A three-phase comprehensive mathematical model of desulfurization in electroslag remelting process

Applied Thermal Engineering 114 (2017) 874–886

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

A three-phase comprehensive mathematical model of desulfurization in electroslag remelting process Qiang Wang a,b, Guangqiang Li a,b,⇑, Zhu He a,b, Baokuan Li c a

The State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan, Hubei 430081, China Key Laboratory for Ferrous Metallurgy and Resources Utilization of Ministry of Education, Wuhan University of Science and Technology, Wuhan, Hubei 430081, China c School of Metallurgy, Northeastern University, Shenyang, Liaoning 110819, China b

h i g h l i g h t s
First developed a three-phase coupled model of desulfurization in ESR process.
The MHD thermal flow in the reactor was clarified.
Distributions of sulfur concentration in the three phases were demonstrated.
An experiment was carried out to validate the simulation.

a r t i c l e

i n f o

Article history: Received 28 June 2016 Revised 6 December 2016 Accepted 8 December 2016 Available online 18 December 2016 Keywords: Electroslag remelting Desulfurization Heat transfer MHD flow Numerical simulation

a b s t r a c t A three-phase comprehensive mathematical model has been established to study the desulfurization behavior in electroslag remelting (ESR) process. The solutions of the mass, momentum, energy, and species conservation equations were simultaneously calculated by the finite volume method. The Joule heating and Lorentz force were fully coupled through solving the Maxwell’s equations with the assistance of the magnetic potential vector. The movements of the air-slag and slag-metal interfaces were described by the volume of fluid (VOF) approach. In order to include the influences of the air, the slag and the electric current on the desulfurization, a thermodynamic and kinetic module was introduced. An experiment was conducted to validate the model. The completely comparison between the measured and simulated data indicates that the model can describe the desulfurization behavior in the ESR process with an acceptable accuracy. The sulfur in the metal would be transferred into the slag under the combined effect of the slag treatment and the electrochemical reaction, and is primarily achieved in the period of the droplet formation. The sulfur in the slag then could be transferred into the air because of the oxidation. The maximum calculated removal ratio in the whole process is around 88%. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Electroslag remelting (ESR) processes have been employed to produce the ultralow sulfur nickel alloy for use in ocean, aeronautics, and pipeline [1]. Fig. 1 shows a schematic of the ESR process. A direct current is passed from the consumable electrode to the baseplate in a water-cooled mold with an open air atmosphere, creating Joule heating in a highly resistive molten slag, whose large amount is enough to melt the electrode. A film of the molten metal is then created at the electrode tip, while a droplet is gradually formed

⇑ Corresponding author at: The State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan, Hubei 430081, China. E-mail address: [email protected] (G. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.12.035 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

within the continuous melting process. The dense metal droplet sinks through the less dense molten slag, forming a liquid metal pool [2]. Due to chemical reaction occurred at slag-metal interface, sulfur dissolved in the metal would be transferred into the slag. Part of sulfur in the slag then would be oxidized by the air at airslag interface, and finally escape into the atmosphere [3]. Kato et al. experimentally investigated the desulfurization rate in direct current ESR processes [3]. The efficiency ranged from approximately 30% to 85% under different conditions. The oxygen partial pressure in atmosphere was found to remarkably influence the sulfur transfer. Desulfurization would be promoted by a higher oxygen partial pressure. Minh and King used chronopotentiometry technique to study the contribution of electrochemical reaction to the desulfurization [4]. The sulfur transfer from metal to slag was shown to be controlled by the electrochemical and the direct

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Nomenclature A ! A a½Al aðAl2 O3 Þ a !½O B CS ca cm cs cp;a cp;m cp;s D E eij F! Fe ! Fs ! F st ! Ft f ½Al f ½S I! J K ksa kss ksm kT L LS MS

specific surface area for reaction (m1) magnetic potential vector (Vs/m) activity of aluminum in the metal activity of alumina in the slag activity of oxygen in the metal magnetic flux density (T) sulfide capacity of the slag mass percent of sulfur in air (%) mass percent of sulfur in metal (%) mass percent of sulfur in slag (%) specific heat of air at constant pressure (J/(kg K)) specific heat of metal at constant pressure (J/(kg K)) specific heat of slag at constant pressure (J/(kg K)) diffusion coefficient of sulfur (m2/s) internal energy of mixture phase (J/m3) interaction coefficient of the element j with respect to the element i Faraday law constant (C/mol) Lorentz force (N/m3) solute buoyancy force (N/m3) surface tension force (N/m3) thermal buoyancy force (N/m3) activity coefficient of the aluminum in the metal activity coefficient of the sulfur in the metal current (A) current density (A/m2) reaction equilibrium constant mass transfer coefficient of sulfur in the air (m/s) mass transfer coefficient of sulfur in the slag (m/s) mass transfer coefficient of sulfur in the metal (m/s) effective thermal conductivity (W/(m K)) latent heat of fusion (J/kg) sulfur partition ratio molar mass of the sulfur (kg/mol)

