**2.1 Liquid metal floating in an electromagnetic field**

A metal with diamagnetic properties can freely hang in a constant magnetic field in the presence of a potential well in it, that is, a region where the tension decreases from the edges to the middle [4–7]. The interaction of a high-frequency magnetic field with a metal leads to the appearance of eddy currents in the latter, which displace the field from the space occupied by the metal, or, in other words, the field inside the metal is weakened by eddy currents. As a result of this, in a variable magnetic field, a nonferromagnetic conductor behaves like a diamagnet in a constant field. Due to the force interaction of eddy currents and the field, the metal is pushed out of the zone with a higher field density to a region with a lower density, that is, into a potential well. If the indicated forces are sufficiently large, then the metal can be raised up despite the action of gravity and held in space in suspension. The simplest field diagram of an inductor system consisting of three parallel wires is shown in **Figure 1**.

It is of interest to consider qualitatively the physical processes that occur during metal levitation. In **Figure 1b**, the location of the metal in the potential well of the magnetic field of three wires with currents is shown. The direction of eddy currents in the surface layer of the metal is opposite to the current in the wires. On the surface of the molten metal, there are special areas characterized by the absence of eddy currents. This is due to the weakening of the magnetic field in these areas. During levitation, in the lower part of the molten metal sample, such areas necessarily exist with almost any configuration of the magnetic field. The absence of eddy currents in such areas eliminates the appearance of electromagnetic pressure, which should lead to the flow of liquid metal through these places, but this does not happen, since the existing surface tension forces compensate for the weakening of the power shell near points 1–4. It should be noted that these forces can balance only a small part of the hydrostatic pressure of the metal and contribute to the retention of the latter at a small height of the liquid column above the indicated surface area. To partially compensate for the weakening of the interaction of eddy currents of the metal with the field, use a two-frequency power supply system of the inductors, which allows you to periodically shift the weakened magnetic field on a larger surface of the metal. Due to the rapid redistribution of the weakened magnetic field and the large inertia of the molten metal, its outflow through singular points does not occur. This technique also contributes to an increase in the mass of the metal. However, the increase in mass is limited by the formation of folds in the lower part of the liquid metal column. The direction of the folds coincides with the magnetic field lines of force, and the depth of the folds is commensurate with the penetration depth Δ.

#### **Figure 1.**

*The magnetic field of three wires with currents (the density of the field lines is shown by the degree of blackness of the shading and fixes the hydrostatic pressure). a—without metal; b—with metal; the numbers 1, 2, 3, and 4 are singular points on the surface of the metal melt.*

**165**

*Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

medium than air

(ball in water)

(ball in the air)

through which the current flows in the opposite direction.

Along with the indicated limitations of the levitation of metallic melts, there are two more. The first is associated with a given vibrational power in the inductor circuit or, in other words, the limiting field strength corresponds to the limiting metal height *h*. The second is due to the presence of a minimum volume of liquid metal, which may still be in a levitation state in the cap, close to the penetration depth or skin layer *h*Δ = Δ. This was confirmed experimentally by melting Al, Sn, Fe, Ti and Cu using frequencies of 500–2500 kHz. It should only be noted that the surface tension coefficient α and the penetration depth Δ depend on temperature. Currents flow in the upper wires in the same direction, so the field between them is weaker than around each. A particularly strong field is created near the lower wire,

In addition to the above-considered features of the interaction of a high-frequency electromagnetic field and a liquid metal during levitation, there is a group of phenomena associated with the stability of the metal [4–7]. The swaying of a droplet hanging in a magnetic field is not a specific property of the liquid state but is caused by the electromagnetic interaction of the metal with the field. Change in the position of the metal relative to the inductor at a constant emf affects the value of the current flowing through the inductor, which causes a change in the force acting on the metal. When conducting experiments with balls of aluminum floating in air, water and oil, the following features of their behavior were discovered:

• Stable equilibrium with respect to finite disturbances in a more viscous

• Stable undamped oscillations with a small amplitude of constant magnitude

• Increasing oscillations with an amplitude exceeding the size of the inductor

The presented nature of the phenomena does not depend on the current value in the inductor (10–30 A) and on the degree of compression of the balls by the magnetic field. The stability of the metal is ensured if the center of curvature of the surface of the melt in its stable state lies outside the volume of the melt. However, this is impossible, since in acute angles, the value of the Laplace pressure of the curved surface of the liquid would reach infinity. The presence of a special configuration in the electromagnetic field of the potential well, as well as a relatively large volume of metal, leads to the extension of the lower part of the ball, and a drop of

A characteristic feature of liquid metal during levitation is intensive mixing inside the drop. A model study carried out with liquid sodium placed in a glass flask, which was in an electromagnetic field (**Figure 2**), showed the existence of turbulent motion of the metal inside the flask [4–7]. Using pitot tubes, as well as photographing methods, melt velocities were measured. It can be seen that the bulk of the liquid in the floating flask moves up. Along the walls of the flask, the metal moves at a much greater speed down. To determine the dependence of the metal velocity on the magnetic field strength, the flask was fixed and the vertical velocity component was measured for various current values. Special experiments without a flask made it possible to conclude that the mixing of the melt during levitation in vacuum or an inert gas is more intense than that described above, since the velocity of the metal on the surface of the drop is not equal to zero. With an increase in the current value in

• Quickly established stable equilibrium (for all studied balls) in oil

metal takes the form of a pear hanging from the cuttings down [7].

#### *Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

*Magnetic Materials and Magnetic Levitation*

is shown in **Figure 1**.

**2. Physical features of electromagnetic levitation**

**2.1 Liquid metal floating in an electromagnetic field**

A metal with diamagnetic properties can freely hang in a constant magnetic field in the presence of a potential well in it, that is, a region where the tension decreases from the edges to the middle [4–7]. The interaction of a high-frequency magnetic field with a metal leads to the appearance of eddy currents in the latter, which displace the field from the space occupied by the metal, or, in other words, the field inside the metal is weakened by eddy currents. As a result of this, in a variable magnetic field, a nonferromagnetic conductor behaves like a diamagnet in a constant field. Due to the force interaction of eddy currents and the field, the metal is pushed out of the zone with a higher field density to a region with a lower density, that is, into a potential well. If the indicated forces are sufficiently large, then the metal can be raised up despite the action of gravity and held in space in suspension. The simplest field diagram of an inductor system consisting of three parallel wires

It is of interest to consider qualitatively the physical processes that occur during metal levitation. In **Figure 1b**, the location of the metal in the potential well of the magnetic field of three wires with currents is shown. The direction of eddy currents in the surface layer of the metal is opposite to the current in the wires. On the surface of the molten metal, there are special areas characterized by the absence of eddy currents. This is due to the weakening of the magnetic field in these areas. During levitation, in the lower part of the molten metal sample, such areas necessarily exist with almost any configuration of the magnetic field. The absence of eddy currents in such areas eliminates the appearance of electromagnetic pressure, which should lead to the flow of liquid metal through these places, but this does not happen, since the existing surface tension forces compensate for the weakening of the power shell near points 1–4. It should be noted that these forces can balance only a small part of the hydrostatic pressure of the metal and contribute to the retention of the latter at a small height of the liquid column above the indicated surface area. To partially compensate for the weakening of the interaction of eddy currents of the metal with the field, use a two-frequency power supply system of the inductors, which allows you to periodically shift the weakened magnetic field on a larger surface of the metal. Due to the rapid redistribution of the weakened magnetic field and the large inertia of the molten metal, its outflow through singular points does not occur. This technique also contributes to an increase in the mass of the metal. However, the increase in mass is limited by the formation of folds in the lower part of the liquid metal column. The direction of the folds coincides with the magnetic field lines of force, and the depth of the folds is commensurate with the penetration depth Δ.

