Effect of Foamy Slag Height on Hot Spots Formation inside the Electric Arc Furnace Based on a Radiation Model

Transcription

1 , pp Effect of Foamy Slag Height on Hot Spots Formation inside the Electric Arc Furnace Based on a Radiation Model J. L. G. SANCHEZ, 1) A. N. CONEJO 2) and M. A. RAMIREZ-ARGAEZ 3) 1) Formerly Graduate Student at Morelia Technological Institute (ITM), Morelia, México. 2) Morelia Technological Institute (ITM), Morelia, México. anconejo@gmail.com 3) Department of Metallurgy, Facultad de Química, Universidad Nacional Autónoma de México (UNAM), México City. marco.ramirez@unam.mx (Received on September 14, 2011; accepted on November 18, 2011) Recent Electric Arc Furnaces are equipped with ultra high power transformers to provide maximum values of electric power and minimize the melting time. The active power is increased by increasing arc length and arc voltage, however in these conditions energy losses due to radiation can also be increased with a consequent decrease in thermal efficiency. The energy radiated from the electric arcs is transferred to the walls inside the EAF promoting hot spots which represent a catastrophic operational condition. This wor reports a radiation model which describes the formation of hot spots as a function of arc length and foamy slag height in an industrial EAF of 210 ton. of nominal capacity. Temperature profiles on the surface of the water cooled panels and values for the incident radiation were computed as a function of foamy slag height, used subsequently to define conditions to eliminate the formation of hot spots. KEY WORDS: hot spots; EAF; water cooled panels; radiation model. 1. Introduction The Electric Arc Furnace (EAF) has evolved into a melting reactor. The faster it melts results in higher productivity. In order to increase the melting rate it is mandatory to increase the arc length. Previously, this practice was prohibited because of the damage of the refractory walls and a restriction to sustain high temperatures, therefore a short arc operation was the common practice. With the development of water cooled panels (WCP) it was possible to sustain higher temperatures and wor with longer arcs at the expense of a lower thermal efficiency of the EAF. In the late 1980 s the promotion of foamy slags allowed to increase both arc length and thermal efficiency. Hot spots formation is due to the concentration of incident radiation in several parts of the furnace, typically in some regions of the shell. Hot spots are potentially dangerous because its formation leads to sudden perforations of the WCP which may create catastrophic situations. Once a WCP is perforated, the leaage of water above the molten bath provides conditions for an explosion. New furnaces are designed with temperature sensors in each WCP, however it is common to observe in steel plants that a wrong maintenance practice leaves this system out of operation. More commonly, the furnace operator relies on its own experience to detect the formation of hot spots. A critical assessment of this phenomenon has not been reported before. Jones et al. 1) summarized the influence of the main process parameters on panel life: (i) The optimum pipe thicness results from a trade off between mechanical strength to withstand high mechanical loads, which require a large thicness and that for maximization of heat transfer to the cooling water, which requires a small thicness. A typical thicness ranges from 8 10 mm. (ii) Boiler tube grade or copper is used for WCP. Steel withstands thermal loads up to 7 million Jm 2 h 1 and copper up to 21 million Jm 2 h 1. Copper panels are only used in those areas subjected to excessive heat loads. (iii) Low water velocities could promote steam generation and then a sharp decrease in heat transfers occurs. Water velocity ranges from ms 1. Fins are welded to WCP to promote the formation of a slag layer of larger thicness. Due to the lower thermal conductivity of slags, about 1.2 Wm 1 K 1, in comparison with 15 Wm 1 K 1 for steel, it serves to reduce the thermal load. The thicness of this layer can vary from 2 5 cm. Reynolds 2) reported a radiation model developed for smelting electric arc furnaces. This model suggests that radiation from the molten bath surface is the principal source of energy loss and not the plasma arc column. Several model parameters were evaluated, for instance, it was found that as the surface emissivity of the refractory drops the surface temperatures in all regions also drop. Guo and Irons 3,4) have reported a radiation model applied to the EAF. In this wor they reported the amount of incident energy to different parts of the furnace, including WCP, slag line and liquid slag. Their wor was used to validate the present mathematical model. The objective of this wor is to develop a mathematical model capable to describe the effect of foamy slag height on hot spots formation which is quantified based on the magnitude of incident radiation and temperature profiles of the inner surfaces of the EAF ISIJ 804

