetic Flux Density From the predicted electric potential it is possible to get a correct description of the current density in the system. The magnetic flux density is calculated from the predicted current using Eq. (17). In Fig. 3 predicted current density profiles (z-component) in the arc column at different distances from the cathode are plotted for a case with a 36 kA arc and a 25 cm arc length. As seen in the figure, the distribution of current in the arc defines the extent of the conduction region (i.e. arc radius) in the system. The extent of the arc re- QA5JA(VA1qA) .........................(26) where VA is the anode fall voltage and qA is the work function (required anode fall to release an electron from the anode). In this investigation VA and qA have both been assumed to be 4V.8) The contributions to heat transfer from convection, radiation, condensation and the Thompson effect were implemented into the code by including source terms in the first cell closest to the anode. The Thompson effect was also in1093 © 2000 ISIJ ISIJ International, Vol. 40 (2000), No. 11 Fig. 4. Azimuthal magnetic field contours in the arc region calculated using Ampere’s law. Arc current is 36 kA and arc length 25 cm. Each line represents 0.02T. Fig. 5. Temperature contours in the arc. Arc current is 36 kA and arc length 25 cm. Each line represents 1 500 K. gion increases with an increased distance from the cathode. Figure 4 shows a plot of the calculated azimuthal magnetic flux density obtained by integration of Eq. (17). As seen in the figure, the radial dependence of the magnetic field, for a given axial location, increases to a maximum value as the radius increases and then follows a decreasing trend. The location of the maximum value shifts to a larger radial distance as the distance from the cathode increases. These resultant contours clearly define two regions associated with low and high electric conduction that make up the arc region. Also note that the local maximum values of the magnetic flux density decrease with an increasing distance from the cathode. Figures 3 and 4 describe the two important variables, the current density and azimuthal magnetic field, in resolving the magnetic problem. Fig. 6. Anode current density at the anode for arc lengths in the range of 15–30 cm. Arc current is 36 kA. 4.2. Heat Transfer and Shear Stress at the Steel Bath As mentioned above, the main purpose of this study was to calculate the boundary conditions at the steel/arc interface in an electric arc furnace. These are necessary to model the EAF steel bath more accurately. These boundary conditions mainly involve heat transfer from the DC arc to the steel bath (anode) and shear stresses at the steel/arc interface. The current distribution at the anode is required to calculate the transport of energy by the electrons due to the Thompson effect (25) and the condensation of electrons (26). The current density at the anode is also needed to calculate the current density inside the steel bath. The arc temperature is important for determining the heat transfer due to radiation as described by Eq. (24). The temperature contours for a 36-kA arc with an arc length of 25 cm are plotted in Fig. 5. As can be seen, the highest arc temperatures are found at a central location close to the cathode. The temperature distribution in the arc is affected by the plasma flow (i.e. turbulent mixing) for a given arc length and current distribution. Hot gas from the cathode region is transported to the bath surface and thereafter spread parallel to the surface, forming an impinging region and a growing boundary layer along the surface of the steel bath. Simultaneously, cold gas is being entrained into the turbulent plasma jet. Steep temperature gradients are observed across the fringes of the jet. As seen in previous figures, most of the current in the arc © 2000 ISIJ is transported in a narrow arc column. This affects the current distribution at the anode, as shown by the results presented in Fig. 6. It is apparent from this figure that a longer arc length results in a somewhat wider distribution of the current, but a lower maximum value. The data also show that the maximum current in the center of the arc decreases with the arc length at the anode. The maximum current at the center of the anode is proportional to approximately 1/(arc length), which implies that the change in maximum current at the anode is greater for a short arc than a long arc for the same change in arc length. The radial gas velocity at the anode plays an important role in determining the convective rate of heat transfer, as indicated by Eq. (23). A plot of velocities at the anode is shown in Fig. 7 for calculations with an arc current of 36 kA. As seen in the figure, the radial velocity is higher for the shortest arc (15 cm) up to a distance of about 5 cm from the center of the arc and then it is higher for the longer arcs (20, 25 and 30 cm). It is also clear that the maximum radial velocity is higher when a shorter arc (15 cm) is used, since the impingement region is closer to the high-pressure region (Maeker pressure) generated in the vicinity of the cathode. This high-pressure region is also