Mathematical model of electromagnetic elimination in tubule with high frequency magnetic field

GUO Qing-tao(郭庆涛), JIN Jun-ze(金俊泽), LI Ting-ju(李廷举),

WANG Tong-min((王同敏), ZHANG Qi(张 琦), CAO Zhi-qiang((曹志强)

State Key Laboratory of Materials Modification by Laser, Ion and Electron Beams,
School of Materials Science and Engineering, Dalian University of Technology, Dalian 116024, China

Received 10 April 2006; accepted 25 April 2006

Abstract: In order to find the ways to improve the elimination efficiency with high frequency magnetic field, a mathematical model of electromagnetic elimination (EME) in the tubule with high frequency magnetic field was set up. The calculated results show that by ignoring the flow of molten metal, when the surface magnetic induction intensity of the metal (B₀) is 0.03 T and the diameter of the tubule is 8 mm, the non-metallic inclusions with 30 μm diameter can be wiped off in 7 s from the center of the molten aluminum, whereas the elimination time of the 5 μm non-metallic inclusions is more than 240 s. When B₀ is 0.03 T, the diameter of the tubule is 8 mm and elimination time is more than 30 s, the elimination efficiency of 5μm, 10 μm and 30 μm non-metallic inclusions is about 60%, 90% and 100%, respectively, the elimination efficiency increases with the decreasing diameter of the tubule. It can be concluded that increasing the magnetic induction intensity or decreasing the diameter of the tubule can decrease the elimination time and improve the elimination efficiency in EME with high frequency magnetic field.

Key words: electromagnetic elimination, non-metallic inclusions, elimination time, elimination efficiency

1 Introduction

Electromagnetic elimination(EME) is a well- established industrial method. A wide variety of specific technical application exists, ranging from separation of uranium isotope in the early nuclear era to separation of aluminum cans and other nonferrous metals from household or industrial waste [1]. In recent years, the methods of EME have been proposed and tested in purifying the molten metals, such as induction coil elimination [2-4], traveling magnetic field elimination [5-7] and high magnetic field elimination [8], etc. Because of its convenience, the method of high frequency field elimination has become one of the main research focuses. SHOJI et al [9] have calculated the electromagnetic force in liquid steel flowing in the circular pipe. However, the model can only calculate the force in about 1.5 δ(δ is the skin depth ).

In this paper, a mathematical model of EME in the tubule with high frequency magnetic field was set up. The electromagnetic body force (EMF) that distributes throughout the metal, elimination velocity and elimination time were deduced respectively. The relationships between EMF and B₀, EMF and (the distance from the metal surface), elimination time and the diameter of the non-metallic inclusions, elimination efficiency and the diameter of the tubule were also analyzed.

2 Theory calculation of mathematical model of EME

2.1 Principle

The electromagnetic body force can be generated by the interaction between alternating magnetic field and induced current in the molten metal. Because of the different electric resistance between the metal and the non-metallic inclusions, non-metallic inclusions will migrate opposite to the EMF and then can be wiped off. This is the theory of EME. Fig.1 shows the schematic diagram of EME with high frequency magnetic field.

Fig.1 Principle of electromagnetic purification with high frequency magnetic field: ● Non-metallic inclusions;

○ Metal

EMF points to the inner of the molten metal in the direction of .

2.2 Electromagnetic body force

The EMF perpendicular to the surface of strand and pointing to the inner of conductor is described as [10]:

         (1)

where is EMF in radial, N/m³;  is the magnetic permeability, H/m and  is the electric current density,  A/m². For electromagnetic solenoid, )， and  are taken into account, so following equation can be obtained,

     (2)

            (3)

Then Eqn.(4) can be obtained,

              (4)

where   is the electrical conductivity of molten metal and f  is the frequency of the electrical source.

2.3 Electromagnetic pinch force

Based on the theory of LEENOV et al [11], electromagnetic pinch force can be calculated with the following equation:

                     (5)

where  is the electromagnetic pinch force;  is electrical conductivity of the non-metallic inclusion and  is the diameter of the non-metallic inclusion. Comparing to , is so little that it can be omitted, Eqn.(6) can be obtained as follows,

                             (6)

It can be seen that the electromagnetic pinch force is in the opposite direction of EMF.

2.4 Viscosity resistance

Viscosity resistance can be calculated with Eqn.(7) [12],

                               (7)

where   is the damping coefficient;  is the area of the inclusions; ;  is the density of the inclusions;  is the relative velocity.