2 2. Mathematical Modelling In order to define the amount of energy radiated from the electrodes and its subsequent distribution into the furnace it is required the development of two models, one model should be able to describe the total amount of energy radiated from the arcs and the second one is the distribution of this energy as a function of the height of the foamy slag. These models are described in this section Arc Model Currently, there are two different approaches to investigate alternate-current (AC) arcs: The Magneto-Fluidynamic Model (MFD) and the Channel Arc Model (CAM). The first model is more complete because it computes velocity, pressure and temperature profiles in the arc region, however if the main purpose is to compute arc temperature, arc radius and instantaneous power, the second model is better due to its simplicity. CAM has been improved by Larsen and Baen 5 10) and full details of this model are given elsewhere. 11) In the following paragraphs, a short description is presented. In a DC arc, voltage and current are constant, contrary to AC arcs. One of the basis of CAM is that AC arcs tend to achieve the steady state conditions of DC arcs, additionally CAM assumes the shape of the arc is a cylinder of uniform temperature with only three modes of energy dissipation: radiation, convection and electron flow. The equations employed to compute power delivery due to radiation, electron flow and convection are given below: P e P r = π R 2 lu 5 T = I Oan + B + U 2e... (1)... (2)... (3) Two parameters which are provided by CAM are the total amount of radiant energy and arc radius. For a standard case using the maximum tap at volts and a large arc of 45 cm, the Channel Arc Model previously reported 11) predicts a total power delivery of MW, from this amount 30.3 MW correspond to energy transferred by radiation, which represents 25% of the total energy. Table 1 summarizes the energy distribution predicted by the Channel Arc Model. These values are used subsequently in the radiation model Radiation Model of Discrete Ordinates (DO) The Computational FluiDynamics (CFD) code FLUENT version was used to develop the radiation model. This code includes four radiation models. The selection of the Table 1. Power distribution with l = 45 cm and V = V. Mechanism Power, MW % Flow of electrons Radiation Convection Total an P = πr 2 ρ v ( h h ) conv DC F radiation model of discrete ordinates (DO) was made due to the following advantages: 12,13) it can analyze a broad range of optical thicnesses, emissivity and dispersion is taen into account, and more important is its capability to wor with localized sources of heating. The computational domain includes only the gas phase above the liquid bath and the inner walls of the furnace interior affected by radiation. The bath surface (slag surface) is the bottom boundary of the computational domain. This domain does not include the slag phase directly, although its height is involved in the model according with the strategy employed to describe the extent of radiation as a function of the slag height. All computations were carried out at steady state due to the following conditions: Heat flow due to radiation from the plasmas is constant for each case of slag height, constant heat flow extraction from all the surfaces affected by the radiation model and a flat slag surface. Assumptions: Radiation among gray surfaces. Air is the gas phase. No chemical reactions in the gaseous phase. The slag surface is flat. There is radiation from the walls, roof and slag surface. Steady state. Governing equations: Continuity equation: This equation satisfies mass conservation which for steady state and incompressible fluids can be written as follows: ( v)= 0... (4) Momentum transport: ( ρvv )= p + ( τ)+ ρg... (5) The term on the left side represents momentum transport due to convection. On the right side the first term represents pressure gradients, the second term represents viscous momentum transport and the third term represents gravity forces. Energy conservation equation: ( v( ρe+ ρ) )= ( eff T)+ Sh... (6) The term on the left side represents energy transport due to convection. On the right side, the first term represents energy transport due to conduction. The second term involves the generation of heat due to chemical reactions or any other thermal source. Heat transfer by radiation is included in this term. The radiation model of Discrete Ordinates (DO) requires the absorption coefficient (a) to be nown a priori and solves the radiative transfer equation over a domain of discrete solid angles. The Radiation Transport Equation (RTE) is given by the following equation: diin(,) r s + ( a+ σ s ) Iin ( r, s) ds 4 4π...(7a) 2 σt σ s = an + Iin ( rs, ) Φ( ss, ) dω π 4π ISIJ

