WO2019175884A1

WO2019175884A1 - Method of optimizing electromagnetic stirring in metallurgical technologies

Info

Publication number: WO2019175884A1
Application number: PCT/IL2019/050291
Authority: WO
Inventors: Arkady Kapusta; Michael Havkin; Boris Mikhailovich; Boris Tilman; Aleksei KUDRIAVTCEV; Kirill SMIRNOV
Original assignee: Nord Israel Research And Development Ltd.
Priority date: 2018-03-14
Filing date: 2019-03-14
Publication date: 2019-09-19

Abstract

Methods and systems are provided for increasing the efficiency of electromagnetic stirring by applying amplitude-frequency modulation (AFM) of travelling or rotating magnetic fields, where the applied modulation frequency is a resonant frequency that maximizes one or more of: an electromagnetic volume force, a current density, a magnetic induction, a motion velocity of the melt, and an amplitude of vector potential of the melt, or of a mold of the melt.

Description

METHOD OF OPTIMIZING ELECTROMAGNETIC STIRRING IN

METALLURGICAL TECHNOLOGIES

RELATED APPLICATION DATA

The present application claims priority from U.S. Provisional Patent Application 62/642,617, filed on March, 14, 2018 and incorporated herein by reference.

FIELD OF THE INVENTION

The present invention is directed to systems and methods of electromagnetic stirring for metallurgical processes.

BACKGROUND OF THE INVENTION

Electromagnetic stirring (EMS) using harmonic rotating or traveling magnetic fields (RMF or TMF) has been used in metallurgy and mechanical engineering in production of ferrous and non-ferrous metals for more than 60 years. The method is widely used on most billet casters and in stationary casting of ingots, in order to improve the macrostructure and chemical homogeneity of ingots and billets, especially for quality steel products.

Experiments on the use of RMF for improving the quality of a continuous cast ingot were performed by E. Pestel [1] and by Junghans and Schaaber [2], who proposed using a three- phase inductor RMF. To increase the efficiency of mixing, they also applied the interruption of current in induction coils with a predetermined period in combination with a change in the direction of rotation of the magnetic field. Subsequently, A. Zibold [3] and A. Kapusta [4] proposed using a frequency-modulated rotating magnetic field (FM RMF) to increase the contribution of the turbulent component to the melt flow in order to increase the mixing efficiency.

Subsequently, F. Beitelman [5] proposed a method for improving macro- and microstructure of metal alloys by creating a superposition of two or more electromagnetic fields of different frequencies and/or amplitudes between multiple stirrers arranged in series. Mikhailovich, Kapusta, and Levy [6] presented methods for optimizing heat and mass transfer using amplitude-frequency modulation (AFM), i.e., a combination of amplitude and frequency modulations, of TMF or RMF. Xiaodon et al. [7] proposed a solution for heat and mass transfer based on multidimensional numerical modeling.

Additional theoretical and experimental developments on using AFM for EMS were described by Dardik [8] and by Feldman [9]. These authors described how AFM RMF increases the turbulent component of melt flow, thereby reducing the time needed for stirring.

Varying parameters of the AFM inductor coil current affects the crystallization process and affects the macro- and microstructure of the continuously cast billets. However, the absence of criteria for optimizing the stirring process means that practitioners must test many AFM parameter combinations to find effective values, which is a significant cost and time burden.

SUMMARY

Embodiments of the present invention provide methods for increasing the stirring intensity of electromagnetic stirring (EMS), using resonant values for amplitude and frequency modulation. Applications include continuous and stationary melt casting of ferrous and non- ferrous metals, as well as out-of-furnace ladle casting. The invention is developed from an analysis of resonant phenomena arising from the interaction of AFM RMF or TMF with electrically conductive media.

Optimization of electromagnetic stirring parameters, in particular of a modulation frequency 0O_f, is based on optimizing magnetic field vector potentials, given boundary conditions for different types of casting. Methods are provided for out-of-furnace processing of metals in ladle and for continuous or stationary billet casting. In the latter case, two options for the AFM RMF impact on the melt are considered: 1) creating conditions for electromagnetic forces resonance in the liquid core directly and 2) excitation of the electromagnetic forces causing resonance in the mold wall, which transfers the mechanical vibrations to the liquid core of the ingot. With the penetration of the AFM magnetic field into the wall of the mold, the Fourier amplitudes of the harmonic spectrum are reduced because of the skin effect. Part of this energy is converted into mechanical oscillations of the mold. Consequently, the following three force factors act on the liquid core of ingot: an average electromagnetic force, exciting large-scale convection of the melt; resonance oscillations of the liquid core of the ingot; and additional mechanical vibrations of the melt, caused by mold vibration, through the solidified portion of ingot.

