Effect of Electromagnetic Field on Three-Phase Flow Behavior

Aug 18, 2010 - Novel Method Based on Electric Field Simulation and Optimization for Designing an Energy-Saving Magnesium Electrolysis Cell. Ze Sun , Y...
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Ind. Eng. Chem. Res. 2010, 49, 10798–10803

Effect of Electromagnetic Field on Three-Phase Flow Behavior Ze Sun,† Ping Li,† Guimin Lu,‡ Bing Li,‡ Jin Wang,† and Jianguo Yu*,†,‡ State Key Lab of Chemical Engineering and National Engineering Research Center for Integrated Utilization of Salt Lake Resources, East China UniVersity of Science and Technology, Shanghai, China

The multiphase transport phenomena frequently take place in metallurgical processes, for example, in the electrolysis process of magnesium, where there exists three-phase flow including liquid magnesium, molten electrolyte, and chlorine gas under the electromagnetic field. In this paper, the three-phase flow behaviors under the electromagnetic field in the advanced diaphragmless electrolytic cell were investigated by CFD simulation. The governing equations of the internal flow field in the electrolytic cell were established, where the standard k-e turbulence model and the VOF multiphase flow model were adopted for the comprehensive description of the flow characteristics of multiphase flow in the electrolytic cell, and the Lorentz force was added to the momentum equation of fluids as the momentum source term to combine the effect of electromagnetic field with the flow field. The order coupling method was adopted for the calculation of the coupled field. The numerical simulation on the three-phase flow field considering the effect of electromagnetic field was done using FLUENT6.3 software, the numerical simulations on the electric field and the magnetic field were carried out using Ansys 11.0 software, respectively, and the connection between the finite element software ANSYS and the control volume software Fluent was built using the user-defined function (UDF). According to the analysis on the distributions of the electromagnetic field and the flow field, the optimum flow circulation in the advanced diaphragmless electrolytic cell was obtained, which is very helpful for the design in the electrolysis process of molten magnesium salt. 1. Introduction The multiphase transport phenomena frequently take place in metallurgical processes. For example, in the electrolysis process of the molten magnesium salt with the highest current above 400 kA, magnesium is produced from the cathode as liquid droplets, while the byproduct chlorine gas is produced from the anode as bubbles, both trying to rise the surface of the electrolyte as a result of the density difference with the electrolyte, so a three-phase flow under the strong electromagnetic field will be formed in the electrolytic cell.1-3 During the electrolysis process, the electrolyte circulation is very important; the better the electrolyte circulation is, the better is the current efficiency in the electrolysis magnesium industry. Many scientists and engineers have paid attention to the flow field in the electrolytic cell. Both the cold model experiments and the numerical simulation were frequently adopted to study the flow field in the electrolytic cell. Holliday4 used the similarity criteria to study the electrolysis magnesium process in low-density electrolytes. Burnakin5 used H2O and argon gas to describe the hydrodynamics of the two-phase flow, where the gas flow rate was controlled with the value of current density from 0.1-0.4 A/cm2 to simulate the working electrolytic cell. Brunakin also built equations for the two-phase flow characteristics for a magnesium electrolytic cell.6 Korobov7 and Shcherbinin8-10 built the mathematical model to simulate the temperature and electric fields in the magnesium electrolytic cell. Agalakov11 investigated the flow field of the multiphase flow in the electrolytic cell by the computational numerical simulation using the two-dimension mathematical model under the following assumptions and simplifications: (i) only focused on the flow phenomena; (ii) two-dimension model; (iii) steady * To whom correspondence should be addressed. E-mail: jgyu@ ecust.edu.cn. Fax: 86-21-64252826. † State Key Lab of Chemical Engineering. ‡ National Engineering Research Center for Integrated Utilization of Salt Lake Resources.