Fig. 1. Schematic of electroslag remelting process.

exchange reactions, while the sulfur reversion from slag to metal was a diffusion controlled, reversible electrochemical process. Unfortunately, the information obtained from experiments was limited due to the opaque reactor and the harsh environment.

_ m p pO2 QJ R Sr1 ; Sr2 t T

melt rate (kg/s) pressure (Pa) oxygen partial pressure (Pa) Joule heating (W/m3) gas constant (J/(mol K)) source term which represented the reaction rate time (s) temperature (K) ! v velocity (m/s) w½Al mass percent of aluminum in metal (%) wðAl2 O3 Þ; wðFeOÞ; wðSiO2 Þ mass percent of aluminum oxide, ferrous oxide and silicon dioxide in the slag (%) x; y; z Cartesian coordinates Greek symbols ai volume fraction l viscosity of mixture phase (Pa s) l0 permeability of vacuum (T m/A) q density of mixture phase (kg/m3) qa density of air (kg/m3) qm density of metal (kg/m3) qs density of slag (kg/m3) n coefficient in Eq. (32) which represents the power efficiency r electrical conductivity of mixture phase (X1 m1) u electrical potential (V)  / mixture phase property /a air property /m metal property slag property /s K optical basicity of the slag

Moreover, the electric current, velocity, and temperature fields were unknown, which seriously influence the sulfur transfer. However, given the difficulty of performing experiments on a real device, along with the continuously increase of computation resources, numerical simulation is an adequate way to provide a deeper insight into desulfurization in this process. Wang et al. established a transient three-dimensional (3D) coupled mathematical model to clarify the flow pattern and the temperature distribution in the ESR process [5,6]. The evolution of the Joule heating as well as the Lorentz force were also taken into consideration. A comprehensive model was developed by Yanke et al. through the application of a modified VOF method [7]. Using the modified VOF technique allowed the simulation of slag skin freezing to the mold, and this model included the effect of the slag skin thickness on melt rate and sump shapes during the remelting process. Fezi et al. numerically investigated the macrosegregation in alloy 625 produced by the ESR [8]. The calculated results indicated that the segregation tended to decrease with decreasing interdendritic liquid velocity and sump depth. Karimi-Sibaki et al. presented a numerical method to investigate the shape of electrode tip and the melt rate in ESR [9]. A dynamic mesh-based approach was invoked to model the dynamic formation of the shape of electrode tip. It was found that the simulated shape of the electrode tip closely matched with the observed shape. The desulfurization however was not taken into account in the above works. Josson et al. [10] proposed a new coupled computational fluid dynamic (CFD) and thermodynamic module to describe the sulfur transfer between the slag and metal in a gas-stirred ladle. A metallurgical

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thermodynamic module was used to represent the desulfurization mechanism. The CFD module was employed to solve relevant thermodynamic parameters affected by the flow pattern and temperature distribution, including sulfide capacity and oxygen activity. Sen et al. [11] established a two-dimensional model for electrochemical magnetohydrodynamics. In their study, the ButlerVolmer module for the kinetic of the heterogeneous electrode reactions was used to obtain the faradaic current density. Sen et al. also studied the interplay of Lorentz force, convection and redox species concentration distribution. Li et al. established a thermodynamic model to understand the oxidation desulfurization of copper slag [12]. The calculated results agreed well with the experimental data. The reaction rate was significantly affected by the diffusion of oxygen in the air, and the diffusion of ferrous sulfide and cuprous sulfide in the slag. As discussed above, desulfurization in the ESR process with an open air atmosphere remains unclear. Few attempts have been made to numerically investigate desulfurization coupled with electromagnetic, velocity, and temperature fields. For these reasons, the authors were motivated to establish a transient 3D comprehensive model to understand sulfur transfer between the metal, the slag and the air. A metallurgical thermodynamic and kinetic module was employed to describe the reaction rate, while the physical fields were simultaneously included. In addition, an experiment was implemented to properly validate the model.