In some embodiments, a modulation frequency oo_f is chosen so that a frequency of amplitude modulation oo_a is equal to or is a multiple of one of the frequencies of harmonics of the Fourier spectrum W_h in the melt.

The maximum amplitude of electromagnetic forces, as represented by the vector potential, provides the high intensity of melt turbulence, suppressing the growth of dendrites and inclusions (clusters) and, thus promotes the fine-grained formation of solidified structure and chemical uniformity of billets, blooms and ingots.

BRIEF DESCRIPTION OF DRAWINGS

For a better understanding of various embodiments of the invention and to show how the same may be carried into effect, reference is made, by way of example, to the accompanying drawings. Structural details of the invention are shown to provide a fundamental understanding of the invention, the description, taken with the drawings, making apparent to those skilled in the art how the several forms of the invention may be embodied in practice. In the accompanying drawings:

Figs. 1 and 2 are schematic illustrations of a ladle for molten ingot casting;

Fig. 3 is a schematic illustration of a liquid core of square ingot section;

Fig. 4 is a schematic illustration of a wall mold with a square cross-section;

Fig. 5 is a schematic illustration of a liquid core of a circular ingot section;

Fig. 6 is a schematic illustration of a wall mold with circular cross-section; and

Fig. 7 is a schematic illustration of a dependence of the vector potential amplitude, in a melt, on the frequency of the driving inductive current. DETAILED DESCRIPTION OF THE INVENTION

Hereinbelow, variables have the meanings indicated in the following table:

When a harmonic RMF of inductor coils interacts with a melt, a field of rotating currents is induced in the latter. Frequency of an induced magnetic field is equal to the frequency of the coil RMF current. The interaction of RMF and induced currents generates electromagnetic volumetric forces, which contain constant and variable components that change with double the frequency of the RMF.

Under the action of a constant force in the melt, a turbulent rotational flow arises. The flow structure in the liquid core is characterized by a quasi-solid core with a boundary layer on a liquid-solid phase transition area. The layer thickness depends on the Reynolds number, Re Re = nRo/v

where W is the angular flow velocity, R₀ is the liquid core radial size, and v is the melt kinematic viscosity.

The variable component of the electromagnetic volumetric forces is partially suppressed by the walls of the crystallizer, if the inductor is located in the crystallizer, or on the crystallized ingot shell, if the inductor is located below the crystallizer. In both cases, the variable component of the electromagnetic volumetric forces stimulates the vibration of the walls of the crystallizer, or the surface of the solid phase of the ingot at a certain frequency.

Thus, the interaction of the RMF induction coils with the melt occurs in two opposite directions. On the one hand, the RMF supports the level of average stream velocity and turbulence generated by these streams. On the other hand, the RMF partially suppresses turbulence and transforms part of the turbulence energy into heat. Consequently, the stability of the state of the hydrodynamic system at a given RMF intensity is the result of these processes interaction.

Because the frequency of the inductor current is chosen from the condition of the maximum angular velocity in the melt, only the total inductor power is a parameter that affects the quality of the ingot. These power values reach 200 kVA and higher at existing plants.

However, increasing the inductor power by harmonic stirring method does not lead to a stirring intensity. The effect of the magnetic field on the melt velocity weakens with increasing intensity due to the asynchronous character of the electrodynamic processes. Simultaneously, the turbulent flow of the melt is suppressed by a strong magnetic field which leads to deterioration of heat and mass transfer intensity in the melt.

Limitations of the RMF stirring method led us to develop the method of resonant electromagnetic impact (REMI), based on optimization of AFM RMF parameters.

When the AFM RMF is generated in the inductor coils, the current amplitude in each phase can be represented by the following formula

I = I₀(l + xsmm_at)sin(m₀t + x8ίhw_ίΐ) (1) where, I₀ is the nominal current, oo_a is the amplitude modulation frequency, w₀ is the carrier frequency (frequency used in the plant),

0O_f is the frequency modulation frequency, x = Dw/oo_f is the frequency modulation index,

Dw is the frequency deviation.