flow; (iv) bubbles with uniform radius; (v) no interaction between bubbles; (vi) flow field without the electromagnetic force. Two circulations of electrolyte were found in the electrolytic cell by the CFD simulation in Agalakov’s work, the main circulation was the circulation of the liquid magnesium transportation, the other one, a small circulation was formed in the electrolytic cell, which would be resulted in the damage of the back wall. The author’s previous work established the mathematical model of flow field and validated the math model using the particle image velocimetry (PIV) data.12 Go¨khan13 used a new style electrolytic cell to produce magnesium and studied the fluid characteristics in the cell. In this paper, the behaviors of three-phase flow under the electromagnetic field in magnesium electrolytic cell are investigated by the simulation of the computational fluid dynamics (CFD). A three-dimensional model is built to study the threephase flow behavior under the electromagnetic field using the commercial software of both ANSYS and Fluent, where the connection between the finite element software ANSYS and the control volume software Fluent was built by in-house code using the user defined function (UDF). 2. Structure of the Electrolytic Cell and Grids for CFD Simulation The advanced diaphragmless magnesium electrolytic cell has the general shape of a cube with the dimension of 2.91 × 1.87 × 1.40 m3, eight installed anodes with the dimension of 0.95 × 1.14 × 0.15 m3, nine cathodes with the dimension of 0.95 × 1.14 × 0.05 m3, and the distance of anode and cathode as 0.07 m. The structure of the electrolytic cell and grids for CFD simulation are shown in Figure 1 and Figure 2, respectively. For the calculation of the electromagnetic field, the finite element method is used, and only half of the electrolytic cell is used to solve this issue because of its symmetry. Therefore, the geometric model for the calculation shown in Figure 1a is

10.1021/ie100513w  2010 American Chemical Society Published on Web 08/18/2010

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Figure 1. Structure of the electrolytic cell (a) and grid division for the simulation of the electromagnetic field (b).

Lorentz force as a source term is added in the momentum equations of fluids to represent the effect of the electromagnetic field on the three-phase flow field. Maxwell equations and Ohm’s law are used to calculate Lorentz force of the electromagnetic field. The electrical conductivity γ and the magnetic permeability λ are constant, respectively. 3.1. Development of the Mathematical Models. In the VOF model,14-16 the velocities of phases are assumed to equilibrate over a small distance, especially in a small size computational cell, and essentially VMg ) VCl ) VSalt ) V. Then, the fields for all variables (pressure, velocity, etc.) are shared by the multiphases and correspond to the volume-averaged values. A single momentum equation is solved throughout the domain, and the resulting velocity field is shared among the phases. The momentum equation, shown below, is dependent on the volume fractions of all phases through the properties of density F and viscosity µ. Figure 2. Grids of the calculation domain for the simulation of the threephase flow field in the electrolytic cell.

the right part of the electrolytic cell, and its mesh division is shown in Figure 1b. In the real computational process, it is impossible to generate grids for infinity to air around the electric cell. Infin111 is employed to calculate the infinity field, and half air sphere with 3-5 times of the cell’s dimension in length were meshed for the electromagnetic field calculation. For the calculation of the three-phase flow field, Fluent software is adopted. The GAMBIT software release 2.2 is used to generate the structure and the unstructured meshes by hexahedral numerical grid. The grids in the electrolytic cell are shown in Figure 2; the total mash number is up to 48978. 3. Mathematical Models and Computational Procedure In the electrolysis process of the molten magnesium salt, magnesium is produced from a cathode as liquid droplets, while the byproduct chlorine gas is produced from an anode as bubbles, both trying to rise the surface of the electrolyte as a result of the density difference with the electrolyte, so threephase flow under the strong electromagnetic field will be formed in the advanced diaphragmless magnesium electrolytic cell. The following assumptions are presumed in the development of the mathematical models: The volume of fluid (VOF) model is used to capture the interface between multiphase fluids.14 The geometric reconstruction scheme is adopted to represent the interface between fluids using a piecewise-linear approach.15,16 The Reynolds-averaged Navier-Stokes (RANS) models are used to calculate the turbulent kinetic energy and the turbulent dissipation rate.17-19

