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Mathematical modeling of electromagnetic separation of inclusions from magnesium melt in a rectangular channel
Materials Letters 61 (2007) 2045 – 2049 www.elsevier.com/locate/matlet
Mathematical modeling of electromagnetic separation of inclusions from magnesium melt in a rectangular channel M. Reza Afshar a,⁎, M. Reza Aboutalebi a , M. Isac b , R.I.L. Guthrie b a
b
Advanced Materials Research Centre (AMRC), Dept. of Materials and Metallurgical Engineering, Iran University of Science and Technology (IUST), Tehran 16846-13114, Iran McGill Metals Processing Centre (MMPC), McGill University, 3610 University Street, Montreal, Canada QC H3A 2B2 Received 26 May 2006; accepted 7 August 2006 Available online 24 August 2006
Abstract A mathematical model was developed to study the effect of electromagnetic forces on non-metallic inclusions removal from magnesium melts passing through a channel. The electromagnetic force exerted on the inclusion induced by a DC current field was calculated. In order to compute the velocity field within the channel, the Navier–Stokes equations were solved numerically. The trajectories of the inclusions were calculated using the equations of motion for inclusions. Parametric studies were carried out to evaluate the effect of various parameters on the inclusions removal efficiency. © 2006 Elsevier B.V. All rights reserved. Keywords: Magnesium; Computer simulation; Removal efficiency; Electromagnetic separation; Metals and alloys
1. Introduction Magnesium, the lightest engineering metal, has received much attention in recent years as a structural material for the aerospace, automotive and electronic industries because of its high specific strength as well as its good castability and machinability [1,2]. One of the most important parameters in controlling the properties of magnesium and its alloys for various applications is melt cleanliness with respect to inclusions. The presence of such inclusions will strongly influence the mechanical properties and corrosion resistance of structural parts [3,4]. Recently electromagnetic separation technique has been considered as a new method to produce metals free from inclusions [5]. Various methods have been proposed, based on different sources of electromagnetic force. These methods were reviewed by Makarov et al. [6]. Mathematical modeling has been widely used to study the electromagnetic separation of inclusions from liquid metals [7–9]. The removal of inclusions ⁎ Corresponding author. Tel.: +98 21 7391 2877; fax: +98 21 7724 0480. E-mail address: [email protected] (M.R. Afshar). 0167-577X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2006.08.012
from magnesium melts using an AC magnetic field was studied by Kulinsky and Agalakov in which the trajectory of inclusions under magnetic field was calculated [10]. In the present study, a three dimensional mathematical model was developed to simulate the flow of liquid magnesium in a rectangular channel, in the presence of an electromagnetic force field caused by the passage of a DC current through the channel. 2. Mathematical model The rectangular vertical channel adopted in this study is shown schematically in Fig. 1. A DC current of constant density, JY (A/m2) in the axial direction of the channel was assumed. A constant inlet velocity, Vin (m/s) for liquid magnesium flowing through the channel was considered. Geometric parameters and the properties of liquid magnesium are presented in Table 1. 2.1. Magnetic and force fields Fig. 1 illustrates the rectangular channel. Considering the channel to be equivalent to a long conductor, the induced
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M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049 Y
Y do (3) The electrical density, J and magnetic flux density B not change with axial distance, y. (4) The electromagnetic field is not affected by the fluid flow.
Based on the above assumptions, the following governing equations were derived from the conservation of mass and momentum for three dimensional steady state Newtonian incompressible flows. Continuity equation: Bðqui Þ ¼0 Bxi
ð3Þ
Momentum equation: Bðquj ui Þ B Bui BP ¼ ðuÞ þ Fei − Bxj Bxj Bxj Bxi Fig. 1. Schematic of computational domains.
magnetic flux density, BY (T) can be calculated according to the Biot–Savart law [11] as follows: Z a Z b Y Y Y l J r B¼ e dz Vdx V ð1Þ 2p 0 r2 0
ð4Þ
Y . Where FYe ¼ JY B The uniform profile for v velocity was assumed at the inlet while the other velocity components (u and w) were set to zero.