2.3. Electromagnetism The electric current was obtained by solving the electrical potential u from the current continuity equation and the Ohm’s law [16]:

 ruÞ ¼ 0 r  ðr

ð3Þ

!

 ru J ¼ r

ð4Þ

At the same time, the magnetic potential vector was introduced to solve the magnetic field: !

!

B ¼rA

ð5Þ !

!

Ampere’s law r  B ¼ l0 J could therefore be rewritten as a Poisson equation: !

!

r2 A ¼ l0 J

ð6Þ

The magnetic field was obtained by combining Eqs. (5) and (6). Lorentz force and Joule heating were then expressed as: !

!

!

Fe ¼ J  B

ð7Þ

! !

QJ ¼

JJ r

ð8Þ

2. Mathematical model 2.4. Fluid flow and heat transfer 2.1. Assumptions In order to keep a reasonable computational time, the model relied on the following assumptions: (1) The domain included the metal, the slag and the air, while the electrode was disregarded [13]. (2) The three fluids were incompressible Newtonian fluid. The densities of the metal, the slag and the air were a function of the temperature, and the slag electrical conductivity depended on the temperature. Other metal, slag and air properties were assumed to be constant [14]. (3) The slag and the metal were assumed to be electrically insulated from the mold [14]. (4) Other elements in the slag and the metal were ignored except the sulfur. (5) Solidification was not taken into account.

2.2. VOF method For modeling three-phase flow, the VOF method was adopted to track three scalar fields in the whole domain [15]: ! @ ai þ r  ðv ai Þ ¼ 0 @t

ð1Þ

In the present work, the three scalars, a1, a2 and a3, stood for the volume fractions of the slag, the metal and the air, respectively, and they were updated at every time step. Meanwhile, the properties of the mixture phase such as electrical conductivity, density and viscosity were related to the volume fractions:

 ¼ / a1 þ / a2 þ / a3 / s m a

ð2Þ

Surface tension force between the air and slag as well as between the metal and the slag was described by the continuum surface force model.

The continuity and time-averaged Navier-Stokes equations were invoked to describe the turbulent movement in the furnace [5–9]:

 ! @q  vÞ ¼ 0 þ r  ðq @t

ð9Þ

!

!  vÞ ! ! ! ! @ðq  ðr v þrv T Þ þ F st  v  v Þ ¼ rp þ r  ½l þ r  ðq @t !

!

!

þ Fe þ Fs þ Ft !

ð10Þ

!

where F st and F e were the surface tension and the Lorentz forces !

!

mentioned above. F s and F t were the solutal and the thermal buoyancy forces determined by the Boussinesq approximation. The RNG k-e turbulence model, which is able to capture the behavior of flows with lower Reynolds numbers, was employed to calculate the turbulent viscosity. An enhanced wall function was used to work with the RNG k-e turbulence model. The energy conservation equation, shared between the three fluids, was [15]: ! @  EÞ þ r  ðv q  EÞ ¼ r  ðkT rTÞ þ Q J ðq @t

ð11Þ

where Q J was the Joule heating. E was the internal energy of the mixture phase, which was solved based on the three fluids’ specific heat and temperature:

E¼

a1 qs cp;s T þ a2 qm cp;m T þ a3 qa cp;a T a1 qs þ a2 qm þ a3 qa

ð12Þ

2.5. Mass transfer of sulfur As mentioned above, the transfer process of sulfur happened at the slag-metal and the air-slag interfaces because of the chemical reaction. The convection and diffusion of the sulfur in the three flu-

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ids were also related to the desulfurization. These phenomena were represented by [17]:

where i and j represented the dissolved elements in the metal. All the elements should be considered except the solvent. w½j repre-

 cs Þ ! @ðq  v cs Þ ¼ r  ða1 Drcs Þ þ Sr1  Sr2 þ r  ðq @t

ð13Þ

 cm Þ ! @ðq  v cm Þ ¼ r  ða2 Drcm Þ  Sr1 þ r  ðq @t

ð14Þ

 ca Þ ! @ðq  v ca Þ ¼ r  ða3 Drca Þ þ Sr2 þ r  ðq @t

sented the mass percent of element j in the metal. eij was the interaction coefficient of the element j with respect to the element i, and the interaction coefficients were taken from relevant Refs. [23–25]. The oxygen activity in the metal, a½O , was supposed to be influenced by the dissolved aluminum and oxygen in the bulk of the metal and the alumina in the slag according to the reaction [10]:

ð15Þ

2½Al þ 3½O ¼ ðAl2 O3 Þ

ð21Þ

The equation was established in the slag, in the metal and in the air, respectively. The three equations were simultaneously solved. The source term Sr1 indicated the mass transfer rate at the slagmetal interface, and the source term Sr2 represented the mass transfer rate at the air-slag interface.