As shown in [10], formula (1) can be represented in the form,

As follows from formula (2), the AFM currents can be represented as a Fourier series in which there are frequency terms of higher and lower values than the carrier frequency. The carrier frequency is the operating frequency of the harmonic current used in the casting technology for a particular electromagnetic stirring setup.

A mathematical study of the Maxwell equation describing the electrodynamic processes in the liquid core of ingots of square and circular cross sections and in the wall of the corresponding crystallizer shows that resonant energy transfer occurs from one of the Fourier harmonics of the AFM RMF or TMF to the melt or to the copper wall of the crystallizer (i.e., the mold).

The resonance conditions are obtained as a result of solving a three-dimensional nonstationary equation for the vector potential a:

where,

V is the mean velocity vector of the turbulent liquid motion;

L is a vector operator, which is written as follows, depending on the spatial symmetry of the object. For a square ingot section, in a Cartesian coordinate system x, y, z:

For a circular ingot section, or the ladle, in cylindrical coordinate system r, f, z :

where D = - , D_* = - -I- - o r or r

Z₀ and R₀ are the height and radius of the liquid phase in active zone respectively.

The active zone is determined as a height of an inductor pole. Boundary conditions for electromagnetic stirring for stirring in ladle or arc furnace and in square and circular section ingots are presented below.

Boundary conditions for different melt configurations

Boundary conditions for stirring in a ladle (Figs. 1 and 2):

Figs. 1 and 2 show a configuration of a typical metallurgical ladle mold 20, having a mold wall 22, to which is affixed an inductor 24, for example, a traveling magnetic field (TMF) inductor. The inductor 24 has inductor poles 26. Inside the mold is the melt 28 (typically a conductive metal liquid), and the slag 30. As indicated, the conductive melt has a height Z₀. Fig. 2 shows the mold from above, including the mold wall 22 and the inductor 24. The mold radius is shown as RQ.

1 3(r a_f)

r dr = q^~q(f)F^*(ΐ), (6)

r=l

where a_f is a component of the vector potential, q(f) is a step function, equal to

1, — (po < f < cpo

q(f) =

(0, — (po > f > cpo’

F^*(ΐ) is a time function, which has the following form accordance to [10]

where f₀ is shown in Fig. 2 as the working angle of the inductor element; c is an amplitude depth modulation, W_h is non-dimensional angular frequency (i.e., velocity), J_{n -} Bessel function of the first kind of order n , x = - frequency modulation index, Dw- frequency

OJf

deviation, oo_{f -} carrier frequency.

Boundary conditions for square ingot section (Figs. 3 and 4):

Figs. 3 and 4 show a configuration of a typical metallurgical square ingot mold 40, in which is a melt 42 (also referred to hereinbelow as the liquid core), surrounding by a square mold wall 44. (The inductor is not shown.) The width of the mold wall is shown as d^.

The boundary conditions are written in the Cartesian coordinate system.

Two variants of the AFM exposure on the ingot liquid core were considered:

a) Magnetic field of the inductor coils current affects directly to the liquid core of the ingot (Fig.2). The z - component of the dimensionless vector potential of magnetic induction ^az _Liqsq h^as the following form on the lateral surface of melt az_Liq_sq - (1 - ίU)F^*(ΐ) (8) x=l

and in the mold center az_Liq_Sq = 0 (9) x=0

b) Magnetic field of the inductor coils current acting on the liquid core through the vibration of the mold wall (Fig.3). The y - component of the vector potential a_z h h following form on the outer mold surface

and on the inner mold wall x = 1— d

where the dimensionless wall thickness of the mold d is defined as d =—

; d^ is the

dimensional thickness of the mold wall; X₀ is half of the dimensional outer mold side length, D_M sq - dimensional depth of magnetic field penetration into the mold wall is determined by the following expression

Boundary conditions for circular ingot section

Figs. 5 and 6 show a configuration of a typical metallurgical circular, or cylindrical ingot mold 60, in which is a melt 62 (also referred to hereinbelow as the liquid core), surrounding by a cylindrical mold wall 64. (The inductor is not shown.) The width of the mold wall is shown as There are two variants of the AFM exposure on the ingot liquid core: a) Magnetic field of the inductor coils current directly affects the liquid core of the ingot (Fig.4). z - component of the vector potential <¾_Liq_cyi h^as following form on the external side surface of ingot d^aZ