where μe is the magnetic permeability of free space, equal to 4π × 10− 7 H/m and Y r is the distance between two points. Leenov and Kolin [12] derived the force acting on a particle ðY F e;p Þ of different electrical conductivities (σp) to that of the conductive fluid (σf) with diameter of dp as: p rf −rp Y Y F e;p ¼ − ð2Þ d3 F 2 2rf −rp p e Where FYe is the electromagnetic force acting on a volume of Y conductive liquid metal (N/m3), which is equal to JY B . In these calculations, the magnetic Reynolds number (Rem = μσf Vina) is much smaller than unity, so the induced electric field is negligible [13]. 2.2. Fluid flow model In the model, the following assumptions were made: (1) The flow is laminar and developing steady state. (2) The thermo-physical properties of liquid magnesium are constant. Table 1 Thermo-physical properties of magnesium, electromagnetic conditions and geometrical parameters Variables
Value
Liquid temperature Density Viscosity Electrical conductivity Inlet velocity Channel height,h Channel dimensions
740 °C 1563 kg m− 3 1.02 × 10− 3 N m− 2 s 12.82 × 105 kg− 1 m− 3 q2 s 0.01–0.04 m/s 0.15 m a = b = 2–5 mm
Y
Fig. 2. Electromagnetic force, F e in the channel's cross section (3 × 3 mm) for a current density of 2000 kA/m2, (a) force acting on the liquid magnesium and (b) force acting on an inclusion.
M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049
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Fully developed conditions were adopted at the outlet. Based on the assumption of non-slip condition at the solid walls, velocities were set to zero at the walls.
et al. [16]. Eqs. (6)–(9) were solved numerically using a fourthorder Runge–Kutta method to calculate the trajectories of inclusion particles in the channel.
2.3. Numerical solution
3. Results and discussion
The governing equations (Eqs. (3) and (4)) associated with the boundary conditions were solved numerically using the controlvolume based finite difference method. The hybrid-scheme was used to discretize the convection terms. In order to couple the velocity field and the pressure in the momentum equations, the well known SIMPLE algorithm suggested by Patankar [14] was adopted. A uniform grid of 42 × 66 × 42 for x, y and z directions respectively was selected for all the numerical simulations reported here. The following convergence criterion was considered for all dependent variables in the present computational study:
Fig. 2 illustrates the electromagnetic force field generated in the cross section of the channel (x–z plane). This figure shows the uniform force density across the channel in which, the resultant force acts towards the center of the channel, reaching maxima at the corners and approaching zero at the center. Imposing the uniform force in a large volume of the melt is rather difficult and this non-uniformity made induced fluid motion. However, a uniform magnetic density can be achieved in a small channel as adopted in this study. In this case circulating flow cannot develop due to the large inertia of the fluid. It is noted that, the electromagnetic force on an inclusion within the melt acts in the opposite direction (Eq. (2)). Fig. 3a shows the trajectory of a magnesium oxide particle (ρ = 3600 kg/m3) with diameter of 50 μm entering the channel at different positions for a current density of 2000 kA/m2. As one can see from this figure, the particle closer to the wall is under the effect of larger electromagnetic forces and therefore collides with the wall near the top of the channel. For particles closer to the center, lower forces are experienced and the inclusion meets the wall further down the channel or even may exit from the bottom of the channel. It is seen in Fig. 3b that by increasing the current density the particle meets the wall at points closer to the top of the channel. The inclusion removal efficiency is influenced by various parameters. The removal efficiency was calculated based on the fraction of cross sectional area where entering inclusions reach the sidewalls. For this purpose, the boundary of the area is identified by the numerical solution of Eqs. (6)–(9).