*The magnetic field of three wires with currents (the density of the field lines is shown by the degree of blackness of the shading and fixes the hydrostatic pressure). a—without metal; b—with metal; the numbers 1, 2, 3, and 4* 

**164**

**Figure 1.**

*are singular points on the surface of the metal melt.*

Along with the indicated limitations of the levitation of metallic melts, there are two more. The first is associated with a given vibrational power in the inductor circuit or, in other words, the limiting field strength corresponds to the limiting metal height *h*. The second is due to the presence of a minimum volume of liquid metal, which may still be in a levitation state in the cap, close to the penetration depth or skin layer *h*Δ = Δ. This was confirmed experimentally by melting Al, Sn, Fe, Ti and Cu using frequencies of 500–2500 kHz. It should only be noted that the surface tension coefficient α and the penetration depth Δ depend on temperature. Currents flow in the upper wires in the same direction, so the field between them is weaker than around each. A particularly strong field is created near the lower wire, through which the current flows in the opposite direction.

In addition to the above-considered features of the interaction of a high-frequency electromagnetic field and a liquid metal during levitation, there is a group of phenomena associated with the stability of the metal [4–7]. The swaying of a droplet hanging in a magnetic field is not a specific property of the liquid state but is caused by the electromagnetic interaction of the metal with the field. Change in the position of the metal relative to the inductor at a constant emf affects the value of the current flowing through the inductor, which causes a change in the force acting on the metal. When conducting experiments with balls of aluminum floating in air, water and oil, the following features of their behavior were discovered:


The presented nature of the phenomena does not depend on the current value in the inductor (10–30 A) and on the degree of compression of the balls by the magnetic field. The stability of the metal is ensured if the center of curvature of the surface of the melt in its stable state lies outside the volume of the melt. However, this is impossible, since in acute angles, the value of the Laplace pressure of the curved surface of the liquid would reach infinity. The presence of a special configuration in the electromagnetic field of the potential well, as well as a relatively large volume of metal, leads to the extension of the lower part of the ball, and a drop of metal takes the form of a pear hanging from the cuttings down [7].

A characteristic feature of liquid metal during levitation is intensive mixing inside the drop. A model study carried out with liquid sodium placed in a glass flask, which was in an electromagnetic field (**Figure 2**), showed the existence of turbulent motion of the metal inside the flask [4–7]. Using pitot tubes, as well as photographing methods, melt velocities were measured. It can be seen that the bulk of the liquid in the floating flask moves up. Along the walls of the flask, the metal moves at a much greater speed down. To determine the dependence of the metal velocity on the magnetic field strength, the flask was fixed and the vertical velocity component was measured for various current values. Special experiments without a flask made it possible to conclude that the mixing of the melt during levitation in vacuum or an inert gas is more intense than that described above, since the velocity of the metal on the surface of the drop is not equal to zero. With an increase in the current value in

#### *Magnetic Materials and Magnetic Levitation*

the inductor, the metal velocity increased. The stability of the metal largely depends on the speed of its rotation. This is usually observed with a spherical shape.

The complete process of heating and melting a sample can consist of four stages. At the first stage, the solid sample rises to fixation in a certain stable position (**Figure 3a**,**b**). This part of the process completely depends on the location of the inductor, field currents and the initial position of the sample. With incorrect levitation parameters, the behavior of the sample may turn out to be unpredictable. The next step is to heat the sample to the melting temperature (**Figure 3b**). At this stage, with increasing temperature, a change in the physical properties of the material itself is possible, which can affect both the change in the electromagnetic field and the stability of the position of the sample. Therefore, this stage is central to the duration and overall effectiveness of the levitation process. The third stage consists of melting the sample (**Figure 3c**). It is known that the melting model is

