Current, Magnetic Field and Joule Heating in Electroslag Remelting Processes

ISIJ International, Vol. 52 (2012), No. 7, pp. 1289–1295

Baokuan LI,1) Fang WANG1) and Fumitaka TSUKIHASHI2)

1) School of Materials and Metallurgy, Northeastern University, Shenyang, 110819 China. 2) Department of Advanced Materials, Graduate School of Frontier Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8561 Japan. (Received on June 14, 2011; accepted on February 16, 2012)

A three-dimensional (3D) finite element model was developed to simulate the current density, magnetic field, electromagnetic force and Joule heating for a system of electrode, slag and ingot in electroslag remelting processes. Especially, the skin effect is taken into account in electromagnetic field model by magnetic vector potential method. Simulated magnetic flux density is compared with experiment and obtained a good agreement. Numerical results show that the skin effect becomes strong and the maximum current occurs at the surface of electrode and ingot, the maximum of electromagnetic force is at the upper surface nearby the electrode in the slag and the maximum of Joule heating is at the interface of electrode and slag. The parametric study shows that the eddy current flow occurs in interior of electrode and ingot if the current frequency is more than 40 Hz. With the increasing of electrode immersion depth or decreasing of slag cap thickness, the maximum of the Joule heating in slag increases. KEY WORDS: electroslag remelting processes; electromagnetic field; Joule heating; numerical simulation.

the voltage measurements in an ESR process. Lately, Patel4) 1. Introduction obtained the electromagnetic field using a current stream- The electroslag remelting (ESR) system has been widely function and an analytical solution. The general stream-func- applied in the special steel industry. Knowledge of current, tion formulation was also modified to account for mold cur- magnetic field and Joule heating, which controls the remelt- rent. However, a current stream-function and an analytical ing processes and quality of the product, is important for the solution are difficult to be extended to analyze the 3D electro- design and operation of the system. Mathematical modeling magnetic field. Kharicha et al.5) developed a numerical model and numerical simulation are of great interest because to predict the electric current paths in an ESR process where experimental study in such a system is extremely difficult. no electric current insulation hypothesis for the mold is made. The principal components of an ESR system as shown in Weber et al.6) developed a two-dimensional axisymmetric Fig. 1 include a consumable electrode, a molten slag pool, transient model of the ESR process, which accounts for cou- a liquid metal pool, a solidified ingot, and a water-cooled pled electromagnetic, fluid flow, heat transfer, and phase mold. The electric current (AC) is passed from the electrode change phenomena. Hernandez-Morales and Mitchell7) had through the molten slag and the liquid metal pool to the reviewed the mathematical models of transport in ESR pro- ingot. The resultant Joule heating of the slag melts the elec- cesses and pointed that the more effort is required before the trode while the droplets formed, fall through the slag and accu- models can be applied to define actual operating conditions. mulate in the liquid metal pool. Because of the water cooling The purpose of the present work is to understand the dis- provided in the mold, solidification occurs continuously at the tribution of current, induced magnetic flux density, and pool-ingot interface. The solidified ingot is the product of the Joule heating in the ESR processes based on the three- ESR process. dimensional model with the primary variables of electro- Dilwari and Szekely1) developed the general framework magnetic field. At the same time, parametric study is also for analyzing the electromagnetic fields in ESR. The elec- conducted with the effect of frequency, electrode immersion tromagnetic field was formulated using the Maxwell equa- depth, and slag cap thickness on current density and Joule tions, and a complete set of boundary conditions was also heating in the ESR system. prescribed. They solved the electromagnetic field problem numerically, using the finite difference technique. Mitchell 2. Mathematical Formulation and Joshi2) predicted the voltage distribution within the slag from the slag resistance, they divided the slag into smaller 2.1. Governing Equations domains, and determined the resistance of each domain in The present work focuses on an ESR system with the elec- the network from its geometry and electrical conductivity. trode, slag and ingot as shown in Fig. 1. Mathematical formu- They also meticulously measured the voltage during an ESR lation with the primary variables of electromagnetic field is process with an 8 cm diameter ingot. Rawson et al.3) reported composed of Maxwell equations, Lorentz law and Joule Law.