∂ (F) + ∇ · (FV b) ) 0 ∂t ∂ (FV b) + ∇ · (FV bb V ) ) -∇P + ∇ · [µ(∇V b + ∇V bT)] + ∂t Fg b+F F

(1)

(2)

The density and viscosity used in continuity eq 1 and momentum eq 2 are calculated by F ) RMgFMg + RClFCl + (1 - RMg - RCl)FSalt

(3)

µ ) RMgµMg + RClµCl + (1 - RMg - RCl)µSalt

(4)

R(x,y,z,t) is the fluid fraction function, which has a value between unity and zero, representing the volume fraction of a cell occupied by a fluid: R(x,y,z,t) ) 0, the cell is empty; R(x,y,z,t) ) 1, the cell is full of a fluid; 0 < R(x,y,z,t) < 1, the cell contains the interface between multiphase fluids. In the model, RMg(x,y,z,t) is assigned for liquid magnesium, and RCl(x,y,z,t) is assigned for chlorine gas. To capture the interface between the phases, the time variation of the volume fraction function is calculated by ∂ (R ) + ∇ · (Rqb V q) ) 0 ∂t q

(5)

where the subscript q ) Mg, Cl, and the volume fraction of primary phase (molten salt) is limited by 3

∑R

q

)1

(6)

q)1

that is, the volume fraction of molten salt is 1 - RCl - RMg.

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The Reynolds-averaged Navier-Stokes (RANS) model is used to calculate the turbulent kinetic energy and turbulent dissipation rate; the k-ε model equations are

[( ) ] [( ) ]

µt ∂k ∂ ∂ ∂ (Fk) + (Fujk) ) + Gk - Fε ∂t ∂xj ∂xj σk0 ∂xj

(7)

µt ∂ε ∂ ∂ ∂ ε (Fε) + (Fu ε) ) + (Cε1Gk - Cε2Fε) ∂t ∂xj j ∂xj σε0 ∂xj k (8) where Gk ) µt(∂ui/∂xj)(∂uj/∂xi + ∂ui/∂xj); Cµ, Cε1, Cε2, σε0, and σk0 are constants, and Cµ ) 0.09, Cε1 ) 1.44, Cε2 ) 1.92, σε0 ) 1.3, and σk0 ) 1.0, respectively. In momentum eq 2, the force term b F is the Lorentz force of the electromagnetic field. The Lorentz forces are not acting on all phases, just on molten salt, for the molten salt with charges. The Lorentz force was zero in the phases of liquid magnesium and chlorine gas, since the Lorentz force equals the magnetic flux density multiplying the current density. But, the flow field changed when the Lorentz force acted on molten salt, for the electrolyte was reacted by the force first and then the interphase drag force changed. The Lorentz force is calculated based on Maxwell equations and Ohm’s law. The differential form of the Maxwell partial differential equations is given as follows: ∇×H)J+ ∇×E)-

∂D ∂t

∂B ∂t

(9) (10)

∇·D ) φ

(11)

∇·B ) 0

(12)

Where H is the magnetic intensity vector, J is the conduction current density, D is the electric displacement vector, E is the electric field intensity vector, B is the magnetic flux density vector, and φ is the free charge volume density. The electromagnetic constitutive equations of simple isotropic material mediums can be expressed as: D ) βE

(13)

B ) λH

(14)

J ) γE

(15)

where β ) βrβ0 is the absolute electric constant, λ ) λrλ0 is the absolute permeability, and γ is the electric conductivity. The Lorentz force is defined as F)J×B

(16)