#### **Figure 2.**

*The motion of the melt in a flask with liquid sodium. 1, 2—inductors with 50 and 700 turns, 3—diaphragm.*

#### **Figure 3.**

*Simplest arrangement of induction melting with one inductor: 1—solid charge, 2—inductor, 3—ceramic stand. a—the solid sample is fixed in a certain stable position in an inductor; b—the sample is heated to Tm; c—melting the sample; d—the sample is melted, and the melt can be heated to a given temperature due to intensive mixing.*

**167**

*Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

**2.2 Solid and liquid metal levitation**

*F* = \_1

*Ps* = \_1

field on its frequency is established for

*x* = *Re*[

referred to a surface unit:

heated to a given temperature due to intensive mixing.

not easy even from a geometric point of view, but it is characterized by the fact that the internal volume of the sample remains solid, while the surface of the sample is covered with a liquid film due to the surface nature of the implementation of Joule losses. At the fourth stage (**Figure 3d**), the sample is melted, and the melt can be

The main problem of levitation is the development of a theory and its applications to the problem of the retention of liquid metal during EML, although over the past decades there have been many studies that are somehow related to the theoretical basis of levitation. The significant advantages of EML compared to other methods of metal melting have led to the rapid spread of this method; however, this had to be done almost by touch, without sufficient theoretical justification. More recently, more or less adequate theoretical foundations of the method have been developed that can be used to optimize the setup parameters for levitation. Historically, for example, in Russia, the development of levitation occurred along the path of using two-coils and multi-coil inductors. This significantly affected the development of the theory. The main reason for using two types of inductors is the different power of high-frequency generators used for levitation. Two-coil inductors require increased power (26–100 kW), while multi-coil inductors are able to operate at lower power (8–15 kW). Theoretical prerequisites and experimental possibilities for using two-coil inductors (with parallel turns) were developed by Alex Vogel and his lab [6, 39]. In accordance with the development of this lab, the effect of an idealized uniform electromagnetic field on the metal half-space is summed up from the electromagnetic force (*F*) and the power absorbed by the metal (*P*s), which goes to heat it. Analytically, this is expressed as the following relationship:

> 2 √

> > 2 √

\_

where *F* is the electromagnetic force acting on the metal and equal to its mass; *μ* is the magnetic permeability of the vacuum; *ρ* is the electrical resistivity of the metal; *f* is the field frequency; and *Ps* is the power transmitted to the metal and

\_

where *H* is the amplitude of the magnetic field on the surface of the half-space. As a result of solving the system of Maxwell equations for a plate in a longitudinal plane-parallel magnetic field, the dependence of the magnetic component of the

> \_ *yBz* 2

and a constant plate thickness. This dependence was hyperbolic in nature. Therefore, with a fixed size of the metal sample, there is a well-defined frequency range at which the metal theoretically levitates in an electromagnetic field. The choice of a specific frequency value is determined by the required sample temperature. In addition, it is also necessary to take into account the configuration of the field in order to determine the nature of the dependence of the lifting force on its parameters. Several similarity criteria were theoretically established (conditions for the equality of electromagnetic pressure and mass, equality at specific points of hydrostatic and Laplace forces, and equality relating the skin effect, circular

*μ*/*f Ps* S (1)

*f H2* (2)

] = Const. (3)

#### *Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

*Magnetic Materials and Magnetic Levitation*

the inductor, the metal velocity increased. The stability of the metal largely depends

*The motion of the melt in a flask with liquid sodium. 1, 2—inductors with 50 and 700 turns, 3—diaphragm.*