3 The previous equation represents an energy balance on radiation intensity for a number of directions ( s ) and positions ( r ) in the angular space 4π, which results from its discretization in a number (N θ N Φ) of solid angles. The first term on the left side represent the change in radiation intensity (I in) with respect to position ( r ) in the direction ( s ), also called incident radiation or irradiation. The second term on the left side represents losses due to absorption and deviation due to the medium. The first term on the right side is the emission of the gas and the last term refers to the scattering addition term which adds radiation from other directions but deviated in the directions of the energy balances. Equation (7a) can be simplified if the medium considered is air because gases with symmetric structures lie oxygen and nitrogen are diathermal which implies that their emissive and absorption capacity is negligible. For air, the absorption coefficient, optical thicness and scattering coefficient are all zero and the refractive index is close to the unity. Based on this assumption, Eq. (7a) becomes: 4π di in (,) r s σ s = Iin(,) r s d... (7b) ds π Ω 4 0 In the present radiation model, all surfaces have been considered gray surfaces to differentiate from an opaque surface which do not allows radiation heat transfer through the walls. In addition to this it is also considered that all the reflected radiation is in a diffusive way. This means that a beam s direction once it leaves the surface is independent of its direction of incidence upon the surface. In this condition the diffusive fraction, f d, is unity. Radiosity represents the total radiation intensity leaving a surface. It is represented by two terms; the radiation emitted by the surface ( = n 2 εt 4 ) and the radiation reflected in a diffusive way = f. d( 1 ε ) Iin One feature of the DO model is that does not involve view factors. A view factor represents a geometric attenuation of radiative exchange occurring between a pair of surfaces due to their relative orientation and shape. Turbulence model -ε : The turbulent inetic energy () and its dissipation rate (ε ) are obtained from the following transport equations: ( ρ t )+ x = x j i ( ρv ) i μt + μ + G + Gb σ x j Y + S ρε M... (8) ( ρε )+ ( ρε vi) t xi = + t μ μ 2 ε ε + C1 ε ( G + C3εGb) C2ερ ε + Sε xj σ ε xj... (9) The turbulent viscosity, μ t, is computed combining and ε : μt ρc 2 = μ ε... (10) Thermal conductivities and emissivities employed in the model for different surfaces are shown in Table 2. Boundary conditions: The boundary conditions in this problem represent measured and computed values for the thermal state of those materials exposed to radiation from the electric arcs: water cooled panels (shell-balcony, roof and 4 th hole), refractories (slag line and delta), gas phase inside the furnace (air inlet from slag door and gas released from the bath) as well as the temperature of the electrodes. (i) Water cooled panels: In order to measure the amount of heat extracted from the water cooled panels it was necessary to measure the inlet and outlet temperatures of water cooling. Those measurements were not made individually but globally for the shell-balcony, roof and 4 th hole (gas outlet). The thermodynamic properties for water correspond to those for a saturated liquid instead of those for a compressed liquid due to the lac of information. The properties for a saturated liquid consider that liquids are virtually incompressible, however this is not the case for the enthalpy. The following equation has been suggested to include the effect of pressure on enthalpy. 16) H( PT, )= H + V ( P )...