G=1

and in the mold center r = 0 a, _Liq_Cyi = 0 (13) r= 0

b) Magnetic field of the inductor coils current acts on the liquid core through the vibration of the mold wall (Fig.5). z - component of the scalar vector potential c _M_cyl h^as the following form on the outer mold surface x = 1 — az_M_cyl =— e^-ilup<t>^*(t), (14) r=l

on the inner mold wall r = 1— d

where the dimensionless wall thickness of the mold d is defined as d = d =

d_ά is the

Xo dimensional thickness of the mold wall; X₀ is half-length of the outer mold side D_M cyl penetration length of magnetic field into the mold wall A_Mcyl

The procedure of solving electrodynamic problems is common to all the above mentioned configurations and includes the following:

1. A double conversion of physical variables which change the form of original equations (1) with appropriate boundary conditions

2. Solving the obtained equations by the Galerkin method by expanding the initial function into a single or double series with respect to the space variables.

3. Obtaining an equation for determining time-dependent expansion coefficients. The solution of this equation and its investigation on the existence of extremums.

4. Obtaining formulas for the calculation of the value of resonance parameters.

Solutions for different configurations

Solving for resonance of the vector potential resonance for the boundary conditions of the different configurations gives formulas for calculating the resonance modulation frequencies for the different configurations. The amplitude w_a and the frequency w₍· modulation are assumed below to be equal to each other, i.e., w_a =

where, oo_a amplitude modulation of the frequency is determined by the formula w_a =—

-,

w₀

f_a is the dimensional frequency of frequency modulation, w₀ is the dimensional frequency of carrier frequency.

Solutions are given for both the resonant mode of the liquid core and resonance of the vibration of the mold wall. Transitions from the resonant mode of the liquid core to the vibration of the mold wall may be performed by setting a duty cycle for the transition. 1. Cylindrical Ladle

y of modulation,

w₀

n is a number of Fourier harmonic oscillations in a packet of a traveling magnetic field (TMF) caused by an amplitude-frequency modulation of a TMF inductor; uation D^*/_¾(F_/C) ⁼ 0, where J_H(y_k) is a Bessel function, 2, 3 ...

estimated melt angular velocity,

w₀ is the carrier angular frequency,

v is the melt kinematic viscosity,

d_z =— , is a normalized melt height,

2R₀

Z_Q, R_Q are ladle height and radius,

s specific electrical conductivity,

m₀ is the magnetic permeability of a vacuum,

x is a pole division of an inductor.

2. Square mold - core mixing

0)_r = (1 - W_h)/(h + 1),

where,

s is the specific electrical conductivity of the melt, and

X₀ is a half of the billet cross-section length size.

3. Square ingot - wall vibration

w = (0.414 P_M - 1 )/(n + 1) , where,

= m₀s_{( o}w₀L'o , where a_Co is electrical conductivity (of copper),

k, l = 1, 2, 3 ...

d = 8_m/X₀ is a dimensionless mold wall thickness,

6_m is the dimensional mold wall thickness,

¾ is the half length of the outer mold cross-section side length.

4. Cylindrical ingot - core mixing

where the relative melt frequency is determined by the formula

a_Liq is a the specific electrical conductivity of the melt,

R_Liq is the internal radius of the mold cross-section,

/?_k are the roots of the equation - - J₁ (/?_kr) = 0,

or _r-i

r is the liquid core radius of the melt,

W₀ is the dimensionless angular velocity of the melt (turbulent melt motion) defined

umber,

B₀ is a magnetic induction in inductor core,

Re , ^(QQRQ P s

h

s and h are the electrical conductivity and dynamic viscosity of the melt, p is a melt density,

5. Cylindrical mold - wall vibration

" = (1 - P_k)/(n - 1) ,

where,

n = 1, 2, 3

¾ are the roots of equation:

where - - is a derivative of the first kind Bessel function of the first order,

dr

— - is a derivative of the second kind Bessel function of the first order, and

or

k, 1 =1, 2, 3 ...