*Simplest arrangement of induction melting with one inductor: 1—solid charge, 2—inductor, 3—ceramic stand. a—the solid sample is fixed in a certain stable position in an inductor; b—the sample is heated to Tm; c—melting the sample; d—the sample is melted, and the melt can be heated to a given temperature due to* 

on the speed of its rotation. This is usually observed with a spherical shape. The complete process of heating and melting a sample can consist of four stages. At the first stage, the solid sample rises to fixation in a certain stable position (**Figure 3a**,**b**). This part of the process completely depends on the location of the inductor, field currents and the initial position of the sample. With incorrect levitation parameters, the behavior of the sample may turn out to be unpredictable. The next step is to heat the sample to the melting temperature (**Figure 3b**). At this stage, with increasing temperature, a change in the physical properties of the material itself is possible, which can affect both the change in the electromagnetic field and the stability of the position of the sample. Therefore, this stage is central to the duration and overall effectiveness of the levitation process. The third stage consists of melting the sample (**Figure 3c**). It is known that the melting model is

**166**

**Figure 3.**

*intensive mixing.*

**Figure 2.**

not easy even from a geometric point of view, but it is characterized by the fact that the internal volume of the sample remains solid, while the surface of the sample is covered with a liquid film due to the surface nature of the implementation of Joule losses. At the fourth stage (**Figure 3d**), the sample is melted, and the melt can be heated to a given temperature due to intensive mixing.

### **2.2 Solid and liquid metal levitation**

The main problem of levitation is the development of a theory and its applications to the problem of the retention of liquid metal during EML, although over the past decades there have been many studies that are somehow related to the theoretical basis of levitation. The significant advantages of EML compared to other methods of metal melting have led to the rapid spread of this method; however, this had to be done almost by touch, without sufficient theoretical justification. More recently, more or less adequate theoretical foundations of the method have been developed that can be used to optimize the setup parameters for levitation. Historically, for example, in Russia, the development of levitation occurred along the path of using two-coils and multi-coil inductors. This significantly affected the development of the theory. The main reason for using two types of inductors is the different power of high-frequency generators used for levitation. Two-coil inductors require increased power (26–100 kW), while multi-coil inductors are able to operate at lower power (8–15 kW). Theoretical prerequisites and experimental possibilities for using two-coil inductors (with parallel turns) were developed by Alex Vogel and his lab [6, 39]. In accordance with the development of this lab, the effect of an idealized uniform electromagnetic field on the metal half-space is summed up from the electromagnetic force (*F*) and the power absorbed by the metal (*P*s), which goes to heat it. Analytically, this is expressed as the following relationship: \_

$$F = \frac{1}{2} \sqrt{\mu / \pi \rho f} \text{ Ps } \mathbb{S} \tag{1}$$

where *F* is the electromagnetic force acting on the metal and equal to its mass; *μ* is the magnetic permeability of the vacuum; *ρ* is the electrical resistivity of the metal; *f* is the field frequency; and *Ps* is the power transmitted to the metal and referred to a surface unit: \_

$$P\mathfrak{r} = \frac{1}{2}\sqrt{\pi\mu\rho f} \text{ H}^2 \tag{2}$$

where *H* is the amplitude of the magnetic field on the surface of the half-space. As a result of solving the system of Maxwell equations for a plate in a longitudinal plane-parallel magnetic field, the dependence of the magnetic component of the field on its frequency is established for

d on its frequency is established for 
$$\beta \text{x = Re} \left[ \frac{\delta yBx}{2} \right] = \text{Const.} \tag{3}$$

and a constant plate thickness. This dependence was hyperbolic in nature. Therefore, with a fixed size of the metal sample, there is a well-defined frequency range at which the metal theoretically levitates in an electromagnetic field. The choice of a specific frequency value is determined by the required sample temperature. In addition, it is also necessary to take into account the configuration of the field in order to determine the nature of the dependence of the lifting force on its parameters. Several similarity criteria were theoretically established (conditions for the equality of electromagnetic pressure and mass, equality at specific points of hydrostatic and Laplace forces, and equality relating the skin effect, circular