(11) The results are shown in Table 3. According with those results the heat flux extracted by the water cooled panels from the shell-balcony corresponds to 80 w/m 2, equal to the 4 th hole and that for the roof is 118 w/m 2. (ii) Refractories: The heat flux for the slag line and delta was computed applying the heat conduction equation for a composite wall. The thicnesses of the woring refractory, safety refractory and steel plate are 350 mm, 76 mm and 32 mm, respectively. It was estimated that the inner surface of the refractory is at K. With the properties reported in Table 4, the heat fluxes through the refractory of the slag line and delta are estimated approximately at 8.9 W/m 2, each one. (iii) Gas phase inside the furnace: The gas phase inside Table 2. Values employed in the model. Zone Material K, W/mK ε Slag door B Body Electrode Graphite WCP Steel Slag refractory MgO Table 3. ( T) f P l sat Measurements on water cooling for WCP. Measurement Shell-balcony 4 th hole Roof Inlet temperature, C Outlet temperature, C Inlet pressure, g/cm Outlet pressure, g/cm Surface, m Water flow, m 3 /hr Heat extracted, W/m ISIJ 806

4 Table 4. Properties of refractories. Thermal conductivity MgO (1 300 K), W/mK 4 Thermal conductivity of steel (350 K), W/mK Total thicness of refractory (MgO), m Steel thicness, m Heat flux for refractory of slag line, W/m Heat flux for refractory of delta, W/m the furnace is comprised of two components, one of them is the gas released from the molten bath and the second one is the air inlet from the slag door. The amount of gas released from the molten metal is mainly due to the decarburization reaction (CO) which is partially combusted to CO 2 in contact with air. The gas flow rate generated was estimated based on a previous thermodynamic model 14) which reported a value of g/s. In order to facilitate the analysis in the radiation model, the gas produced was assumed to be pure air at K. The amount of air entering from the slag door is initially unnown, however with the pressure differences (at the slag door, furnace interior and outlet pressure at the 4 th hole) it can be computed. The pressure inside the furnace and slag door is atmospheric ( Pa) and the outlet pressure at the 4 th hole, measured with a pitot tube was 6 mm H 2O equivalent to 58.8 Pa. The main contribution to radiation is due to the solid or liquid surfaces and the gas participates to a lower extent in the radiation exchange and therefore the real gas composition would not affect to a large extent the estimated values of both the incident radiation and temperature fields assuming pure air. (iv) Temperature at the surface of the electrodes: The temperature at the surface of the electrodes was taen from the model reported by Guo and Irons 3,4) for an electrode of 610 mm with an operating current of 61.5 A. The actual values of the furnace of the current model are 711 mm and 85 A, respectively. Figure 1 shows the temperature profile in the axial direction. Fig. 1. Fig. 2. Temperature profile of electrodes in the axial direction. Computational domain of the industrial EAF, including electrodes, arcs and 4 th hole. 3. Results and Analysis The computational domain corresponds to an industrial Electric Arc Furnace of 210 tons. of nominal capacity. A mesh sensibility analysis in the computational domain was carried out using a mesh range from thousand nodes. It was found that the results were ept constant with a mesh size of nodes. This was the mesh size employed in the computations. Figures 2 and 3 display the computational domain. Figure 2 displays the three plasmas. The rectangular shaded region corresponds to the slag door. The electrode closest to the slag door is named phase 1. Figure 3 includes the remaining surfaces involved in the radiation model; shell, balcony, roof, delta and 4 th hole. In order to evaluate the influence of foamy slag height on hot spots formation it was needed to define a criterion to represent slag depth in the radiation model. The Channel Arc Model (CAM) predicts an arc radius of 5.96 cm, then if an arc length of 45 cm is used it yields a surface area per plasma of m 2 and a total surface area of 0.5 m 2 for the three plasmas. Previously it was indicated that CAM Fig. 3. Boundary surfaces in the EAF. also predicts the fraction of energy radiated from the plasmas, corresponding to MW. This energy can be lost completely if the plasma is not covered, but on the other hand, it can be fully recovered to heat the liquid phases underneath the three plasmas if the slag height is 45 cm. Using this criterion, it was defined the fraction of energy radiated as a function of slag height, as reported in Table Model Validation It is highly complicated to tae actual measurements on incident radiation in a real process. This situation motivated to compare our model with a similar one developed by Guo and Irons. 3,4) These authors reported the energy absorbed by different parts of the furnace exposed to the arcs as a func ISIJ