The equilibrium constant for this reaction could be expressed [26]:

2.6. Thermodynamic and kinetic module

where a½Al was the aluminum activity in the metal, which was estimated by the aluminum activity coefficient:

lg K ¼ lg

aðAl2 O3 Þ 45300  11:62 ¼ T a2½Al  a3½O

An auxiliary metallurgical thermodynamic and kinetic module was developed to estimate the reaction rates. The sulfur transfer between the metal and the slag was first discussed:

a½Al ¼ f ½Al  w½Al

½S þ 2e ¼ ðS2 Þ

as:

ð16Þ

where [ ] and () stood for the matter in the metal and the slag. Thus [S] indicates the sulfur atom in the metal, and (S2) expresses the sulfur ion in the slag. Electrons participating in the reaction were supposed to be provided by the oxygen ion in the slag and the current. The source term Sr1 therefore could be divided into two parts. The first part was related to the slag properties, and the second part was determined by the current density. The first part of the source term Sr1 was written in terms of the two-film theory [18–20]:

S1 ¼


 kss ksm qs Ls cs A cm  ksm qm þ kss qs Ls Ls

ð17Þ

where kss and ksm were the mass transfer coefficient of sulfur in the slag and the metal, respectively. Ls was the sulfur partition ratio, which expresses the measure of the ratio of the sulfur concentration in the slag to that in the metal at the slag-metal interface. It represents the thermodynamics ability of desulfurization of the slag, and was calculated [21]:

lg LS ¼ 

935 þ 1:375 þ lg C S þ lg f ½S  lg a½O T

ð18Þ

where a½O and f ½S were the activity of oxygen and the activity coefficient of the sulfur in the metal, respectively. C S was the sulfide capacity of the slag, which was proposed by Fincham and Richardson, and was used to quantitatively evaluate the sulfur dissolution capacity of a slag [22]. Young’s model was employed here to calculate the sulfide capacity [20]:

lg C S ¼ 13:913 þ 42:84K  23:82K2  11:710  2:223 T 102 wðSiO2 Þ  2:275  102 wðAl2 O3 Þ

K < 0:8

K  2587 lg C S ¼ 0:6261 þ 0:4804K þ 0:7197K2 þ 1697 T T

þ5:144  104 wðFeOÞ

K P 0:8 ð19Þ

where K was the optical basicity of the slag. The optical basicity is a measure of the electron donor power of the oxides in the slag. According to the dilute solution model, the activity coefficient of the sulfur f ½S was a function of the metal composition [23]:

lg f ½i ¼

X

eij  w½j

ð20Þ

ð22Þ

ð23Þ

Substituting Eq. (23) into Eq. (22), a½O therefore was rewritten

lg a½O ¼

1 2 2 15100 lg aðAl2 O3 Þ  lg f ½Al  lg w½Al  þ 3:87 3 3 3 T

ð24Þ

The aluminum activity coefficient, f ½Al , was calculated using Eq. (20). aðAl2 O3 Þ was the alumina activity in the slag, which varies with the temperature. The slag temperature in the ESR process was mainly within the range of 1700–2100 K [5–8]. The thermodynamic commercial software FactSage was then adopted to estimate the evolution of the aðAl2 O3 Þ within this temperature range. A fitting function was proposed:

aðAl2 O3 Þ ¼ 0:3  3:99342e0:0018T

ð25Þ

Now let us considered the electrons provided by the current. The mass transfer rate of the sulfur was related to the intensity of the electron flow [27]: !

S2 ¼ 

jJj  MS nF

ð26Þ

!

where j J j was the magnitude of the local current density. n referred to the number of the electrons entering in the reaction (in this case n = 2). MS was the molar mass of the sulfur. Moreover, the rate would have a positive or a negative sign, which depends on the movement direction of the electron. When the electrons flow from the metal to the slag, sulfur in the metal would capture two electrons and then enter into the slag as sulfur ion. The sulfur content in the metal therefore reduces, and the rate is positive. But if the electrons migrate in an opposite direction, sulfur in the slag would go into the metal as sulfur atom but lose two electrons. As a result, the sulfur concentration in the metal rises, and the rate is negative. As mentioned above, sulfur in the slag would be oxidized by the oxygen in the air. Due to the high temperature, the oxidation reaction was extremely fast. The oxygen partial pressure, the driving force of the reaction, was found to be the rate-limiting step. The sulfur mass transfer rate at the air-slag interface was [12,28,29]:

Sr2 ¼

ksa M S p cs RT O2

ð27Þ

where ksa was the sulfur mass transfer coefficient in the air, and pO2 was the local oxygen partial pressure.