frequency, etc.) and experiments were conducted to simulate the vaporization of liquid aluminum on molten sodium. In an elongated two-coil inductor "with a parallel reverse coil," a group of droplets that did not merge with each other due to the existence of repulsive forces between them was stably held, which is obviously explained by the interaction of currents of the opposite direction flowing at the ends of the droplets. For levitation, when the field is inhomogeneous and the metal moves in it, falling into zones of various configurations, coefficient *A* characterizing the configuration of the field is introduced into Eq. (2), and then Eq. (2) takes the form: \_

$$F = \frac{1}{2} \sqrt{\mu / \pi \rho f} \text{ APs} \tag{4}$$

The validity of this equation is verified experimentally, provided that the inductor has a coefficient value *A* = Const. The power supplied to the copper, molybdenum and niobium balls with a diameter of 15 mm at different field frequencies using the same two-coil inductor "with side parallel coils" was calorimetrically determined. This power was compared with the calculated one [6, 39]. The difference between *P*s obtained by calculation and experimentally was 23%, which proves the good reliability of Eq. (4), in which the values of coefficient *A* are determined experimentally for each type of inductor. For a two-coil inductor "with two side parallel coils," the values of *A* vary from 0.7 to 0.9, whereas for a two-coil inductor "with two parallel coils," these values vary from 0.4 to 0.7. For a two-coil inductor "with sequential switching of coils," the values of coefficient *A* are in the range of 0.2–0.7. Ensuring the transmission of a given power *P*s depends on the mass of the metal, the surface of the sample, the frequency of the field and a certain coefficient *A*. Coefficient *A* characterizes the degree of heterogeneity of the electromagnetic field. It was experimentally established that the greater its heterogeneity, the smaller the coefficient *A*, which theoretically can tend to zero, but it is practically impossible to get it less than 0.2. The heterogeneity at the surface of the melt is different; therefore, the value of *A* also depends on the position of the melt in the inductor and on the size and shape of the sample. When considering the behavior of a metal ball in the field of a two-coil inductor system, the functional dependence of coefficient *A* on the position of the sample was determined [6]. All investigated metals are divided into three groups in accordance with the interval of fixed temperature:

Group 1: Al, Fe, Co, Ni, Cu, Rh, Hf, Ir (tr > tm); Group 2: Ti, Zr, Nb, Mo, Ru (tr > t*m*, depending on *I* and *U*); Group 3: Ta, W, Re, Os (tr < tm).

For a two-coil inductor, the dependence of *r/A* on the volume of various metals is obtained, which varies similarly to the previously described [6]. The increased temperature range corresponds to the optimal volume of the metal, so that depending on the frequency, a different temperature of the metal melt is set, and in vacuum, the temperature is always higher. It is promising to obtain a stable melt temperature using two fields: holding and heating, but not for all metals equally. For Group 1 metals, the use of two-frequency heating is excluded, since the confining field overheats these metals much higher than *t*m. For metals of Groups 2 and 3, two-frequency heating is practically possible.

The interdependence between the values of the function *F* characterizing the skin effect and the power *P* transmitted to the metal is extremely important. Obtaining a given metal temperature is achieved either by changing the frequency of the generator or by choosing the shape of the inductor. A change in the current in the inductor cannot lead to a direct change in the *P/F* ratio, since both quantities depend on *I* 2 . With increasing current, the metal in the inductor rises and falls into the area with a lower electromagnetic field strength and a large gradient of it. This means that temperature can only be controlled to a limited extent.

**169**

*Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

metal during levitation.

**2.3 Temperature of the levitated melt**

with a given height and physical properties.