responsible for another high-pressure zone at the point of impingement with the anode. The point of maximum velocity shifts to an increased radial distance with an increase in arc length due 1094 ISIJ International, Vol. 40 (2000), No. 11 Fig. 9. Heat transfer from the arc to the steel bath due to i) convection, ii) radiation, iii) Thompson effect, and iv) condensation of electrons. Arc current is 36 kA and arc length is 25 cm. Fig. 7. Radial gas velocity at the anode for arc lengths in the range of 15–30 cm. Arc current is 36 kA. Table 3. Heat transfer from the described heat transfer mechanisms to a circular region with a 0.5 m radius. Fig. 8. Radial temperature at the anode for arc lengths in the range of 15–30 cm. Arc current is 36 kA. to greater expansion of the jet. Beyond the core region, at radial distances larger than 5 cm, the reverse trend is observed, i.e. higher velocities correspond to longer arc lengths. The gas temperature at the anode is also important for the convective heat transfer as can be seen in Eq. (23). Computed radial temperature profiles at the anode are presented in Fig. 8 for different arc lengths but the same current of 36 kA. As seen in this figure, the maximum temperature at the center of the arc is higher for the case with the shortest arc (15 cm). This is due to the shorter distance to the hot region below the cathode and, as a consequence, a higher current density at the center (as indicated in Fig. 4). Farther away, the radial temperature is higher for longer arcs (20, 25 and 30 cm), which in part is also a consequence of the differences in current distribution. Furthermore, the fluid temperature profile in the vicinity of the anode is affected by the different mechanisms that determine the overall rate of heat transfer at the anode (i.e. convection, radiation, anode fall and Thompson effect). The contributions from the different heat transfer mechanisms to the total heat transfer to the steel bath for a 36 kA arc with an arc length of 25 cm are compared in Fig. 9. The results suggest that the rate of heat transfer from the arc to the steel bath are dominated by convection and radiation for Fig. 10. Cumulative heat transfer to the anode for arc lengths in the range of 15–30 cm. Arc current is 36 kA. the range of conditions studied in this work even though the maximum heat transfer for the contribution from the condensation of electrons is higher at the center. In Table 3 the amount of heat transferred to the steel bath from the described heat transfer mechanisms is compared. Here it can be seen that convective and radiative heat transfer are of the same order of magnitude for a given arc length and current. The contribution from anode fall is about half of that and the contribution from the Thompson effect is one order of magnitude less. A longer arc results in a larger change in the radiative than the convective contribution to heat transfer, which also means that a larger part of the total heat transfer will be from radiation. In Fig. 10 the cumulative heat transfer rate from a 36-kA arc to a circular region of 0.5 m radius at the anode is plotted for different arc lengths. Here it can be seen that a longer arc provides a higher total heat flow (W) to the 1095 © 2000 ISIJ ISIJ International, Vol. 40 (2000), No. 11 arc length. If the arc efficiency is plotted as a function of power generation in the arc, which is done in Fig. 12, it can be seen that the arc efficiency is higher for lower power, which means that the use of shorter arcs and lower currents will result in higher efficiency. This can be described by the correlation: Earc54.61 · 1027 Pg125.9 ...................(27) where Pg is the power generation and Earc is the arc efficiency. This implies that if lower energy consumption is preferred, shorter arcs and/or lower currents are beneficial, but if higher heating rates are desired, longer arcs and/or higher currents are needed. 5. Fig. 11. Shear stress at the anode for arc lengths in the range of 15–30 cm. Arc current is 36 kA. A mathematical model of a DC arc has been developed. The DC arc model was used to predict the heat transfer from the arc to the steel bath in an EAF, and also the shear stresses caused by the arc at the bath surface. Calculations were made for three currents, 36, 40 and 44 kA, and four arc lengths, 15, 20, 25 and 30 cm. In the calculation of the heat transfer the contributions from radiation, convection, condensation and energy transported by electrons were taken into account. Contrary to results presented by previous researchers, in the present work it is found that radiation and convection dominate the rate of heat transfer from the arc to the steel bath, even though the contribution due to electrons is higher at the center of the system, under the conditions studied. For longer arcs, radiation becomes even more dominant as compared with the other mechanisms. This contradiction regarding which mechanisms that is dominating the heat transfer can probably be contributed to the fact that previous researchers did apply a specified current distribution, or did not include the induced current, in the model. As opposed to the model presented here where the