K plex problem. This can be achieved by the complex formal- KK∂D Ampere’s law: ∇×HJ = + ...... (1) ism. A harmonic analysis provides two sets of solution: the ∂ t real and imaginary components of a complex solution. Thus K K ∂ the force can be obtained as the sum of “real” and “imaginary” Faraday’s law: ∇× = − B ...... (2) E 1 ∂t forces. The time average force is F=+() JB JB . In a K 2 rr ii Gauss’s law: ∇⋅D = ρ ...... (3) similar manner the time averaged Joule heat can be obtained K 1 Continuity of Magnetic flux density:∇⋅B =0 ...... (4) as: Q=+() JE JE . The time averaged values of theses 2 rr ii The displacement electric current is much lower than the quadratic quantities can be obtained as the sum of real and electric conduction as the frequency is less than 50 Hz. So imaginary set solutions. that Ampere’s law changes: The data required for running the model are mainly the KK thermophysical properties of slag and metal, the geometry ∇×HJ = ...... (5) KKK K of the furnace (slag, ingot, electrode, and mold) and the Ohm’s law: J=σ () EvB +× ,...... (6) melting sequence, i.e., the melt rate and electric current pro- K KK files. All properties were compared and completed with Lorentz’s law: F=× JB...... (7) information reported in the literature. For the melt consid- Joule’s law: Magnetic diffusion dominates and fluid flow ered, the electrode material was die steel (H13), and a clas- does not influence the magnetic field, because sical CaF2-based slag with several oxide additions is used. the magnetic Reynolds number remains very The main properties required for the simulation are listed in low.KK Thus, the simplified Ohm’s law applies. Table 1. The commercial software package Ansys 12.0 is JE= σ , used to simulate the electromagnetic field and Joule heating. Joule heat (Joule heat generation rate) is expressed: 2 σ Table 1. Physical properties, geometrical and operating conditions K Q (Jt /) K 2 ω == =σ E ...... (8) of the ESR system. t t K K K Parameter Value where, H : magneticG field, J :currentK density, D : electricK flux density, F : Lorentz force, B : magnetic flux density, E : Electrode (diameter/height), m 0.225/0.5 electric field, ρ: electric charge density, σ : electric conduc- Ingot (diameter/height), m 0.45/1 K K tivity, μ : magnetic permeability, v : velocity, ω : Joule heat Slag (diameter/height), m 0.45/0.15, 0.2, 0.23 generation rate, t: time. Air volume (diameter/height), m 2/2 Current amplitude, A 10 000 2.1.1. Harmonic Analysis Voltage, V 50 In an electromagnetic analysis field, the quantity such as K K K K Frequency, Hz 10, 20, 30, 40, 50 B , H , E and J depends on the space r and time t variables. –1 –1 5 Electric conductivity of ingot, σm, Ω m 7.14 × 10 However, the electromagnetic force and Joule heat are not σ Ω–1 –1 σ same as above field quantity because they include a time- Electric conductivity of slag, s, m ln s = –6 769/T + 8.818 constant term. To minimize the computational cost, the orig- Magnetic permeability of slag, μ, H/m 1.26 × 10–6 inally 4 (3 space + 1 time) dimensional real problem can be Depth of electrode in slag, m 0, 0.03, 0.05 reduced to a 3 (space) dimensional with phase angle com- Density of electrode, kg/m3 7 800 Density of slag, kg/m3 2 800 Density of ingot, kg/m3 7 800

Fig. 1. Schematic of electroslag remelting (ESR) processes. Fig. 2. Finite element mesh in ESR system (electrode, slag, ingot).