5 tion of the foamy slag height. Figure 4 shows the results using the model developed in this wor for the conditions employed in the wor of Guo and Irons (P = 98 MW, l = 45.2 cm). The agreement with the results reported by those authors is quite good with only minor differences in spite of the large differences in radiated energy. Guo and Irons assumed 80% of energy radiated. In this wor the computed energy radiated was only 25%. This value resulted for the following conditions: arc length of 45 cm, tap voltage of V and 120 MW of active power. Both models predict a sharp decline in heat radiated to the slag when the slag height increases. The main difference is that in Guo and Irons model it is observed a large increase in heat absorbed by the side panels as the slag height decreases, which is correct, however in the current model this effect is less pronounced Hot Spots Formation and Incident Radiation (i) Shell: The formation of hot spots mainly occurs in the shell. This is because the shell is the solid surface closest to the electric arcs. The triangular design of the pitch circle creates a geometry which naturally defines the formation of Table 5. Radiation intensity as a function of slag height. Slag height, cm Arc covered, % Exposed surface, m 2 /arc Radiation, MW/arc Total radiation, MW Fig. 4. Model validation with another model reported by Guo and Irons. 3,4) Fig. 5. Incident radiation in W/m 2 in the shell as a function of slag height ISIJ 808

6 three hot spots, commonly nown as hot spot for phase 1, 2 or 3. Figure 5 describes the magnitude of the incident radiation as a function of slag height and Fig. 6 describes the surface temperatures in the shell also as a function of slag height. Maximum values in both radiation intensity and surface temperatures are obtained with the bare steel surface. In this condition their respective average values are 4.6 MW/m 2 and K (1 927 C). This temperature would immediately lead to perforation of the WCP. It is observed that the magnitude of the incident radiation decreases and becomes uniform all over the shell until the fraction of plasma coverage is increased above 75% (i.e cm). When this condition is reached the average incident radiation is decreased to approximately 2.0 MW/m 2. The presence of hot spots is clearly observed when that fraction is below 50%, which corresponds to approximately 3.0 MW/m 2 in incident radiation. The formation of hot spots is best visualized in terms of temperatures. Taing into account that boiler tube grade A steel is used to manufacture the WCP, the maximum temperature to avoid hot spots is in the order of K (1 530 C). This temperature serves as a reference to measure Fig. 6. Hot spots formation in the shell as a function of slag height. Temperature in K. Fig. 7. Hot spot formation as a function of slag height for phase 1. Temperature in K ISIJ

7 the formation of hot spots. It is quite clear, from Fig. 6 that hot spots are present when the slag height covers 50% of the arc length. The current model does not include the coating of slag layer on WCP, however, it is anticipated that this coating would help to reduce the effect of radiation on hot spots formation. Figure 7 describes hot spots formation for phase 1. Table 6 completes such description. In this table, it is shown the size of the hot spot, the incident radiation and hot spot temperature as a function of slag height. These results clearly indicate that in order to eep under control hot spot formation in a metallurgical practice with long arc operation (45 cm) the foamy slag must cover at least 75% of the height. If the slag height is equal or less than 50%, hot spot formation is unavoidable. In terms of incident radiation, it should be ept lower than 2.07 MW/m 2 in order to prevent hot spot formation. In Figs. 5 and 6 it is observed the formation of hot spots on the shell adjacent to the balcony. This result represents what could happen if the EAF were melting scrap, however, in the case of continuous charging of Direct Reduced Iron (DRI), the stream of DRI falling into the EAF creates a wall which receives the heat directed towards that zone maintaining low temperatures and removing the formation of those hot spots. Table 6. h s, cm area (m 2 ) Evolution of hot spots as a function of slag height. Approx. diameter (m) Average incident radiation MW/m 2 Maximum Temperature, C Table 7. Incident radiation and average temperatures for roof and delta as a function of slag height. Bare surface Foamy slag Incident radiation, MW/m Roof temperature, K Delta temperature, K Fig. 8. Temperature profiles for the roof, delta and 4 th hole sections as a function of foamy slag height. Temperature in K ISIJ 810