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2.7. Boundary conditions A zero potential was imposed at the bottom, while a potential gradient was applied at the inlet [16]:

@u ¼I @z

ð28Þ

The top surface, the electrode wall and the mold wall were assumed to be electrically insulated:

@u ¼0 @n

ð29Þ

The magnetic flux density was continuous at the top surface, the inlet and the bottom, and was negligibly small at the walls [2]:

Top surface; inlet and bottom : Ax ¼ Ay ¼

@Az ¼0 @z

ð30Þ

Walls : Ax ¼ Ay ¼ Az ¼ 0

ð31Þ

where Ax , Ay and Az was the magnetic potential vector along the x, y and z axis, respectively. At the inlet, a varied mass flow rate was adopted, which was determined by the Joule heating [30]:

_ ¼ m

nQ J L

ð32Þ

where n represented the power efficiency. Since power efficiency varies based on operation conditions, it was difficult to estimate its exact value. A reasonable power efficiency was obtained from literature, but was then adjusted according to the conditions of our experiment [5,6,30]. An outflow boundary condition was correspondingly used at the bottom. A no-slip condition was applied to both the electrode and mold walls. Zero shear stress wall was adopted on the top surface. A constant sulfur mass percent, which was equal to that in the electrode, was supplied at the inlet, and the sulfur was allowed to flow out from the bottom and the top surface. Additionally, a zero flux was used at the mold wall.

Table 1 Physical properties, geometrical and operating conditions. Parameter

Value

Physical properties of metal Reference density, kg/m3 Viscosity, Pa s Latent heat of fusion, kJ/kg Thermal conductivity, W/m K Specific heat, J/kg K Electrical conductivity, X1 m1 Magnetic permeability, H/m Thermal coefficient of cubical expansion, K1

7800 0.0061 270 30.52 752 7.14  105 1.257  106 2.1  104

Physical properties of slag Reference density, kg/m3 Viscosity, Pa s Thermal conductivity, W/m K Specific heat, J/kg K Electrical conductivity, X1 m1 Magnetic permeability, H/m Thermal coefficient of cubical expansion, K1 Optical basicity

2800 0.025 10.46 1255 ln r ¼ 6769=T þ 8:818 1.257  106 4.1  104 0.62

Physical properties of air Reference density, kg/m3 Viscosity, Pa s Thermal conductivity, W/m K Specific heat, J/kg K Thermal coefficient of cubical expansion, K1

1.29 1.8  105 0.036 1006.43 3.7  103

Geometry Electrode diameter, m Mold diameter, m Air layer height, m Slag pool height, m Metal pool height, m

0.055 0.12 0.03 0.06 0.03

Operating condition Current, A

1500

Air

Slag Metal Z

X Y

(a)

Table 2 Compositions of the electrode, %. C

Si

Mn

Cr

P

S

Ni

Mo

Fe

0.08

0.75

1.8

18

0.008

0.011

5.1

2.1

Bal.

Table 3 Compositions of the slag, %. CaF2

Al2O3

70

30

(b) Fig. 2. Mesh model and boundaries: (a) mesh model, (b) boundaries.

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To simplify the consideration of the melting process, the temperature of the metal at the inlet was given by a parabolic profile. This parabolic profile had an approximate 30 K superheat and a peripheral boundary temperature close to the metal liquidus temperature [31]. A fixed temperature was adopted at the electrode wall, which was measured in the experiment. Equivalent heat transfer coefficients were applied to the top surface and the mold wall [5–8]. The detailed physical properties, the geometrical and operating conditions were listed in Table 1. The chemical compositions of the electrode and the slag were displayed in Tables 2 and 3.

The commercial software ANSYS-FLUENT 12.1 was employed to run the simulation [32]. The governing equations for the electromagnetism, three-phase flow, heat transfer, and solute transport were integrated over each control volume and solved simultaneously, using an iterative procedure. The introduction of the magnetic potential vector, and the development of the thermodynamic and kinetic module were implemented by the userdefined functions. The widely used SIMPLE algorithm was

0.15

Joule heating (W/m3) 2.21E+08 1.61E+08 1.19E+08 1.00E+08 8.18E+07 6.81E+07 4.78E+07 2.81E+07

3. Solution procedure

0.12

Z (m)

0.09

0.06

0.03

0 -0.06

-0.03

0

X (m)

0.03

0.06

Fig. 3. Distributions of electric current streamlines and Joule heating at 900.0 s.