3.2. Computational Procedure. For the calculation of the three phases flow field under the electromagnetic field, we adopt the calculation method of the coupled fields. First, the electric field is calculated and the distribution of the current density is obtained. Second, on the basis of the distribution of the current density, the magnetic field is calculated, and the Lorentz forces term is obtained. Third, the three phases flow field under the electromagnetic field is obtained by solving the continuity and momentum equations where Lorentz forces term as the source term is added to the momentum equation. The solution of the electromagnetic field is carried out using the magnetic vector potential (MVP) method in ANSYS 11.0 software. The finite elements-type SOLID 69 is used to get the

Table 1. Properties of Materials Used for the Calculation of the Electromagnetic Field properties

electrolyte

graghite

cast steel

magnetic permeability 1.0 1.0 1.0 electrical 4.0 × 10-3 8.5 × 10-6 1.3 × 10-7 conductivity (Ω · m)

refractory brick air 1.0 ∞

1.0 ∞

electric field, and SOLID 97 is used to get the electromagnetic field. The Lorentz force can be calculated from the results of current distribution and electromagnetic field. Then, the distribution of the Lorentz force is exported in a special document which is called by the in-house code of User Defined Function as the source term in the momentum equations. The flow field distribution is obtained using Fluent 6.3 software. To solve the governing partial differential equations, Navier-Stokes equations, for the conservation of mass, momentum, and scalars such as turbulence in integral form, a control-volume technique and SIMPLE procedure are used. The momentum source term, Lorentz forces, is read by the user defined function (UDF). All simulations are performed on a personal computer equipped with two processors, Intel core 6300 and 2 GB main memory running under the Windows operating system. In the VOF model, the geometric reconstruction scheme represents the interface between multiphase fluids using a piecewise-linear approach.15 The first step in this reconstruction scheme, is to calculate the position of the linear interface relative to the center of each partially filled cell, based on information about the volume fraction and its derivatives in the cell. The second step is to calculate the advection amount of fluid through each face using the computed linear interface representation and information about the normal and tangential velocity distribution on the face. The third step is to calculate the volume fraction in each cell using the balance of fluxes calculated during the previous step. 4. Results and Discussion Distribution of the Electromagnetic Field. The distribution of the electromagnetic field in the magnesium electrolytic cell was calculated using ANSYS 11.0 software. The properties of materials used for the simulation are listed in Table 1, where the materials include graphite anode, cast steel cathode, refractory brick, air, and electrolyte with the composition of NaCl 35%, KCl 25%, CaCl2 20%, and MgCl2 20%. The magnesium electrolytic cell works at 700 °C with 120 KA current density and 4.95 V voltages. Figure 3a shows the contour of the electric field in the electrolytic cell. The electric field energy is concentrated in the electrolysis compartment, especially in the space between anodes and cathodes. From anode to the top of cathode, the value of voltage decreases from maximum to zero. So, in the collection compartment of magnesium electrolytic cell, magnesium is not affected by the electric field because of magnesium without charge. This favors the collection of magnesium in the collection compartment. Figure 3b shows the contour of voltage on the vertical plane paralleling to the work surfaces of electrodes. In the picture, the maximum value of voltage is a constant on the surface of cathode. The value of voltage declines rapidly when away from the surface of cathode. The simulation results in Figure 4 show that the distributions of both the magnetic field and the Lorentz force are vertically symmetrical along the electrolysis compartment, decreasing gradually from the end toward the middle, and the maximum Lorentz force reaching 0.0135 N at the top corner of electrolysis compartment. Figure 5 shows the vectors of the Lorentz forces

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Figure 3. Contours of the electric field in the electrolytic cell.

Figure 4. Contours of the magnetic field and the Lorentz forces in the electrolytic cell.