current including the induced part, is calculated from the solved electric potential equation. The model is also useful in determining the role of the various operating parameters (current, separation length, electrode and bath dimensions, etc.) in the behavior of the system. Specifically, it is possible to calculate the heat efficiency of the system for a given set of operating conditions. The shear stress at the steel bath was also calculated since it has significant influence on the surface velocities in the model of the EAF. Predictions show that the shear stress is proportional to the radial gas velocity, which is expected. The overall conclusion from this study is that it is possible to predict heat transfer and shear stress at the anode (steel bath) in a DC EAF caused by an arc. Therefore, it is possible to calculate boundary conditions at the steel surface/arc interface, which can be used to model the DC EAF. The effect of the heat transfer and shear stress on the heat and fluid flow in the steel bath is discussed in the Part II report on this work.13) Fig. 12. Arc efficiency as a function of input power for arc lengths in the range of 15–30 cm and three different currents (36, 40 and 44 kA). anode even though a shorter arc gives a higher maximum heat flux (W/m2) at the center of the arc, which leads to higher heat transfer directly below the electrode. In order to model the effect of the arc on the velocities close to the steel surface in the EAF it is also necessary to calculate the shear stresses generated by the plasma at the anode. Results from predictions are shown in Fig. 11. It can be seen that the shear stress is greater for the 15 cm arc than for the other arc lengths up to a distance of about 5 cm from the arc center. Beyond the central region the shear stress profiles in the 20, 25 and 30 cm arcs are of similar magnitude. The shear stress is proportional to the radial velocity gradient in the axial direction (dv/dz). Therefore, the larger maximum radial velocity for the 15 cm arc in Fig. 7 gives an indication of why the maximum shear stress is higher for shorter arcs. One interesting characteristic of the heat transfer from an electric arc is the arc efficiency. The arc efficiency is defined here as the amount (percentage) of generated power in the arc that is transferred as heat to the steel bath. The generated power is the product of voltage drop and current in the arc. The arc efficiency is affected by generated power and arc length. The generated power is in turn affected by © 2000 ISIJ Conclusions Acknowledgements The authors wish to thank the Swedish Steel Producers Association (Jernkontoret) and the committee TO23, for financial support of part of this work. Furthermore, M. R. 1096 ISIJ International, Vol. 40 (2000), No. 11 and G. T. would like to express their thanks to CONACYT (Mexican government) for financial support by awarding a scholarship to M. R. during his graduate work at MIT. 14) 15) REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 16) M. Ushio, J. Szekely and C. W. Chang: Ironmaking Steelmaking, 6 (1981), 279–286. J. Szekely, J. McKelliget and M. Choudhary: Ironmaking Steelmaking, 10 (1983), 169–179. J. McKelliget and J. Szekely: 5th Arc furnace meeting, Budapest, Hungary, (1985). R. T. C. Choo, J. Szekely and R. C. Westhoff : Metall. Trans. B, 23B (1992). F. Qian, B. Farouk and R. Mutharasan: Metall. Trans. B, 26B (1995), 1057. H. L. Löken Larsen and J. A. Bakken: TPP-4, (1996). J. Alexis: Proc. of 56th Electric Furnace Conf., ISS, Warrendale, PA, (1998), 319–326. J. W. McKelliget and J. Szekely: Metall. Trans. A, 17A (1986), 1139. B. Andresen: NTH, Trondheim, Norway, MI-report (1995): 34. P. Jönsson, PhD Thesis, MIT, Boston, USA (1993). J. W. McKelliget and J. Szekely: J. Phys. D., 16 (1986), 1007. J. Szekely, J. McKelliget and M. Choudhary: Ironmaking Steelmaking, 10 (1983), 4. M. A. Ramirez, G. Trapaga, J. Alexis and P. Jönsson: “Modeling of a 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 1097 DC Electric Arc Furnace Part II: Mixing in the Bath”, to be submitted to ISIJ. B. Bowman: J. Phys. D., 5 (1983), 1422. A. B. Murphy and C. J. Arundell: Plasma Chem. Plasma Process., 14 (1994), No. 4, 451–490. A. B. Murphy: Plasma Chem. Plasma Process., 15, (1995), No. 2, 279–307. K. C. Hsu and E. Pfender: J. Appl. Phys., 54 (1983), 256. K. C. Hsu, K. Etemadi and E. Pfender: J. Appl. Phys., 54 (1983), 1293. K. C. Hsu and E. Pfender: J. Appl. Phys., 54 (1983), 4359. W. F. Hughes and F. J. Young: The electromagnetodynamics of fluids., Wiley, New York, (1966). K. A. Ernst, J. G. Kopainsky and H. H. Maecker: IEEE Trans. Plasma Sci., 1 (1973), No. 4, 3. L. E. Cram: J. Phys. D: Appl. Phys., 18 (1985), 401. E. Launder and D. B. Spalding: Mathematical models of turbulence, Academic Press, London, (1972). Handbook of Physico-chemical properties at high temperatures, Eds. by Y. Kawai and Y. Shiraishi, Tokyo, (1988). G. R. Jordan, B. Bowman and D. Wakelam: J. Phys. D., (1970), 1089–1099. K. C. Hsu and E. Pfender: J. Appl. Phys., 54 (1983), 3818. W. Finkelnburg and S. M. Segal: Phys. Rev., 83 (1951), 582–585. F. R. G. Eckert and E. Pfender: Adv. Heat Transfer, (1967), 4, 229–316. © 2000 ISIJ