Analysis on the characteristics of pulsed electromagnetic force and the fluctuation behavior of molten metal free surface under pulsed magnetic field

A self-made line laser liquid level measurement system was used to measure the fluctuation of molten metal free surface under pulsed magnetic field. The electromagnetic characteristics of pulsed electromagnetic force were mathematically analyzed. Results showed that the electromagnetic force presents oscillatory attenuation during a single pulse period. The electromagnetic force was composed of turning force f t u r n and non-turning force f n o n t u r n . The direction of f t u r n was always consistent with the melt circumferential direction, which turned the f n o n t u r n consisted of electromagnetic pull and push forces, which caused the melt to oscillate. Under the pulsed magnetic field, the free surface formed a meniscus with a high middle, low side structure. With increased pulsed field intensity, the center surface of molten melt had an oscillation of ±3.52 mm at 0.187 T. The wave power density had only two spectral peaks (at 0.60 and 3.36 Hz) without a magnetic field. Under pulsed magnetic field, four spectral peaks were found at 0.40, 3.00, 6.50 and 13.00 Hz.


Introduction
Pulsed magnetic field applied to the solidification process of metals can significantly refine metal grains [1][2][3][4], which can improve the mechanical properties of metal and reduce macroscopic segregation and thermal cracking tendency. By imposing a magnetic field, the electromagnetic casting technology overcomes the contact pressure between the molten metal and the mold inner wall, which can control the initial solidification process and improve the casting quality [5].
Pulsed magnetic field technology can be mainly divided into high-and low-frequency pulse magnetic fields. When a low-frequency pulsed magnetic field is used, the melt is oscillated and stirred [6]. When the pulse discharge frequency coincides with a certain natural frequency of the solidification system, the resonance effect is induced and the effect on the pulse electromagnetic force is enhanced [7]. When a high frequency pulsed magnetic field is applied, the magnetic sound waves are produced in the melt [8]. Moreover, when the magnetic field is pulsing, the electromagnetic wave is either a sine or cosine [9]. However, the pulsed magnetic field presents attenuation with increasing time. Therefore, the electromagnetic signal must not be set as a sine or cosine.
Some free surface fluctuation is beneficial for promoting interface quality [10]. However, if the fluctuation of free surface is too sharp, it will likely cause the entrapped slag and gas to melt. Free surface under an alternating magnetic field has been studied [11][12][13]. Some studies focused on the calculation methods for melt shape [14]. The alternating electromagnetic force results in an unstable surface. The characteristics of fluctuation behavior on molten metal free surface under pulsed magnetic field have rarely been studied. This paper focus on the characteristics of pulsed electromagnetic force and the fluctuation behavior of molten metal free surface under pulsed magnetic field. Based on the collected pulse signal data, the law of impulse signals is fitted by a nonlinear fitting method. The expressions of low-frequency pulsed electromagnetic oscillation and stirring terms are deduced by mathematical analysis. The characteristics of low-frequency pulsed electromagnetic oscillation are analyzed. The effect of pulsed magnetic field on the wave behavior of molten metal surface is also studied.

Experimental
In this experiment, a self-made line laser liquid level measurement system was used to measure the fluctuations of different positions on the molten metal surface under pulsed magnetic field. The measurement of magnetic induction intensity was based on small coil electromagnetic induction method. Liquid level displacement was measured using the line laser (Oxlasers, OX-R100L-5) measuring system (figure 1). First, the liquid level fluctuation was recorded by a high-speed camera (Phantom, v2640). Second, image data processing was preformed using C# programming to obtain the fluctuation data recorded by the camera. As shown in figure 1, the relationship between liquid level height h and beam displacement l on the light curtains was [ ( ) ] / q q q q = + h l 2 cos 2 sin cot . At q = 0, 1 / = h l 2 could then be obtained. Therefore, the liquid level fluctuation could be obtained by measuring the beam displacement l. Low-melting GaInSn alloy was used to simulate molten steel. Figure 2 shows the schematic of the experimental apparatus with pulsed magnetic field. A self-made pulsed power with a frequency of 1-100 Hz and a voltage reaching 3000 V was used to produce the pulsed electricity. The GaInSn alloy was heated to 100°C and then poured into a 32 mm-wide, 50 mm-tall graphite crucible. The graphite crucible was kept warm by magnesia heat preservation. The magnetic field intensity was measured by electromagnetic induction. The test coil was placed in the middle of the experimental apparatus, where the graphite crucible was located. A signal conditioner was used to enlarge the measured signal. The oscilloscope connected with a computer was applied to record the measured signal. The induction coil had 100 turns and a diameter of 150 mm. The physical property of GaInSn alloy and molten steel are shown in table 1.