Ru ¼
X
jau up −
X
anb unb −bjb1 10−3
ð5Þ
2.4. Trajectories of inclusion particles Computation of the flow field in the channel allows one to investigate the factors affecting the location of inclusion particles. The motion of a particle in the channel can be predicted on the basis of Newton's second law of motion. There, the forces acting on the particle include drag in steady translation, added mass, gravity, buoyancy and electromagnetic forces: ðqp þ CA qÞ
dup 3 CD qðup −uÞjUR j 3 ¼− − Fe;x 4 4 dp dt
ð6Þ
ðqp þ CA qÞ
dvp 3 CD qðvp −vÞjUR j ¼ ðqp −qÞg− 4 dp dt
ð7Þ
ðqp þ CA qÞ
dwp 3 CD qðwp −wÞjUR j 3 ¼− − Fe;z 4 4 dp dt
ð8Þ
Where UR ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðup −uÞ2 þ ðvp −vÞ2 þ ðwp −wÞ2 ;
dx up ¼ ; dt
dy dz vp ¼ ; wp ¼ dt dt
ð9Þ
In the above equations up, vp and wp represent the velocity components of the particle in the x, y and z directions, while u, ν and w are the corresponding liquid velocity components in the x, y and z directions respectively. UR is the resultant relative velocity between particle and liquid. CA is added mass coefficient, which depends on the particle velocity and acceleration. Its value was assumed to be 0.5 for this study [15]. The value of CD depends upon the particle Reynolds number (Rep = ρURDp /μ), Dp and μ are the particle diameter and viscosity respectively. In the present work, semi-empirical relations between CD and Rep were used to determine the standard drag coefficient of particle translation as given by Bird
Fig. 3. Trajectory of magnesium oxide inclusion with dp = 20 μm in the channel, Y (a) different entering position at constant current density of J ¼ 2500 A=m2 and (b) different current density, Vin = 0.02 m/s.
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M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049
inclusions subjected to electromagnetic force and thus, the rate of increase in the removal efficiency decreases. It is also seen from Fig. 4a that the removal efficiency for large particles is higher than that for small particles and most of the large particles (dp N 50 μm) could be removed efficiently. On the other hand, the small particles are difficult to separate effectively. However, the value of removal efficiency for small particles (dp b 50 μm) can be increased by decreasing the flow rate of liquid magnesium and the diameter of the channel or by increasing the length of the channel. Another important parameter affecting inclusion removal efficiency is channel size, which is given in Fig. 4b for magnesium oxide inclusion with dp = 25 μm. As can be seen, by decreasing the channel size, the removal efficiency increases. By decreasing the size of the channel, the distance between the particle entrance position and the wall decreases. The effect of velocity of liquid magnesium on inclusion removal efficiency is given in Fig. 4c for magnesium oxide inclusion with dp = 50 μm. As can be seen, by increasing the melt velocity the removal efficiency decreases. By increasing the inlet velocity of melt, the relative velocity between the particle and liquid magnesium increases and the time for the electromagnetic force affecting it decreases.
4. Conclusions A mathematical model has been developed to simulate the steady state fluid flow of liquid magnesium in a vertical rectangular channel. The electromagnetic force acting on inclusion particles in magnesium melts was calculated based on a DC current passing through the channel and the induced magnetic field. On the basis of the flow field and electromagnetic force fields, the trajectory of particles was also obtained by numerically solving the equation of motion, taking into account drag, added mass, buoyancy, gravity and electromagnetic forces. The results show that by increasing the current density the melt feels larger electromagnetic force and the particles are pushed towards the channel sidewalls more effectively. It is confirmed that the removal efficiency of inclusions depends effectively on current density, melt inlet velocity and channel size. The removal efficiency increases as current density increases and channel size decreases. The removal efficiency also depends on the inclusion particle size. The removal efficiency for small particles is lower than large particles. It can be increased by decreasing the flow rate of liquid magnesium and the size of the channel or by increasing the length of the channel. Fig. 4. Effect of: (a) inclusions size, (b) channel size and (c) inlet velocity, on the efficiency of removal of magnesium oxide.