2.2. Mesh Generation Where, Im: length of current vector, ψ: initial phase angle, In order to ensure the mesh quality, the finite element vol- ω: rotational speed, t: time. ume is controlled by manual operation. The length of cell Figures 3(a), 3(b) and 3(c) show the time dependence for the electrode, slag and ingot is set to be 0.02 m. The current density vector at main section of ESR system at 0°, structured mesh is used for electrode, slag and ingot, and 45°, 90° initial phase angle while frequency is 50 Hz. Elec- unstructured mesh is used for surrounding air. Total cells are trical current flows into the electrode, via the slag cap, and 90 500. Figure 2 shows the finite element model for the then enter into ingot. The skin effect is remarkable in elec- ESR system. trode, and maximum of current density is on surface, and gradual decrease from surface to center of electrode. As the result of electrical conductivity of slag is very low, current 3. Solution and Boundary Conditions density distribution changes obviously while current enter Magnetic vector method (solid 97 element) is used to the slag cap. Maximum of current density is at interface of solve the electromagnetic field and Joule heating. The mag- electrode, it is benefic for the occurrence of Joule heating. netic flux density, current density, and electromagnetic force When the current enters the ingot, the skin effect takes place are calculated by the magnetic vector method. The Joule owing to the high electric conductivity of ingot. When the heating is calculated by the current density. initial phase angle increases from 0° to 90°, the eddy current At the electrode top (electrode/stub contact) and ingot becomes strong as shown in Fig. 3(c). Figure 4(a) illustrates bottom, the radial current is assumed to be zero. Ampere’s the effect of frequency on the current density distribution in law is applied on the nonimmersed electrode lateral surface. electrode at 45° initial phase angle, it is seen that the skin The immersed part of the electrode is in contact with the effect is very remarkable while frequency is 50 Hz. Alternat- slag; thus, at both vertical and horizontal slag-metal inter- ing current distribution depends on diameter of inductor, fre- faces, the continuity of the tangential electric field is quency and phase angle. The total current is the sum of applied imposed. The same condition is used at the slag/liquid pool and induced current in ingot, i.e. J=++ JJJ222 , thus it interface. At the slag-free surface, the axial current is obvi- zrθ ously assumed to be zero. The solidified slag skin insulates is root-mean-value. applied current distribution is decided electrically the slag and ingot from the mold. Table 1 shows 22 the geometrical, physical properties and operation condi- by the skin depth, δ == = 0.084 m. μωσ μ σ⋅ π tions. In the calculation, the electric conductivity 175 s/m 2 f was deduced through the formulas, when the average tem- Where, f is frequency. Radius of ingot is 0.225 m. Phase perature of slag is assumed as 1 853°C. angle is 45° for alternating current of the 40 Hz and 50 Hz, applied current is very small at center zone of ingot but the induced current is relatively strong, so that the current den- 4. Results and Discussion sity of 40 Hz and 50 Hz are large at center portion as shown 4.1. Current Density Distribution in Fig. 4(b), where the alternating current phase angle is not As the applied source current is alternative, it is expressed nπ (0, ±1, ±2, ……). The eddy current in ingot is observed by sinusoidal function, while frequency more than 40 Hz in Fig. 4(b). As the result of axisymmetrical distribution of current density, Jθ = 0. J i = I sin(ωt + ψ), ...... (9) r m is also little in the slag cap. With the increasing of phase

Fig. 3. Simulated electric current density in ESR system, (a) 0° phase angle, (b) 45° phase angle and (c) 90° phase angle.

Fig. 6. Simulated vector summation of magnetic flux density in air and inner of electrode along the height, frequency is 50 Hz.

Fig. 4. Effects of the frequency on vector summation of current density at (a) 0.3 m above surface of slag in electrode and (b) 0.2 m below interface of slag/metal in ingot in the ESR system.

Fig. 7. Comparison of simulated vector summation of magnetic flux density with measured in air and inner of electrode, frequency is 50 Hz.

associated with the applied current at 45° initial phase angle is shown in Figs. 5(a) and 5(b), direction of magnetic flux density is clockwise, which is accords with right-hand rule of applied current. Figure 6 displays variation of time dependence magnetic flux density of 50 Hz frequency at 45° initial phase angle at different horizontal location including electrode, slag, ingot and air. Maximum value is located at surface of electrode or ingot, and value in electrode is more than that of ingot, since eddy current in ingot is strong. Comparison of predicted time dependence magnetic flux density labeled by lines with measured in air and 0.1 m above interface of slag/metal labeled by symbols is shown in Fig. 7 at 45° initial phase angle. The experiment was car- Fig. 5. Magnetic flux density on (a) surface of electrode, slag and ried out at the facilities of standard alloy H13. The slag com- ingot, (b) interface of slag/metal, maximum is 0.0162 T, position was 70%CaF3 + 30%Al2O3. The geometrical and frequency is 50 Hz. operate parameter was the same as the simulated case. The magnetic flux density was measured by Gauss meter SHT- angle, the induced current becomes strong, overlap of 6. It is observed that magnetic flux density varies with the applied and induced current (reverse direction) cause there range of 0 to 0.008 T. The value at 0.5 m from ingot cen- is a minimum value in current Jz, hence the J has also a min- terline is about 0.003 T. A good agreement is obtained imum value. between numerical results and experiment. The effect of frequency on time dependence magnetic flux density in ingot 4.2. Magnetic Flux Density is shown in Fig. 8, skin effect is also observed with the The predicted time dependence magnetic flux density increasing of frequency. The magnetic flux density in Fig. 8

Fig. 10. Simulated time averaged electromagnetic forces along the height of ingot, frequency is 50 Hz. Fig. 8. Effects of the frequency on vector summation of magnetic flux density at 0.2 m below interface of slag/ingot.