On the assumption that the flow of the molten metal is laminar flow, the inclusion Reynolds number is defined as:

                                (8)

                                  (9)

where  is viscosity of molten metal. Then  can be described as

                             (10)

2.5 Elimination velocity

When viscosity resistance equals to electromagnetic pinch force, the maximal velocity () of the inclusions in molten metal can be obtained: 

                             (11)

Substituting Eqn.(4) into Eqn.(11) and Eqn.(12) can be obtained as

    (12)

2.6 Elimination time

Fig.2 shows the schematic diagram of the movement of the non-metallic inclusion in the molten metal. EMF parallels to the direction of X and the flow of the molten metal is in the direction of Z.

When the inclusion begin to move, V₀ approximates to zero so that F_D can be ignored, then the acceleration of

Fig.2 Schematic diagram of movement of non-metallic inclusion in molten metal

the inclusion can be described as

                                 (13)

The locomotive time of the inclusion from X₀ to X₁ is:

          (14)      

If , then

                       (15)                 

                     (16)

In the calculation, we can’t get reasonable results in case of too large V_i it will lead to . Then V_i must be calculated once more, as follows:

                                 (17)

                              (18)

The total time is:

                                  (19)

3 Calculation results and discussion

Calculation parameters are listed in Table 1.

Fig.3 shows that EMF increases with the increase of B₀ and decreases with the increase of the distance from

Table 1 Calculation parameters

Fig.3 Effects of B₀ on distribution of EMF in molten aluminum when f=20 kHz

the molten aluminum surface when f is 20 kHz. Fig.4 shows the distribution of EMF in the molten metal with the varying of frequency when B₀=0.03 T. EMF of the metal surface is increased with the increase of the frequency while the skin depth is decreased.

Fig.4 Effects of frequency on distribution of EMF in molten aluminum when B₀=0.03 T

Because EMF increases with the increase of B₀, higher elimination efficiency can be got if B₀ is increased. However with the increase of B₀, J is increased, and then the temperature of the molten metal is increased with it. So B₀ can’t be increased too much in order to keep the temperature of the metal.

Fig.5 shows the elimination time in molten aluminum with high frequency magnetic field when diameter of the tubule is 8 mm.It can be seen from Fig.5 that elimination time of the inclusions increases with the increase of the distance from the melt surface and decreases with the increase of the diameter of the inclusions. The elimination time of the inclusions with 5 μm diameters near the center of the sample is more than 250 s, but 30 μm inclusions can be eliminated in 7 s when B₀ is 0.03 T and f is 20 kHz.

Fig.6 shows the elimination efficiency of the various dimension inclusions with high frequency magnetic field. The elimination efficiency increases with the increase of the diameter of the inclusions and the elimination time. When B₀ is 0.03 T, the diameter of the tubule is 8 mm and elimination time is more than 30 s, the elimination efficiency of 5 μm, 10 μm and 30 μm non-metallic inclusions is about 60%, 90% and 100%,  respectively.

Fig.5 Relationship between elimination time and position of  various dimension inclusions in liquid aluminum

Fig.6 Relationship between elimination efficiency and time of  various dimension inclusions in aluminum melt(B₀=0.03 T)

Fig.7 shows the relationship between elimination efficiency and the diameter of the tubule. It can be seen that elimination efficiency increases with the decrease of the diameter of the tubule. Elimination efficiency is about 100% with 10 μm non-metallic inclusions when the diameter of the tubule is 6 mm and elimination time is 20 s, while it decreases to 65% when the diameter of

Fig.7 Elimination efficiency in various diameter tubule with high frequency magnetic field when B₀=0.03 T and d_p=10 μm

the tubule is 10 mm.

TAKASHI et al [13,14] have discussed the effects of the turbulence in the EME. Thin tubule can restrain the turbulence and increase elimination efficiency in EME. With the decrease of the diameter of the tubule, the flux through the tubule can be decreased as well. Therefore a bundle of tubules should be applied in order to increase the flux in the EME.

4 Conclusions

1) Electromagnetic body force increases with the increase of the magnetic induction intensity; the force density of the molten metal surface is increased with the increase of the frequency while the skin depth is decreased.

2) The non-metallic inclusions with the diameter 30 μm can be wiped off in 7 s from the center of the molten aluminum while the elimination time of the 5 μm non-metallic inclusions is more than 240 s when B₀ is 0.03 T and the diameter of the tubule is 8 mm.

3) When B₀ is 0.03 T, the diameter of the tubule is 8 mm and elimination time is more than 30 s, the elimination efficiency of 5, 10 and 30 μm non-metallic inclusions is about 60%, 90% and 100%, respectively.

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(Edited by ZHAO Jun)

Foundation item: Projects (50474055; 50274018) supported by the National Natural Science Foundation of China; project (20052176) supported by the Natural Science Foundation of Liaoning, China

Corresponding author: GUO Qing-tao; Tel: +86-13604259195; E-mail: mystone_2002@tom.com