References The effect of current density on the removal efficiency of the inclusion was studied for the magnesium oxide particle with different sizes at a constant inlet velocity (Vin = 0.02 m/s). As shown in the diagram in Fig. 4, an increase in current density results in an increase in the electromagnetic force acting on the particle. At lower current densities, only the particles entering the channel near the wall are removed. With increases in current density, the electromagnetic force acting on the particles increases, covering a larger area, causing, removal efficiency to increase rapidly. At higher current density the electromagnetic force in most of the cross section is high enough to push the particles towards the sidewalls leading to a lower number of
[1] D. Eliezer, E. Aghion, F.H. (Sam) Fores, Adv. Perform. Mater. 5 (1998) 201. [2] B.L. Mordike, T. Ebert, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 302 (2001) 37. [3] T.S. Shih, W.S. Liu, Y.J. Chen, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 305 (2002) 152. [4] J.D. Hanawalt, C.F. Nelson, J.A. Peloubet, Trans. AIME 147 (1942) 73. [5] N. El-Kaddah, Proceedings of Emerging Separation Technology for Metals II, TMS, Warrendale, 1996, p. 203. [6] S. Makarov, L. Ludwig, D. Apelian, IEEE Trans. Magn. 36 (2001) 2015. [7] S. Taniguchi, K. Brimacombe, ISIJ Int. 34 (1994) 722.
M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049 [8] S. Makarov, R. Ludwig, J. Resnick, D. Apelian, J. Nondestr. Eval. 18 (1999) 99. [9] R.I.L. Guthrie, M. Li, Metall. Mater. Trans., B, Proc. Metall. Mater. Proc. Sci. 32B (2001) 1067. [10] A.I. Kulinsky, V.V. Agalakov, Proc. Inter. Non-Ferrous Processing and Tech. Conf. ASM Inter, Metal Park (1997) 535. [11] M. Zahn, Electromagnetic Field Theory: a Problem Solving Approach, Wiley, New York, 1979, p. 322. [12] D. Leenov, A. Kolin, J. Chem. Phys. 22 (1954) 683.
2049
[13] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, New York, 2001, p. 117. [14] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, 1980, p. 126. [15] J. Szekely, Fluid Flow Phenomena in Metals Processing, Academic Press, New York, 1979, p. 256. [16] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, p. 193.
Mathematical modeling of electromagnetic separation of inclusions from magnesium melt in a rectangular channel M. Reza Afshar a,⁎, M. Reza Aboutalebi a , M. Isac b , R.I.L. Guthrie b a
b
Advanced Materials Research Centre (AMRC), Dept. of Materials and Metallurgical Engineering, Iran University of Science and Technology (IUST), Tehran 16846-13114, Iran McGill Metals Processing Centre (MMPC), McGill University, 3610 University Street, Montreal, Canada QC H3A 2B2 Received 26 May 2006; accepted 7 August 2006 Available online 24 August 2006
Abstract A mathematical model was developed to study the effect of electromagnetic forces on non-metallic inclusions removal from magnesium melts passing through a channel. The electromagnetic force exerted on the inclusion induced by a DC current field was calculated. In order to compute the velocity field within the channel, the Navier–Stokes equations were solved numerically. The trajectories of the inclusions were calculated using the equations of motion for inclusions. Parametric studies were carried out to evaluate the effect of various parameters on the inclusions removal efficiency. © 2006 Elsevier B.V. All rights reserved. Keywords: Magnesium; Computer simulation; Removal efficiency; Electromagnetic separation; Metals and alloys
1. Introduction Magnesium, the lightest engineering metal, has received much attention in recent years as a structural material for the aerospace, automotive and electronic industries because of its high specific strength as well as its good castability and machinability [1,2]. One of the most important parameters in controlling the properties of magnesium and its alloys for various applications is melt cleanliness with respect to inclusions. The presence of such inclusions will strongly influence the mechanical properties and corrosion resistance of structural parts [3,4]. Recently electromagnetic separation technique has been considered as a new method to produce metals free from inclusions [5]. Various methods have been proposed, based on different sources of electromagnetic force. These methods were reviewed by Makarov et al. [6]. Mathematical modeling has been widely used to study the electromagnetic separation of inclusions from liquid metals [7–9]. The removal of inclusions ⁎ Corresponding author. Tel.: +98 21 7391 2877; fax: +98 21 7724 0480. E-mail address: [email protected] (M.R. Afshar). 0167-577X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2006.08.012
from magnesium melts using an AC magnetic field was studied by Kulinsky and Agalakov in which the trajectory of inclusions under magnetic field was calculated [10]. In the present study, a three dimensional mathematical model was developed to simulate the flow of liquid magnesium in a rectangular channel, in the presence of an electromagnetic force field caused by the passage of a DC current through the channel. 2. Mathematical model The rectangular vertical channel adopted in this study is shown schematically in Fig. 1. A DC current of constant density, JY (A/m2) in the axial direction of the channel was assumed. A constant inlet velocity, Vin (m/s) for liquid magnesium flowing through the channel was considered. Geometric parameters and the properties of liquid magnesium are presented in Table 1. 2.1. Magnetic and force fields Fig. 1 illustrates the rectangular channel. Considering the channel to be equivalent to a long conductor, the induced
2046
M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049 Y
Y do (3) The electrical density, J and magnetic flux density B not change with axial distance, y. (4) The electromagnetic field is not affected by the fluid flow.