Examples of implementation

The electromagnetic force volume f equals the vectorial multiplication of real components of the current density ReJ and the magnetic induction Reb f = Re] x Reb

The current density f and the magnetic induction b are proportional to the A_nl, the amplitude of the vector potential a, which is also proportional to the motion velocity of the melt, V. Consequently, resonant amplitudes of the vector potential and the variable component of electromagnetic volume forces f are proportional (though the frequency of change of the field of forces is equal to double the vector potential frequency). The dependence of the vector potential amplitude A_ni on the frequency modulation W_h of the TMF or RMF also characterizes changes in the electromagnetic volume force f.

Fig. 7 shows a graph 100 of A_ni vs. W_h for an exemplary melt configuration. The maximum value of A_ni indicates the optimal modulation harmonic W_h, from which to derive the optimal modulation frequency m_fcl or f_mres (i.e., the "resonance" frequency of modulation). Example 1: Steel Ingot

w/ = (! - W_h)/(^h + 1)

Steel ingot is a square cross section 300x300mm,

copper mold wall thickness of l2mm,

d = 0.074,

w₀=314 rad/s.

The frequency of the resonant Fourier harmonic is determined by the following formula

W-nres = 0.41 * P

Fourier harmonic, n = 1 and modulation frequency of f_mres is equal to

27.8 Hz

Example 2: Steel ingot is a circular cross section

Diameter 300mm. The frequency of the resonant Fourier harmonic is determined by the following formula

0.1,

where, 2; w = m₀sw₀/?₀ ² = 0.64, m₀=3l.4rad/s. Therefore, at k=l, 1=1 R_w= 5

=0.86; Fourier harmonic number n = 4 and dimensional modulation frequency of f_mres is equal to

4.3 Hz

As follows from the boundary conditions (6 - 14), the AFM TMF or RMF is excited by means of superposition of two nonlinear rotating magnetic fields, when one of them is frequency- modulated and the other one is amplitude modulated. Formally, resonance frequencies W_h are equal for each field. However, the frequency spectra for the same W_h are different, since the carrier frequencies of the nearest neighboring harmonics differ by the magnitude amplitude modulation oo_a.

Due to the processes of dissipation in the melt or in the mold, the resonance curves in all the considered cases smoothly change near the resonance (Fig. 6). Thus, a large number of Fourier harmonics have amplitude close to the resonant amplitude, which ensures the stability of turbulent flows and a high intensity of heat and mass transfer.

In practical application of REMI method, the frequency modulation of the frequency oo_f is chosen so that the harmonics w_a will be equal to one of the frequency of spectrum W_h.

The condition defining of amplitude modulation oo_a for the positive value n, has a following form

wa

^f-res (

ni -n₂ 16)

If the module \n₁— n₂ | = 1,

w a-res = w 'f—res (17)

The REMI method initiates the resonance of electromagnetic oscillations processes in the melt and crystallizer and creates the conditions for its maintenance. The excitation of nonlinear oscillations in the cross section of the liquid core or the mold wall is the energy source of the forced nonlinear waves propagating along the axis of the ingot. This effect is an additional factor to maintain the turbulent nature of the flow in those areas of the liquid core of the ingot, where convection currents are weak or absent. Due to nonlinearity and the presence of rotational flow, waves propagating along the axis of the ingot transfer kinetic energy, strengthening heat and mass transfer to the crystallization front and along the liquid core. This should have a positive effect on reducing temperature and concentration gradients throughout the melt volume and, as a result, on the chemical and structural homogeneity of the final products.

The application of this method in electromagnetic stirring processes can effect: reduction in power consumption, more uniform temperature distribution in the melt, more uniform chemical distribution in the melt, reduction of the axial porosity of the ingot, an increase in the proportion of crystals in the ingot cross section, and improved quality of the ingot surface.

REFERENCES

1 . E. Pestel et ah,“Production of Fine Grained Metal Castings,” U.S. Pat. No. 2963758, 1960.

2. S. Junghans, O. Schaaber, "Casting processes, in particular continuous casting method and Plant," Patent of Federal Republic of Germany, 1954, No. DE911425C.

3. A. Zibold et al. Author certificate N 1208888, USSR, 1984.

4. A.Kapusta et al. Author certificate N 1577452, USSR, 1985.

5. F. Beitelman et al. Modulated electromagnetic stirring of metals at advanced stage of solidification. EP 2268431 A4, Jul 12, 2017.