Analysis of pulsed electromagnetic force
The fitting results of pulsed magnetic induction intensity versus time are shown in figure 3(a). The intensity of pulsed magnetic field presents oscillatory decay with time. The fitting cure shows a good fit with the measured data of pulsed magnetic field intensity.
The pulsed magnetic induction intensity versus time can be fitted as followings: where, t is the time; and B is the pulsed magnetic induction intensity. i, a i , b i and ω are the related parameters. The fitting results are listed in table 2. According to Hua [9], if the magnetic field only has the z component, then the diffusion equation of the magnetic field will be as follows: 1 m m is the permeability, and σ is the conductivity. The boundary condition of the magnetic field is Combined with equations (2) and (3), the instantaneous form of the magnetic field can be as follows: where, δ is the skin layer thickness, / d m ws = 2 , m and w is the angular frequency. According to the Maxwell equation, the Lorentz force applied to a conductive melt per unit volume is as follows: where, the first part in equation (5) is turning force f , turn which drives fluid rotation, and the second part is nonturning force f , nonturn which drives fluid oscillation. When only the magnetic field in the z component is considered, the turning force f turn is as follows: [15], the turning force f turn can be set as follows: In addition, the non-turning force f nonturn is as follows: Substituting equation (4) into equations (7) and (8) can obtain the following equations: L k e 1 9 turn m When the value of electromagnetic pulse frequency w is very small, / d » L 1, the turning force f turn and nonturning force f nonturn work. While the w is sufficiently large, /  d L 1, the non-turning force f nonturn dominates. When the value of w approximates / v L, s /  d L 1, the turning force f turn dominates. The result / d = L 2.139 can be obtained by correlation calculation. Therefore, turning force f turn and non-turning force f nonturn work on the molten melt under pulsed magnetic field.
The used melt is GaInSn alloy, and the diameter of the molten pool is 32 mm. The conductivity and relative permeability of melt are 3.4×10 6 S/m and 1, respectively. The nephogram of non-turning force f nonturn (figure 4) can be obtained by considering the melt symmetry and substituting those parameters into equation (10). The non-turning force f nonturn in three representative positions, namely, the surface (x=0 m), radial quarter (x=0.008 m), and center (x=0.016 m) of molten metal are illustrated in figure 5. The nonturning force causes the electromagnetic oscillation. At negative f nonturn value, its orientation is from center to edge of the melt and presents pull force. At positive f nonturn value, its orientation is from the edge to the center of the melt and presents push force. As shown in figures 4 and 5, the electromagnetic oscillation force in the melt surface is largest, while the closer to the internal melt, the smaller the electromagnetic force is due to the skin effect. Electromagnetic force decays with increasing discharge time. With increasing pulse discharge time, the electromagnetic force undergoes an alternate evolution from push force to pull force. The action time of pull force is longer than that of push force. Figure 6 shows that the maximum value of electromagnetic oscillation pull and push forces vary with the increasing time. The maximum value of pull force is larger than the push force. In addition, the maximum values in both the pull force and push forces decreases with increasing charge time. Table 3 shows the non-turning force f nonturn in different distances from the edge of melt in the radial. The maximum value of pull force in the melt surface (x=0 m) is −309,261 mN m −3 , while the maximum of push force is 288,031.9 mN m −3 . In the center of melt, the maximum values of pull and push forces are −105,823 mN m −3 and 101609.4 mN m −3 , respectively. Therefore, the pull force of non-turning force is dominate compared with the push force. The calculated turning force and non-turning force have the same order of magnitude compared with those in other works [9]. The combination of gravity, surface tension, and non-turning forces acts on the molten melt. When the non-turning force is sufficiently high, the melt is derived far from the wall of molten pool. Given that the non-turning force presents oscillatory attenuation with increasing discharging time, the molten melt acted on by gravity and surface tension is pushed back to the wall of the molten pool, resulting in the reciprocating oscillation of molten melt.
The calculation process of turning force is similar to that of the non-turning force. When the relative parameters are substituted into equation (9), the nephogram of turning force f turn is obtained ( figure 7). The turning force f turn in three representative positions, namely, the surface (x=0 m), radial quarter (x=0.008 m), and center (x=0.016 m) of molten metal, are illustrated in figure 8. The electromagnetic turning force in the melt surface is largest, while the closer to the internal melt, the lesser the electromagnetic force. The value of f turn decreases with increasing charge time while maintaining the radial direction. Figure 9 shows that the maximum   The maximum value of f turn is larger than that in the f , nonturn and this finding demonstrates that the turning force is dominate in working on molten melt compared with the non-turning force. Figure 10 shows the 3D diagram of molten melt fluctuation varying with time at different intensities of magnetization. The free surface of molten melt is not stationary without pulsed magnetic field. The free surface of molten melt maintains the oscillation state under the pulsed magnetic field, and the molten melt in the middle of the pool fluctuates sharply. The surface of molten melt in the middle bumps up and is sunken at the edge. This phenomenon illustrates that the free surface of molten melt has a meniscus [16]. The amplitude of surface oscillations increases with increasing magnetic field intensity [17]. The fluctuation amplitude is largest at 0.187 T magnetic field intensity. The non-turning force urges molten melt to reciprocate oscillation. When the nonturning force reaches maximum, the molten melt in the middle of the surface locates the highest position. With the combination of non-turning force and turning force, the molten melt in the surface presents reciprocating oscillation and rotary movement. The fluctuation of molten melt is related to the increasing electromagnetic force by increasing the pulsed current [18]. However, this fluctuation causes the unstable deformation of the  meniscus and even deteriorates the initial solidification status, which negatively affects the improvement of improving the surface quality of casting [19]. Figure 11 shows the waveform of the oscillations obtained at the center of molten surface with different magnetization intensities. The surface in the center of the molten pool has a fluctuation amplitude of±1.13 mm without pulsed field action. This wave maybe produced by uneven heating. Fluctuation increases with pulsed field intensity. The molten metal in the surface center has an oscillation of±3.52 mm at 0.187 T.