In addition to the dependences of the lifting force described above on the frequency and intensity of the electromagnetic field, as well as on its configuration, there are specific conditions for a limited choice of the field frequency for holding a liquid metal, which is characterized by a change in shape. It was previously noted that metal during levitation is located in a potential well in which at least one singular point or area of a weakened field necessarily exists through which the metal does not pour out only due to surface tension on a curved surface. In areas of a weakened field, the pressure of a liquid metal column is balanced by the difference in surface tension values on the curved surfaces of the lower and upper parts of the metal [6]. The use of a two-coil inductor at the selected frequency and configuration of the electromagnetic field revealed the need to study the relationship between the voltage applied to the inductor and the behavior of the liquid metal during levitation. Experimentally, the lowering of a drop was recorded with a decrease in voltage. When a certain value was reached, the metal began to flow out of the inductor. The release of the melt can be controlled by increasing the area of the weakened field while maintaining the strength of the magnetic component of the field, necessary to hold the bulk of the metal. It was experimentally established that in a two-coil inductor there are three zones of the electromagnetic field [6]. In the first zone, the solid metal hangs, and the liquid merges independent of the capillary constant. In the second zone, the liquid metal hangs unstably; its degree of stability depends on the volume of the metal and does not depend on the capillary constant. In the third zone, the position of the metal in the inductor is associated with the presence of volume dependence. The use of a multi-coil inductor at the selected frequency and configuration of the electromagnetic field revealed the need to study the relationship between the voltage applied to the inductor and the behavior of the liquid

Along with the retained metal melt, the production and regulation of its temperature are of great importance. For two-coil inductors, the theoretical foundations and technological designs that provide the necessary heating of the samples were considered in [6, 7]. The validity of the functional dependence (4), which relates the electromagnetic lifting force to the power absorbed by the metal, has been proved experimentally. In the steady state in vacuum, the power transmitted to the metal is equal to the radiated power. For most metals, the temperature dependence of the power *P*s radiated from a unit surface is well known (**Figure 4**). This power is usually determined by the reverse calculation. It is known that ensuring the transmission of a given power *P*s depends on a number of factors: the mass of the melt, the surface of the sample, the frequency of the field and coefficient A. The mass of the melt during levitation in a multi-coil inductor can be determined by knowing the minimum frequency of the electromagnetic field that implements the levitation of a metal melt

Obtaining high temperatures during the levitation of metal melts has no fundamental obstacles. In practice, this is accomplished by choosing, for example, a multi-coil inductor, the conical part of which is open at a small angle and whose field has a small tension gradient. To obtain the necessary lifting force, a sufficiently large current is needed, due to which the metal is heated. Due to the fact that the power increases proportionally with frequency, and the lifting force is much less dependent on it, the field frequency is increased to obtain a higher temperature. In each case, the field frequency must be chosen so that the value *X* = *R*/Δ (here *R* is the radius of the sphere and Δ is the thickness of the skin layer) is more than 10 and the value of *F* does not depend on the change in ρ. Otherwise, during heating, the

#### *Electromagnetic Levitation of Metal Melts DOI: http://dx.doi.org/10.5772/intechopen.92230*

*Magnetic Materials and Magnetic Levitation*

*F* = \_1

frequency, etc.) and experiments were conducted to simulate the vaporization of liquid aluminum on molten sodium. In an elongated two-coil inductor "with a parallel reverse coil," a group of droplets that did not merge with each other due to the existence of repulsive forces between them was stably held, which is obviously explained by the interaction of currents of the opposite direction flowing at the ends of the droplets. For levitation, when the field is inhomogeneous and the metal moves in it, falling into zones of various configurations, coefficient *A* characterizing the configuration of the field is introduced into Eq. (2), and then Eq. (2) takes the form:

> 2 √

The validity of this equation is verified experimentally, provided that the inductor has a coefficient value *A* = Const. The power supplied to the copper, molybdenum and niobium balls with a diameter of 15 mm at different field frequencies using the same two-coil inductor "with side parallel coils" was calorimetrically determined. This power was compared with the calculated one [6, 39]. The difference between *P*s obtained by calculation and experimentally was 23%, which proves the good reliability of Eq. (4), in which the values of coefficient *A* are determined experimentally for each type of inductor. For a two-coil inductor "with two side parallel coils," the values of *A* vary from 0.7 to 0.9, whereas for a two-coil inductor "with two parallel coils," these values vary from 0.4 to 0.7. For a two-coil inductor "with sequential switching of coils," the values of coefficient *A* are in the range of 0.2–0.7. Ensuring the transmission of a given power *P*s depends on the mass of the metal, the surface of the sample, the frequency of the field and a certain coefficient *A*. Coefficient *A* characterizes the degree of heterogeneity of the electromagnetic field. It was experimentally established that the greater its heterogeneity, the smaller the coefficient *A*, which theoretically can tend to zero, but it is practically impossible to get it less than 0.2. The heterogeneity at the surface of the melt is different; therefore, the value of *A* also depends on the position of the melt in the inductor and on the size and shape of the sample. When considering the behavior of a metal ball in the field of a two-coil inductor system, the functional dependence of coefficient *A* on the position of the sample was determined [6]. All investigated metals are divided into three groups in accordance with the interval of

Group 1: Al, Fe, Co, Ni, Cu, Rh, Hf, Ir (tr > tm); Group 2: Ti, Zr, Nb, Mo, Ru

The interdependence between the values of the function *F* characterizing the skin effect and the power *P* transmitted to the metal is extremely important. Obtaining a given metal temperature is achieved either by changing the frequency of the generator or by choosing the shape of the inductor. A change in the current in the inductor cannot lead to a direct change in the *P/F* ratio, since both quantities

the area with a lower electromagnetic field strength and a large gradient of it. This

means that temperature can only be controlled to a limited extent.

. With increasing current, the metal in the inductor rises and falls into

For a two-coil inductor, the dependence of *r/A* on the volume of various metals is obtained, which varies similarly to the previously described [6]. The increased temperature range corresponds to the optimal volume of the metal, so that depending on the frequency, a different temperature of the metal melt is set, and in vacuum, the temperature is always higher. It is promising to obtain a stable melt temperature using two fields: holding and heating, but not for all metals equally. For Group 1 metals, the use of two-frequency heating is excluded, since the confining field overheats these metals much higher than *t*m. For metals of Groups 2 and 3,

(tr > t*m*, depending on *I* and *U*); Group 3: Ta, W, Re, Os (tr < tm).

two-frequency heating is practically possible.

\_

*μ*/*f APs* (4)

**168**

depend on *I*

2

fixed temperature:

In addition to the dependences of the lifting force described above on the frequency and intensity of the electromagnetic field, as well as on its configuration, there are specific conditions for a limited choice of the field frequency for holding a liquid metal, which is characterized by a change in shape. It was previously noted that metal during levitation is located in a potential well in which at least one singular point or area of a weakened field necessarily exists through which the metal does not pour out only due to surface tension on a curved surface. In areas of a weakened field, the pressure of a liquid metal column is balanced by the difference in surface tension values on the curved surfaces of the lower and upper parts of the metal [6]. The use of a two-coil inductor at the selected frequency and configuration of the electromagnetic field revealed the need to study the relationship between the voltage applied to the inductor and the behavior of the liquid metal during levitation. Experimentally, the lowering of a drop was recorded with a decrease in voltage. When a certain value was reached, the metal began to flow out of the inductor. The release of the melt can be controlled by increasing the area of the weakened field while maintaining the strength of the magnetic component of the field, necessary to hold the bulk of the metal. It was experimentally established that in a two-coil inductor there are three zones of the electromagnetic field [6]. In the first zone, the solid metal hangs, and the liquid merges independent of the capillary constant. In the second zone, the liquid metal hangs unstably; its degree of stability depends on the volume of the metal and does not depend on the capillary constant. In the third zone, the position of the metal in the inductor is associated with the presence of volume dependence. The use of a multi-coil inductor at the selected frequency and configuration of the electromagnetic field revealed the need to study the relationship between the voltage applied to the inductor and the behavior of the liquid metal during levitation.