Fig. 11. Simulated time averaged Joule heating in the slag of ESR system, frequency is 50 Hz.

Fig. 9. Simulated time averaged electromagnetic forces on main section in ESR system, maximum is 1 356 N/m3, frequency 4.4. Joule Heating in the Slag is 50 Hz. One of the roles of the slag in ESR is to produce the nec- essary Joule heating to melt electrode. The Joule heating per unit volume is determined from the current density and is vector summation, i.e. B=++ BBB222 , thus it is root- zrθ inversely proportional to the electrical conductivity of slag, mean-value. Bz, and Br is approximately to be zero in an axi- hence the ESR slag with lower electrical conductivity would symmetrical magnetic field, Bθ values depend on the local cur- provide higher Joule heating to melt the electrode. Figure rent densities inside electrode, so that B has the smallest value 11 exhibits the predicted distribution of time-average Joule in minimum J position and center line of electrode. heating, it is observed that maximum is distributed at interface of electrode and slag, and minimum is in corner of slag/ 4.3. Electromagnetic Force mold. This is because the small current density appears in Figure 9 shows the time-average electromagnetic force that zone. The vertical distance between the tip of electrode distribution at main section of ESR system. It is seen the and middle horizontal section of slag shortened when the magnitude of electromagnetic force depends on the current electrode immersion depth increased. As the maximum cur- density and direction of electromagnetic force at all position rent density occurred at the tip of electrode where can be is inward. Maximum value of forces at slag layer is located observed in Fig. 11, the maximum time-average Joule heat- at interface of electrode/slag surface. Electromagnetic force ing along the diameter in middle horizontal section of slag in ingot decreases evidently due to falling of current density. increased. Effect of electrode immersion depth on the time- Distribution of electromagnetic force in ingot is shown in average Joule heating is shown in Fig. 12, when the applied Fig. 10, maximum is located at surface of ingot. current is 10 kA, maximum of Joule heating is 23.55 MW/m3

Fig. 12. Effects of electrode immersion depth on time averaged Fig. 14. Contours of voltage drop in the slag of ESR system, fre- Joule heating in slag, slag cap thickness is 0.2 m, frequency is 50 Hz. quency is 50 Hz.

Fig. 15. Comparison of predicted voltage drop with measured at Fig. 13. Effects of slag cap thickness on time averaged Joule heat- the centerline of mold and 1/2 radius of the electrode in ing in slag, electrode immersion depth is 0.03 m, fre- slag, frequency is 50 Hz. quency is 50 Hz. while the electrode immersion depth is 0.05 m. Increment of 5. Conclusions maximum is 7.9 MW/m3 or 33.6% when the electrode immersion depth increases from 0 to 0.05 m. Figure 13 (1) A finite element model was successfully applied to illustrates effect of slag cap thickness on the Joule heating, simulate the electromagnetic field and Joule heating in the when slag cap thickness is 0.15 m, the maximum Joule heat- ESR system. Some prediction results of important model ing is 31.83 MW/m3. The maximum Joule heating reduces have been detailed, such as the current density distribution, 16.93 MW/ m3 or 53.2% while slag cap thickness increases magnetic flux density, Joule heating and voltage drop in the from 0.15 m to 0.23 m. slag. Predicted results were compared to the experimental measurement, i.e., magnetic flux density and voltage drop in 4.5. Voltage Drop Across the Slag slag cap. Computed results are in very good agreement with The applied current flow results in a voltage drop across the measurements. the slag cap. Figure 14 shows the predicted voltage distri- (2) Effects of frequency on the current density in elec- bution at main section of the ESR system, there is a maxi- trode and ingot, respectively, are examined. The skin effect mum at the electrode-slag interface, which is accord with is remarkable for the alternating current in electrode. The the Joule heating. Kawakami et al.8) had ever measured the current density at surface of electrode and ingot is maximum voltage drop in slag cap in a small ESR system. In order to and gradually decreases from surface to center of electrode. verify the correctness of predicted voltage, comparison of Maximum of current density in slag cap is at interface of predicted voltage drop using the present model with mea- electrode/slag, it is benefic for the occurrence of Joule heat- sured by Kawakami et al.8) applying the same physical prop- ing. When the current enters the ingot, the skin effect takes erties, configuration and operating condition is shown in place owing to the high electric conductivity of ingot. With Fig. 15, a good agreement of both of them is obtained. the increasing of initial phase angle from 0° to 90°, the eddy current becomes stronger and stronger in the ingot. More-

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