Effect of pulsed magnetic field on the fluctuation of molten melt
Fourier transform is often used to analyze the regularity of wave motion on free surface. The law of liquid level fluctuation under pulsed magnetic field can be reflected by its power spectral density distribution. Figure 12  shows the power spectral density obtained at different magnetic field intensities. The power spectral density is distributed symmetrically and increases with increasing of magnetic field intensity. This result illustrates that the displacement in the central point fluctuates violently with increasing magnetic field intensity ( figure 10).   where, m and n are the azimuthal and radial numbers, respectively; r is the liquid density; g is the gravitational acceleration; r and H are the radius and depth of the pool, respectively; s s is the surface tension; and k , mn is the nth zero of the first derivative of the mth-order Bessel function of the first kind. The calculated eigenfrequency frequencies are listed in table 4.
The statistical results of power spectral density in the center fluctuation under different magnetic field intensities are summarized in table 5. Only two spectral peaks (at 0.60 and 3.36 Hz) of the fluctuation power density are observed without pulsed magnetic field. The first peak (at 0.6 Hz) approaches the eigenfrequency (table 4) calculated by equation (12). When applied in pulsed magnetic field, four spectral peaks are observed (appear at 0.40, 3.00, 6.50 and 13.00 Hz). With increasing magnetic field intensity, the position of the four spectral peaks almost remains constant, but the power spectral density increases with magnetic field intensity. Compared with those without magnetic field, the dominant frequencies of the oscillations at the center are reduced by imposing a pulsed magnetic field. The wave frequency of the molten melt in the center of molten pool is different with the frequency of pulsed magnetic field. This result is attributed to the wave of melt being associated with the combined action of pulsed electromagnetic, surface tension and gravity recombination. The melt flow at the free surface must be controlled to obtain stable free surface [19]. Because the wave of melt increases with increasing magnetic field intensity, and these phenomenon results in the large liquid level deformation. The magnetic field should be controlled appropriately to decrease the fluctuation degree of the liquid level. The increasing fluctuation of liquid level causes dross and gas on the surface of molten melt. These may be involved in casting embryo, which reduces the casting embryo quality.

Results
Based on the measuring pulsed magnetic intensity signal, the pulsed electromagnetic force was deduced by mathematical analysis. The effect of pulsed magnetic field on the wave behavior of molten metal surface was also studied. The following conclusions were drawn: (2) The molten melt in surface fluctuated disorderly without electromagnetic field. Applied by the electromagnetic field, it formed a meniscus. The fluctuation increased with increasing pulsed field intensity. The molten metal in center of the surface had an oscillation of ±3.52 mm at 0.187 T.