Volume fraction of slag

0.15

0.9 0.8 0.12

0.7 0.5 0.3 0.2

0.09

Z (m)

0.1

0.06

0.03

0 -0.06

-0.03

0

0.03

0.06

X (m) Fig. 4. Phase distribution (phase above the slag is air, and the phase below the slag is metal) and inward Lorentz force field at 900.0 s, and the maximum force is around 620 N/m3.

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employed for calculating the Navier-Stokes equations. All the equations were discretized by the second order upwind scheme for a higher accuracy. Before advancing, the iterative procedure continued until all normalized unscaled residuals were less than 106. The physical domain was discretized with a structured mesh. Mesh independence was thoroughly tested. Three families of meshes were generated with 209,000, 361,000, and 520,000 control volumes, respectively. After a typical simulation, we carefully compared velocity and temperature of some points in

Temperature (K)

the domain. The deviation of simulated results between the first and second mesh was about 7 pct, while approximately 3 pct between the second and third mesh. Furthermore, the value of y + within the first layer grid of the three meshes was equal to 1. Therefore, considering the expensive computation, the second mesh was retained for the rest of the present work. Fig. 2 shows the mesh model and boundaries. Due to the complexity of the coupling calculation, the time step was kept small to ensure that the above convergence criteria were fulfilled. Using 8 cores of

0.15

2026 2020 2015 0.12

2007 1986 1881 1600

0.09

1300

Z (m)

1434

0.06

0.03

0 -0.06

-0.03

0

0.03

0.06

X (m) Fig. 5. Flow streamlines and temperature distribution at 900.0 s, and the highest temperature is about 2030 K.

Fig. 6. Mass transfer rate of sulfur at the slag-metal interfaces at 900.0 s: (a) caused by the slag treatment, (b) caused by the electrochemical reaction.

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4.00 GHz, one typical case’s calculations took approximately 400 CPU hours.

5. Results and discussions 5.1. Electromagnetic phenomena

4. Experiment

Fig. 3 illustrates the electric current streamlines and Joule heating distribution with a current of 1500 A at 900 s. Once the current enters the slag, it spreads around and then flows downward. The metal greatly affects the current distribution due to its higher electrical conductivity. The current first travel through the metal, and then flow into the slag. As a result, a higher current density is observed in the metal droplet and also in the slag that around the droplet. According to Eq. (8), the Joule heating is determined by the current density and the electrical resistivity. The electrical resistivity of the slag is supposed to be 1000 times larger than that of the metal. Most Joule heating therefore is created by the slag, and a higher Joule heating is found in the slag that close to the

An experiment was carried out by using a mold with an open air atmosphere. The inner diameter, height, and mold wall thickness were 120 mm, 600 mm and 65 mm, respectively. The current used in the experiment was 1500 A. The consumable electrode was the stainless steel with a 55 mm diameter. The slag composition was calcium fluoride, 70 mass pct, and aluminum oxide, 30 mass pct. The slag cap thickness remained constant at 60 mm. The temperature of the electrode wall was measured by an infrared thermometer, and the temperature of the slag was measured by a disposable W3Re/W25Re thermocouple. The mass percent of sulfur in the slag was analyzed by a carbon sulfur analyzer.

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-0.03

0

X (m)

X (m)

X (m)

(b)

(d)

(f)

Fig. 7. Evolution of sulfur mass percent distribution in the metal with time: (a) t = 900.0 s, (b) t = 904.2 s, (c) t = 906.3 s, (d) t = 908.7 s, (e) t = 909.2 s, (f) t = 909.6 s.

Q. Wang et al. / Applied Thermal Engineering 114 (2017) 874–886

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Fig. 8. Evolution of sulfur mass percent distribution in the slag with time: (a) t = 900.0 s, (b) t = 904.2 s, (c) t = 906.3 s, (d) t = 908.7 s, (e) t = 909.2 s, (f) t = 909.6 s.

Fig. 9. Mass transfer rate of sulfur at the air-slag interface at 900.0 s.