Figure 5. Vectors of the Lorentz forces in a vertical plane between the collection compartment and the electrolysis compartments in the electrolytic cell.

in a vertical plane between the collection compartment and the electrolysis compartment in the electrolytic cell. As shown in Figure 5, vectors of Lorentz forces pointed to the electrolysis compartment from the collection compartment. These directions of Lorentz forces will favor the circulation of electrolyte in the electrolytic cell. Table 2 lists the typical discrete values of Lorentz forces calculated by ANSYS 11.0 software, these values will be added to the fluid momentum equations (eq 2) as a source term, in order to couple the effect of the electromagnetic field on the three-phase flow behavior. This is done using an inhouse UDF code during simulations. Three-Phase Flow Field Behavior under the Electromagnetic Field. Table 3 lists the properties of fluids (molten salt electrolyte, chlorine gas, liquid magnesium at 700 °C) used in the simulation of the flow field in the electrolytic cell. Figure 6 shows the typical simulation results for the three-phase flow

field under the electromagnetic field. From Figure 6, it is found that there exist three kinds of the fluid circulations in the electrolytic cell, the fluid circulation A and D near the side walls of the electrolytic cell, the fluid circulation B in the middle of the electrolytic cell, and the fluid circulation C between the B and D circulations of fluid. According to the simulation results of the flow field, we can see that the whole circulation distributes symmetrically along the electrolytic cell. Electrolyte in a pair of the cathode and anode does parallel movement between the surfaces of the electrodes, such as the fluid circulation B. They are the best circulations in the electrolytic cell, since the circulation has the shortest distance from the electrolysis compartment to the collection compartment. Liquid magnesium can be delivered effectively to the collection compartment with the circulation B of electrolyte. Other circulation between the cathode and anode deflect to the side wall, instead of parallel to the surface of cathode and anode, such as the circulation C of electrolyte. They are worse than the circulation B in the process of magnesium production. The worse circulations also include the circulation A and D, since they are near the side wall of the electrolytic cell. Since the side wall has effects on circulations, the circulations A and D have to run a longer distance in the electrolytic cell. That will decrease the efficiency of current. Therefore, the structure optimization of the electrolytic cell should be designed to ensure that the electrolyte circulates parallel to the working surfaces of electrodes. In Figure 7, from the view of the side wall in the electrolytic cell, the main characteristics of the flow field obtained by our simulation results (a) is similar with that of Agalakov’s results (b). That confirms that our math model is correct. Two main circulations appear as shown in Figure 7, one is big and the

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Table 2. Distribution of the Lorentz Force in the Electrolytic Cell X (m)

Y (m) -1

-0.900 × 10 -0.400 × 10-1 0.100 × 10-1 ... -0.302 -0.302

Z (m)

Fx (N) -1

0.275 × 10 0.275 × 101 0.275 × 101 ... 0.284 × 101 0.289 × 101

0.934 × 10 0.935 × 10-1 0.936 × 10-1 ... 0.520 0.520

1

Table 3. Properties of Fluids Used in the Simulation of Three-Phase Flow Field properties

electrolyte

chlorine

magnesium

density (kg/cm3) viscosity (kg/(m · s))

1620 1.68 × 10-3

0.9 2.93 × 10-5

1500 1.04 × 10-3

other is small. During production, the big circulation can transport magnesium from the electrolysis compartment to the collection compartment. The small circulation is near the back wall. It will damage the back wall, and the phenomena have been tested by an existing industrial electrolytic cell in China.

Fy (N) -2

0.206 × 10 0.167 × 10-2 0.117 × 10-2 ... 0.262 × 10-5 0.263 × 10-5

Fz (N) -17

0.857 × 10 0.693 × 10-17 0.481 × 10-17 ... -0.234 × 10-5 -0.373 × 10-5

-0.124 × 10-2 -0.111 × 10-2 -0.815 × 10-3 ... 0.437 × 10-6 0.115 × 100.5

parallel to the working surfaces of electrodes, not parallel to the working surfaces of electrodes, and near the side wall of the electrolytic cell. The circulations parallel to the working surfaces of electrodes can transport liquid magnesium efficiently from the electrolysis compartment to the collocation compartment. It should be emphasized that the structure optimization of the electrolytic cell should be done to ensure that the electrolyte circulation is parallel to the working surfaces of electrodes and to increase the current efficiency in the electrolysis process of molten magnesium salt. Acknowledgment