Based on the above assumptions, the following governing equations were derived from the conservation of mass and momentum for three dimensional steady state Newtonian incompressible flows. Continuity equation: Bðqui Þ ¼0 Bxi
ð3Þ
Momentum equation: Bðquj ui Þ B Bui BP ¼ ðuÞ þ Fei − Bxj Bxj Bxj Bxi Fig. 1. Schematic of computational domains.
magnetic flux density, BY (T) can be calculated according to the Biot–Savart law [11] as follows: Z a Z b Y Y Y l J r B¼ e dz Vdx V ð1Þ 2p 0 r2 0
ð4Þ
Y . Where FYe ¼ JY B The uniform profile for v velocity was assumed at the inlet while the other velocity components (u and w) were set to zero.
where μe is the magnetic permeability of free space, equal to 4π × 10− 7 H/m and Y r is the distance between two points. Leenov and Kolin [12] derived the force acting on a particle ðY F e;p Þ of different electrical conductivities (σp) to that of the conductive fluid (σf) with diameter of dp as: p rf −rp Y Y F e;p ¼ − ð2Þ d3 F 2 2rf −rp p e Where FYe is the electromagnetic force acting on a volume of Y conductive liquid metal (N/m3), which is equal to JY B . In these calculations, the magnetic Reynolds number (Rem = μσf Vina) is much smaller than unity, so the induced electric field is negligible [13]. 2.2. Fluid flow model In the model, the following assumptions were made: (1) The flow is laminar and developing steady state. (2) The thermo-physical properties of liquid magnesium are constant. Table 1 Thermo-physical properties of magnesium, electromagnetic conditions and geometrical parameters Variables
Value
Liquid temperature Density Viscosity Electrical conductivity Inlet velocity Channel height,h Channel dimensions
740 °C 1563 kg m− 3 1.02 × 10− 3 N m− 2 s 12.82 × 105 kg− 1 m− 3 q2 s 0.01–0.04 m/s 0.15 m a = b = 2–5 mm
Y
Fig. 2. Electromagnetic force, F e in the channel's cross section (3 × 3 mm) for a current density of 2000 kA/m2, (a) force acting on the liquid magnesium and (b) force acting on an inclusion.
M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049
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Fully developed conditions were adopted at the outlet. Based on the assumption of non-slip condition at the solid walls, velocities were set to zero at the walls.
et al. [16]. Eqs. (6)–(9) were solved numerically using a fourthorder Runge–Kutta method to calculate the trajectories of inclusion particles in the channel.