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outer edge of the inlet and the droplet tip. Due to a sparser current streamline, the minimum Joule heating locates at the outer side of the top slag layer. The interaction between the self-induced magnetic field and the current generates a Lorentz force, as shown in Fig. 4. The radial Lorentz force points inward creating a pinch effect on the melt. 5.2. Flow and temperature fields Fig. 5 represents the flow pattern and the temperature distribution. Two pairs of vortexes are found in the slag. The heat extracted by the cooling water results in a descent of the slag in the vicinity of the mold wall. If the right side of Fig. 5 is observed, it is shown that this causes a stable clockwise circulation. Meanwhile, the inward Lorentz force along with the falling droplet create a counterclockwise cell at the center of the slag. Due to a larger Joule effect, the slag around the outer side of the inlet is much hotter, with a highest temperature of approximately 2030 K. The hotter

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slag, in turn, would heat the colder metal droplet during the falling process. Thus, the temperature of the slag in the middle is lower. On the other hand, the cold air, just above the air-slag interface, would be heated by the hotter slag. As a consequence, it would float up from the middle of the channel and take away lots of heat. The air then moves downward through both sides of the channel, because the heat in this area is reduced by the cooling water and the electrode, resulting in the sinking of the colder air. In the metal pool, the temperature difference drives the liquid metal along the slag-metal interface toward the outer edge and, from there, down along the mold wall. Lots of heat is therefore transferred to the cooling water. The temperature of the liquid metal at the middle is higher than that at the two sides.

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sulfur mass percent distribution in the air with time. Once the sulfur is shifted into the air, it would constantly move upward along with the air flow under the effect of the thermal buoyancy. And the sulfur content reduces with the increase of height. Fig. 11 demonstrates the comparison of the slag temperature between the simulation and the measurement. It is clear that the temperatures of the two points fluctuate over time, and the range is around 10 K. The temperature of point 1 is higher than that of point 2, because point 1 is closer to the region where the Joule heating is larger. A reasonable agreement between the experimental results and the simulated results is obtained. The discrepancies that exist can be explained by the uncertainty about the thermal boundary conditions as well as the material properties. Fig. 12 shows the evolution of the sulfur mass percent of the two points. The sulfur concentration has an increasing tend as expected, but the increase rate gradually reduces. It is mainly because the sulfur in the slag tend to be saturated, which weakens the desulfurization ability of the slag. Given the results obtained, the predicated sulfur content match with the measured data within an acceptable range of accuracy. It can be noted that the increasing of the measured sulfur content falls behind that of the simulated value at the later stage of the process, which indicates that oxidative desulfurization capacity of the air is underestimated in the present work.

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Temperature (K)

(a) displays the sulfur mass transfer rate induced by the slag treatment at the metal droplet-slag interface. The sulfur content in the droplet is high enough to generate a concentration driving force for the desulfurization. When the droplet grows, the sulfur in the droplet would be continuously transferred into the slag, and thus the desulfurization rate at the metal droplet-slag interface is positive. The sulfur concentration in the metal pool however is too low to push the sulfur transfer, and as a result the rate at the metal pool-slag interface is zero. Fig. 6(b) indicates the mass transfer rate induced by the electrochemical reaction. As stated above, the electrons flow from bottom to inlet in the furnace. Sulfur atoms in the metal pool typically become ions and go into the slag when the electrons travel through the metal pool-slag interface. Hence, the desulfurization rate at this interface is positive. Meanwhile, a negative desulfurization rate is found at the metal droplet-slag interface. Because the electrons would separate themselves from sulfur ions in the slag, and the sulfur would come back to the metal droplet in atom status, resulting in the increase of the sulfur concentration. Fig. 7 shows the distribution of the sulfur mass percent in the metal at different time instants. As mentioned above, the liquid film of the metal would be forced to gather together at the center of the inlet by the inward Lorentz force and gravity. The sulfur in the metal droplet is continuously transferred into the slag during this process. With the growing of the droplet, the sulfur content at the droplet tip obviously gets lower as shown in Fig. 7(c). We can see that the sulfur content in the droplet tip is almost the half of its original content. After a neck is formed, the velocity within the droplet increases due to the increased Lorentz pinch force at the neck. Subjected to the Lorentz force, surface tension force and gravity, the droplet is then detached from the inlet and falls into the metal pool. The droplet changes its shape in the falling process, and moreover the convection in the detached droplet would redistribute the sulfur. We can see that the sulfur concentration in the center of the droplet is higher than that in the outside layer. When the droplet hits the metal pool-slag interface, the sulfur quickly spreads out in the metal pool. Its distribution is then controlled by the flow pattern displayed in Fig. 7(f). The flow of the metal drags the sulfur moving to the outer side of the mold along the interface. Due to the influence of the cooling water, the sulfur then turns downward at the wall, and finally flows out from the outlet. From the time the droplet formation occurs, it takes about 8.7 s for the droplet to enter into the metal pool, and the formation of the droplet clearly occupies most of the time. The desulfurization is supposed to be achieved mostly in the period of the droplet formation. As mentioned above, the sulfur would move to the outer side of the mold along with the metal, and then flows out. It can be concluded that less time is left for the desulfurization in the metal pool. Fig. 8 illustrates the evolution of the sulfur mass percent distribution in the slag with time. Because the desulfurization is mostly accomplished during the droplet formation, the slag in the vicinity of the inlet therefore always has the highest sulfur concentration. The sulfur then gradually expand to the rest of the slag pool along with the fluid flow. Moreover, the sulfur concentration in the slag that around the metal droplet remarkably rises during the falling of the droplet. Sulfur then moves upward under the effect of the wake stream of the droplet. Fig. 9 shows the sulfur mass transfer rate at the air-slag interface. It is obvious that the transfer rate at the interface close to the electrode wall is much larger as a result of the higher sulfur concentration in the slag. And this transfer rate is approximately 100 times smaller than that at the slag-metal interface. It can be deduced that the slag plays a dominant role in the desulfurization, and the effect of the air is small. Fig. 10 displays the variation of the