8 (i) Roof and delta: Table 7 summarizes the magnitude of the incident radiation as well as temperatures of the roof and delta sections of the furnace for two opposite conditions, a bare steel surface and fully covered arcs with foamy slag. From these results it is observed that foamy slag decreases the incident radiation from 2.6 to 1.6 MW/m 2 and also decreases the temperature of both delta and WCP. Figure 8 displays the temperature profiles of the roof, delta and 4 th hole. It is observed that for conditions of full arc coverage the average temperature of the water cooled panels is 1127 C and that for the delta section is C, on the opposite extreme, for arcs fully uncovered, temperature Fig. 9. Incident radiation in the Shell and roof as a function of the foamy slag height. Fig. 10. Averages temperatures of the shell and roof as a function of the foamy slag height. Fig. 11. Incident radiation in W/m 2 on the slag surface as a function of slag height ISIJ

9 increases to and C, respectively, which represents a temperature increment of 400 C and 160 C, for WCP and delta, respectively. Figures 9 and 10 define the relationship between foamy slag height and the magnitude of both average incident radiation and average temperature on the water cooled panels of the shell and roof. It is clearly observed that as the slag height increases, both incident radiation and temperatures on the inner surfaces decreases. (iii) Slag surface: The slag surface receives the largest amount of incident radiation because of its direct contact with the arcs. Table 8 shows the magnitude of the incident radiation. The magnitude of the maximum values of incident radiation are quite large, MW/m 2, however this occurs in the small regions in contact with the arcs and at a short distance abruptly falls. The magnitude of the maximum incident radiation remains almost constant, independently of slag height, because of the nature of the arc and the exponential decay in temperature at short distances from the center of the plasma. On the other hand, the average incident radiation linearly decreases with slag height. Figure 11 shows the incident radiation on the slag surface as a function of slag height. The average values decrease from 5.3 MW/m 2 Table 8. Maximum (I max) and average (I av) Incident radiation on the slag surface (MW/m 2 ). h s, cm I max I av Table 9. Energy losses through the WCP, h s = 0. WCP Energy losses (MW) % 4 th hole Balcony Shell Roof for the bare surface to 2.5 MW/m 2 for the arcs fully covered. The large difference between the maximum and average values is because the region with maximum radiation intensity occurs only in a small area around the electrodes which is small in comparison with the total slag surface area (the arc radius is approximately 6 cm). In Fig. 11 the scale of colors was limited to a maximum of 15 MW/m 2 for all cases in order to have a better visualization of the total slag surface area. For a slag height of 45 cm the plasmas are fully submerged and those regions do not receive radiation. As the slag height decreases the values in radiation intensity increases. A white color means that the actual values are above the current scale which explains the white color in the region of the electrodes. Using a larger scale, up to 47 MW/m 2, would affect a proper visualization of the slag surface. It is to be noted that the current model is unable to predict the temperature profiles on the slag surface because of an imposed boundary condition. Table 9 summarizes energy losses in the EAF through all the WCP for the extreme conditions of a bare surface. In total, 10% of the energy radiated by the arcs is lost by the WCP. The refractory from the slag line also receives radiant energy from the arcs, however due to their insulating properties the energy lost is negligible. Figure 12 summarizes the magnitude of the incident radiation on different parts of the EAF as a function of slag height. In spite that this plot underestimates the case for the bare surface (0 cm slag height) and reports low values of incident radiation, it is clearly observed that the slag surface receives the largest amount of incident radiation of all parts of the furnace. In second place is the refractory of the slag line. When the slag height increases, the incident radiation sharply decreases and for conditions of arcs fully covered, the magnitude of the incident radiation is almost similar for the slag surface and that for the WCP of the shell and refractory of the slag line. 4. Conclusion A radiation model capable to describe hot spots formation for an industrial Electric Arc Furnace with long arc operation