2.3. Numerical solution
3. Results and discussion
The governing equations (Eqs. (3) and (4)) associated with the boundary conditions were solved numerically using the controlvolume based finite difference method. The hybrid-scheme was used to discretize the convection terms. In order to couple the velocity field and the pressure in the momentum equations, the well known SIMPLE algorithm suggested by Patankar [14] was adopted. A uniform grid of 42 × 66 × 42 for x, y and z directions respectively was selected for all the numerical simulations reported here. The following convergence criterion was considered for all dependent variables in the present computational study:
Fig. 2 illustrates the electromagnetic force field generated in the cross section of the channel (x–z plane). This figure shows the uniform force density across the channel in which, the resultant force acts towards the center of the channel, reaching maxima at the corners and approaching zero at the center. Imposing the uniform force in a large volume of the melt is rather difficult and this non-uniformity made induced fluid motion. However, a uniform magnetic density can be achieved in a small channel as adopted in this study. In this case circulating flow cannot develop due to the large inertia of the fluid. It is noted that, the electromagnetic force on an inclusion within the melt acts in the opposite direction (Eq. (2)). Fig. 3a shows the trajectory of a magnesium oxide particle (ρ = 3600 kg/m3) with diameter of 50 μm entering the channel at different positions for a current density of 2000 kA/m2. As one can see from this figure, the particle closer to the wall is under the effect of larger electromagnetic forces and therefore collides with the wall near the top of the channel. For particles closer to the center, lower forces are experienced and the inclusion meets the wall further down the channel or even may exit from the bottom of the channel. It is seen in Fig. 3b that by increasing the current density the particle meets the wall at points closer to the top of the channel. The inclusion removal efficiency is influenced by various parameters. The removal efficiency was calculated based on the fraction of cross sectional area where entering inclusions reach the sidewalls. For this purpose, the boundary of the area is identified by the numerical solution of Eqs. (6)–(9).
Ru ¼
X
jau up −
X
anb unb −bjb1 10−3
ð5Þ
2.4. Trajectories of inclusion particles Computation of the flow field in the channel allows one to investigate the factors affecting the location of inclusion particles. The motion of a particle in the channel can be predicted on the basis of Newton's second law of motion. There, the forces acting on the particle include drag in steady translation, added mass, gravity, buoyancy and electromagnetic forces: ðqp þ CA qÞ
dup 3 CD qðup −uÞjUR j 3 ¼− − Fe;x 4 4 dp dt
ð6Þ
ðqp þ CA qÞ
dvp 3 CD qðvp −vÞjUR j ¼ ðqp −qÞg− 4 dp dt
ð7Þ
ðqp þ CA qÞ
dwp 3 CD qðwp −wÞjUR j 3 ¼− − Fe;z 4 4 dp dt
ð8Þ
Where UR ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðup −uÞ2 þ ðvp −vÞ2 þ ðwp −wÞ2 ;
dx up ¼ ; dt
dy dz vp ¼ ; wp ¼ dt dt
ð9Þ
In the above equations up, vp and wp represent the velocity components of the particle in the x, y and z directions, while u, ν and w are the corresponding liquid velocity components in the x, y and z directions respectively. UR is the resultant relative velocity between particle and liquid. CA is added mass coefficient, which depends on the particle velocity and acceleration. Its value was assumed to be 0.5 for this study [15]. The value of CD depends upon the particle Reynolds number (Rep = ρURDp /μ), Dp and μ are the particle diameter and viscosity respectively. In the present work, semi-empirical relations between CD and Rep were used to determine the standard drag coefficient of particle translation as given by Bird
Fig. 3. Trajectory of magnesium oxide inclusion with dp = 20 μm in the channel, Y (a) different entering position at constant current density of J ¼ 2500 A=m2 and (b) different current density, Vin = 0.02 m/s.