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travel through the metal droplet-slag interface, the sulfur ion would reenter into the metal as sulfur atom but lose two electrons. Hence, the desulfurization rate at this interface is negative. Sulfur atoms in the metal pool typically become ions and go into the slag when the electrons go across the metal pool-slag interface. The desulfurization rate therefore is positive. The redistribution of the sulfur between the metal and the slag is primarily achieved in the period of the droplet formation. Due to the oxidation of the air, the sulfur in the slag then could be taken away by the air. The desulfurization rate is approximately 100 times smaller than that at the slag-metal interface. The slag plays a dominant role in the desulfurization, and the effect of the air is relatively small. The maximum calculated removal ratio in the whole process is around 88%.

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Time (s) Fig. 13. Comparison of the sulfur removal ratio between the simulation and the measurement.

Fig. 13 illustrates the compare between the simulated and measured sulfur removal ratio, which is defined as 1 minus the ratio of the total sulfur content in the refined metal to that in the original metal. The maximum calculated removal ratio during the whole process is about 88%. The calculated removal ratio first increases and then slightly decreases due to the saturated sulfur in the slag. Due to the underestimation of the oxidative desulfurization caused by the air as mentioned above, the measured removal ratio is larger than the calculated removal ratio at the later stage. The above completely comparison between the measured and simulated data indicates that the model can describe the desulfurization behavior in the ESR process with an acceptable accuracy. 6. Conclusions A three-phase comprehensive mathematical model has been established to study the desulfurization behavior in the ESR process. The solutions of the mass, momentum, energy, and species conservation equations were simultaneously calculated by the finite volume method. The Joule heating and Lorentz force were fully coupled through solving Maxwell’s equations with the assistance of the magnetic potential vector. The movements of the airslag and slag-metal interfaces were described by using the VOF approach. In order to include the influences of the air, the slag and the electric current on the desulfurization, a thermodynamic and kinetic module was introduced. An experiment has been carried out to verify the model. The predicated values of the temperature, the sulfur content in the slag, and the removal ratio have been found to agree reasonably with the corresponding measured data. The current at the center of the inlet tends to first move to the middle, and then flows downward traveling through the droplet. A higher Joule heating is created by the slag around the droplet. The inward Lorentz force generates a pinch effect on the droplet, and promotes the growing of the droplet. A counterclockwise cell is created by the Lorentz force and the falling droplet at the center of the slag. The slag in the vicinity of the mold lateral wall flows in a clockwise direction under the effect of the cooling water. The desulfurization mechanism in the ESR process includes the slag treatment and the electrochemical reaction. A positive desulfurization rate, caused by the slag treatment, is always observed at the metal droplet-slag as well as the metal pool-slag interfaces. The desulfurization rate, induced by the electrochemical reaction, however is opposite at both metal-slag interfaces. When the electrons

Acknowledgements The authors’ gratitude goes to National Natural Science Foundation of China (Grant No. 51210007), the Key Program of Joint Funds of the National Natural Science Foundation of China and the Government of Liaoning Province (Grant No. U1508214), and China Postdoctoral Science Foundation Funded Project (Grant No. 2016M600620).

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