5. Conclusions Three-phase flow field under the electromagnetic field in the electrolytic cell of molten salt for magnesium production can be predicted by CFD simulation using both ANSYS software and Fluent software connected by the in-house code. According to the simulation results, we find that (1) the electric field concentrates in the electrolysis compartment; the distributions of the magnetic field and Lorentz force are vertically symmetrical along the electrolysis compartment, decreasing gradually from the end toward the middle of the electrolytic cell. (2) Electrolyte circulations are divided into three different types:

The authors acknowledge the financial support provided by National Natural Science Foundation of China (No. 50874048). Nomenclature H ) magnetic intensity vector, A/m J ) conduction current density, A/m2 D ) electric displacement vector, N/v · m E ) electric field intensity vector, V B ) magnetic flux density vector, T mpq ) mass exchange from qth phase to pth phase mqp ) mass exchange from pth phase to qth phase SRq ) source term, kg F ) Lorentz force, N k ) turbulent kinetic energy, m2/s2 Greek Letters φ ) free charge volume density, C/m3 γ ) electric conductivity, S/m φ ) absolute electric constant, F/m β0 ) vacuum electric constant, F/m λ ) absolute permeability, N/A2 λ0 ) magnetic constant, N/A2 Rq ) volume fraction of qth phase in discrete volume Fq ) density of qth phase, kg/m3 ε ) turbulent dissipation rate m2/s3

Figure 6. Streamlines of the three phase flow field under the electromagnetic field in the electrolytic cell in the view of the X direction from the collection compartment.

Note Added after ASAP Publication: The version of this paper that was published online August 18, 2010 was missing citations for references 17-19. The revised version was published August 24, 2010. Literature Cited

Figure 7. Comparisons of the simulation results of the electrolyte circulation with Agalakov’s results.

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Ind. Eng. Chem. Res., Vol. 49, No. 21, 2010 (6) Burnakin, V. V.; Polyakov, P. V.; Shestakov, V. M.; Kolesnikov, V. A. Analytical calculation of characteristics of two-phase flows in magnesium electrolytic cells. TsVetn. Met. 1979, 8, 73. (7) Korobov, M. A. Mathematical model of a magnesium electrolytic cell. TsVetn. Met. 1983, 5, 53. (8) Scherbinin, S. A.; Yakovleva, G. A.; Fazylov, A. F. Mathematical model of thermal and electric fields in a magnesium electrolytic cell. TsVetn. Met. 1994, 4, 60. (9) Shcherbinin, S. A.; Yakovleva, G. A.; Kazylov, A. K. Numerical study of thermal and electric fields of a magnesium electrolytic cell. TsVetn. Met. 1994, 6, 68. (10) Shcherbinin, S. A.; Fazylov, A. R.; Yakovleva, G. A. Mathematical simulation of three-dimensional thermal and electric fields of magnesium electrolyzer. TsVetn. Met. 1997, 5, 79. (11) Agalakov, V. V.; Shcherbinin, S. A.; Yakovleva, G. A. Mathematical simulation of gas-hydrodynamic processes in a magnesium electrolyzer. TsVetn. Met. 1997, 7, 74. (12) Sun, Z.; Zhang, H. N.; Li, P.; Li, B.; Lu, G. M.; Yu, J. G. Modeling and simulation of the flow field in the electrolysis of magnesium. J. Miner., Met., Mater. Soc. 2009, 61, 29. (13) Go¨khan, D.; Karakaya, I˙. Electrolytic magnesium production and its hydrodynamics by using an Mg-Pb alloy cathode. J. Alloy. Compd. 2008, 465, 255.

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ReceiVed for reView March 7, 2010 ReVised manuscript receiVed July 24, 2010 Accepted July 30, 2010 IE100513W