of 45 cm and a maximum tap voltage of volts as a function of foamy slag height has been developed. This model can predict the magnitude of the incident radiation and temperature profiles of the inner surfaces of the EAF. The results of this model indicate the evolution of hot spots formation in terms of size and temperatures of the hot spots. It is evident that hot spots can be suppressed with a metallurgical practice oriented to sustain foamy slag during the melting process. The thermal efficiency of the electric arc furnace can be increased if the magnitude of incident radiation is decreased with a foamy slag which covers at least 75% of the arcs, equivalent to 2.07 MW/m 2 of incident radiation. If the conditions to suppress hot spots formation are taen into account during the operation of the EAF, up to 25% of the total energy delivered from the power system can be recovered and used for melting purposes. Fig. 12. Incident radiation at various surfaces as a function of slag height ISIJ 812

10 REFERENCES 1) J. A. T. Jones, B. Bowman and P. A. Lefran: The Maing, Shaping and Treating of Steel, Ch. 10, 11th ed, ed. by R. J. Fruehan, AISE Steel Foundation, Pittsburgh PA, (1998), ) Q. Reynolds: Min. Eng., 15 (2002), ) D. Guo and G. Irons: 3rd Int. Conf. on CFD in the Minerals and Process Industries. CSIRO, Melbourne Australia, (2003), ) D. Guo and G. Irons: AISTech Proc. I, AIST, Warrendale, PA, (2004), ) H. L. Larsen: PhD thesis, Norwegian University of Science and Technology, (1996). 6) H. L. Larsen and J. A. Baen: 4th European Conf. on Thermal Plasma Processes (TPP-4), Athens, Greece, (1996) ) H. L. Larsen, G. A. Sœvarsdottir and J. A. Baen: 54th Electric Furnace Conf. Proc., ISS-AISE, Warrendale, PA, (1996), ) G. A. Sœvarsdottir, H. L. Larsen and J. A. Baen: 5th European Conf. on Thermal Plasma Processes (TPP-5), Beijing, China, (1998) ) G. A. Sœvarsdottir, H. L. Larsen and J. A. Baen: J. High Temp. Material Processes, 3 (1999) 1. 10) G. A. Sœvarsdottir, J. A. Baen, V. G. Sevastyaneno and L. Gu: Electric Furnace Conf. Proc., (2000), ) J. L. G. Sanchez, M. A. Ramírez-Argaez and A. N. Conejo: Steel Res. Int., 80 (2009), No. 2, ) G. D. Raithby and E. H. Chui: J. Heat Trans., 112 (1990), ) E. H. Chui and G. D. Raithby: Numer. Heat Transfer. B, 23 (1993), ) J. G. G. Cárdenas: MSc Thesis, Morelia Technological Institute, Morelia México, (2008) (in Spanish). 15) F. P. Incropera and D. P. DeWitt: Heat Transfer fundamentals, 4th Ed, Ed. Pearson, México, (1999), 45, 852 (in Spanish). 16) Y. Cengel and M. Boles: Thermodynamics, An engineering approach, 4th ed., McGraw-Hill, New Yor, (2002), 84. List of symbols a Absorption coefficient, in m 1 C 1ε, C 2ε, C 3ε Constants (1.44, 1.92 and 0.09 respectively) e Electron charge, in C f d Diffusive fraction G Turbulent inetic energy due to velocity gradients G b Generation of turbulent inetic energy due to buoyancy forces H (T) liquid saturation enthalpy at a given temperature h arc specific enthalpy, in Jg 1 h F Surroundings specific enthalpy, in Jg 1 I arc current (in arc model), in amperes I in Change in Radiation intensity with respect to position (in DO model) Turbulent inetic energy B Boltzmann s constant, in JK 1 eff Effective thermal conductivity l arc length, in meters n Refractive index (N θ N Φ) Number of solid angles. O an wor function for the anode, in Volts P electric power, in watts P r Radiation power from arc, in watts P e Electronic power from arc, in watts P conv Convective power from arc, in watts P l Pressure for the saturated liquid P sat Saturation pressure at a given temperature R Arc radius (AC or DC), in meters. r Position vector s Scattering direction vector s Path length S, S h, S ε Source terms T Arc temperature, in K T Local temperature in K U an Anode drop voltage, in volts u Arc radiation density in Wm 3 V f Liquid saturation specific volume at a given temperature Y M Fluctuation on the ratio of compressible turbulence to the global dissipation rate C μ Constant in the turbulent model Time, in s Velocity vector, in ms 1 t v g Gravitational constant, 9.81 ms 1 E Internal energy of gas atmosphere, J g 1 v i Velocity Einstein notation, in ms 1 x i Space coordinate in Einstein notation, in m Gree symbols ε Emissivity ε Dissipation rate of turbulent inetic energy Ω Solid angle ρ Gas density of air, in gm 3 ρdc Gas density of DC arc, in gm 3 Φ Phase function σ Stefan-Boltzmann constant σ s Scattering coefficient in m 1 ν Average gas velocity, in ms 1 σ, σ ε Prandtl turbulent numbers for and ε; 1.0 and 1.3 respectively μ Viscosity, in Kg m 1 s 1 μ t Turbulent viscosity, in Kg m 1 s 1 τ Stress tensor, in N m ISIJ