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M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049
inclusions subjected to electromagnetic force and thus, the rate of increase in the removal efficiency decreases. It is also seen from Fig. 4a that the removal efficiency for large particles is higher than that for small particles and most of the large particles (dp N 50 μm) could be removed efficiently. On the other hand, the small particles are difficult to separate effectively. However, the value of removal efficiency for small particles (dp b 50 μm) can be increased by decreasing the flow rate of liquid magnesium and the diameter of the channel or by increasing the length of the channel. Another important parameter affecting inclusion removal efficiency is channel size, which is given in Fig. 4b for magnesium oxide inclusion with dp = 25 μm. As can be seen, by decreasing the channel size, the removal efficiency increases. By decreasing the size of the channel, the distance between the particle entrance position and the wall decreases. The effect of velocity of liquid magnesium on inclusion removal efficiency is given in Fig. 4c for magnesium oxide inclusion with dp = 50 μm. As can be seen, by increasing the melt velocity the removal efficiency decreases. By increasing the inlet velocity of melt, the relative velocity between the particle and liquid magnesium increases and the time for the electromagnetic force affecting it decreases.
4. Conclusions A mathematical model has been developed to simulate the steady state fluid flow of liquid magnesium in a vertical rectangular channel. The electromagnetic force acting on inclusion particles in magnesium melts was calculated based on a DC current passing through the channel and the induced magnetic field. On the basis of the flow field and electromagnetic force fields, the trajectory of particles was also obtained by numerically solving the equation of motion, taking into account drag, added mass, buoyancy, gravity and electromagnetic forces. The results show that by increasing the current density the melt feels larger electromagnetic force and the particles are pushed towards the channel sidewalls more effectively. It is confirmed that the removal efficiency of inclusions depends effectively on current density, melt inlet velocity and channel size. The removal efficiency increases as current density increases and channel size decreases. The removal efficiency also depends on the inclusion particle size. The removal efficiency for small particles is lower than large particles. It can be increased by decreasing the flow rate of liquid magnesium and the size of the channel or by increasing the length of the channel. Fig. 4. Effect of: (a) inclusions size, (b) channel size and (c) inlet velocity, on the efficiency of removal of magnesium oxide.
References The effect of current density on the removal efficiency of the inclusion was studied for the magnesium oxide particle with different sizes at a constant inlet velocity (Vin = 0.02 m/s). As shown in the diagram in Fig. 4, an increase in current density results in an increase in the electromagnetic force acting on the particle. At lower current densities, only the particles entering the channel near the wall are removed. With increases in current density, the electromagnetic force acting on the particles increases, covering a larger area, causing, removal efficiency to increase rapidly. At higher current density the electromagnetic force in most of the cross section is high enough to push the particles towards the sidewalls leading to a lower number of
[1] D. Eliezer, E. Aghion, F.H. (Sam) Fores, Adv. Perform. Mater. 5 (1998) 201. [2] B.L. Mordike, T. Ebert, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 302 (2001) 37. [3] T.S. Shih, W.S. Liu, Y.J. Chen, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 305 (2002) 152. [4] J.D. Hanawalt, C.F. Nelson, J.A. Peloubet, Trans. AIME 147 (1942) 73. [5] N. El-Kaddah, Proceedings of Emerging Separation Technology for Metals II, TMS, Warrendale, 1996, p. 203. [6] S. Makarov, L. Ludwig, D. Apelian, IEEE Trans. Magn. 36 (2001) 2015. [7] S. Taniguchi, K. Brimacombe, ISIJ Int. 34 (1994) 722.
M.R. Afshar et al. / Materials Letters 61 (2007) 2045–2049 [8] S. Makarov, R. Ludwig, J. Resnick, D. Apelian, J. Nondestr. Eval. 18 (1999) 99. [9] R.I.L. Guthrie, M. Li, Metall. Mater. Trans., B, Proc. Metall. Mater. Proc. Sci. 32B (2001) 1067. [10] A.I. Kulinsky, V.V. Agalakov, Proc. Inter. Non-Ferrous Processing and Tech. Conf. ASM Inter, Metal Park (1997) 535. [11] M. Zahn, Electromagnetic Field Theory: a Problem Solving Approach, Wiley, New York, 1979, p. 322. [12] D. Leenov, A. Kolin, J. Chem. Phys. 22 (1954) 683.
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[13] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, New York, 2001, p. 117. [14] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, 1980, p. 126. [15] J. Szekely, Fluid Flow Phenomena in Metals Processing, Academic Press, New York, 1979, p. 256. [16] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, p. 193.