6. B. Mikhailovich, A. Kapusta, A. Fevy, Proc. of EPM 2015, https://hal.archives- ouvertes.fr/hal-0l3339l3.

7. W. Xiaodon et al, International Journal of Energy and Environmental Engineering, 6 (2015) 367-373.

8. Dardik I., et al. "Systems and methods of electromagnetic influence on electroconducting continuum," US Patent 7,350,559, April 1, 2008.

9. M. Feldman et al, AISTech - Iron and Steel Technology Conference Proceedings, 2 (2016) 1473-1481. 10. A. Angot, Complements de Mathematiques a l'Usage des Ingenieurs de l'electrotechnique et des telecommunications. Paris, 1957.

Claims

1. A method for optimizing electromagnetic stirring of a melt of electrically conductive fluids, comprising applying a modulation frequency

to a carrier frequency w₀ of an inductive current applied by an inductor to the melt, wherein the modulation frequency

maximizes one or more of: an electromagnetic volume force f, a current density j, a magnetic induction b, a motion velocity, V, and an amplitude of vector potential, A_nj, of the melt, or of a mold wall of the melt.

2. The method according to claim 1, wherein the modulation frequency

for the melt in a cylindrical ladle configuration is calculated as:

= (P_k\ + Q + W₀— l)/(n + 1), wherein,

w_ww i_{s a} frequency of modulation,

¹ w₀

n is a number of Fourier harmonic oscillations in a packet of a traveling magnetic field (TMF) caused by an amplitude-frequency modulation of a TMF inductor;

y_k are the roots of equation D ^*/_¾(T_/C) ⁼

where J_H(y_k) is a Bessel function,

k= Vl + l², k, 1 =1, 2, 3 ...

^{Q =} si

= m₀sw₀Ro ,

W₀ = n 10^X/R_ow_o , estimated melt angular velocity,

w₀ is the carrier angular frequency,

v is the melt kinematic viscosity,

d_z = -2- , is a normalized melt height,

2R₀

Z_Q, R_Q are ladle height and radius,

s specific electrical conductivity,

m₀ is the magnetic permeability of a vacuum,

t is a pole division of an inductor.

3. The method according to claim 1 , wherein the modulation frequency w₍· for continuous or stationary ingots cast in a square cross-section mold is calculated as

wherein angular velocity of the RMF W_h is defined as

and wherein,

n = 1 , 2, 3 ... is a Fourier harmonic number;

k = 0, 1, 2 ..., 1 = 1, 2, 3,

_ 2

>_Liqsq ⁼ Ro^swo Xu_q is a dimensionless frequency of EMF oscillations in the melt, created by an amplitude-frequency modulation of an RMF inductor,

s is the specific electrical conductivity of the melt, and

X₀ is a half of the billet cross-section length size.

4. The method according to claim 1, wherein the modulation frequency

for a vibration of a square cross-section mold wall for continuous or stationary casting is calculated as w = (0.414 P_kl - 1 )/(n + 1) , wherein,

a_Co is electrical conductivity (of copper), k, l = 1, 2, 3 ... ,

d = S>_m/X₀ is a dimensionless mold wall thickness,

6_m is the dimensional mold wall thickness, and

¾ is the half length of the outer mold cross-section side length.

5, The method according to claim 1, wherein the resonant modulation frequency w₍· for continuous or stationary ingots casting with a circular cross-section, is calculated as

wherein,

n = 1, 2, 3

P_k is defined

is determined by the formula = Ro^s^^wo Rh_q’

a_Liq is a the specific electrical conductivity of the melt,

R_Liq - internal radius of the mold cross-section,

and /?_k are the roots of equation

where r is the liquid core radius of the melt,

W₀ is the dimensionless angular velocity of the melt defined as

number,

B₀ is a magnetic induction in inductor core,

s and h are the electrical conductivity and dynamic viscosity of the melt, and p is a melt density,

6. The method according to claim 5 for stimulating vibration of a cylindrical mold wall under the action of electromagnetic forces with a resonance dimensionless frequency of the frequency modulation w^, which is determined by the following formula

“r = irS¹ -^p^·

wherein,

n = 1, 2, 3 ... ; P_k is determined as

w = m₀sw₀K₀ ², s is the specific electrical conductivity of the mold wall,

R₀ is an external radius of the mold cross-section,

<Tk are the roots of equation: -

~r- is a derivative of the first kind Bessel function of the first order, and

or

QY

— - is a derivative of the second kind Bessel function of the first order.

or

7. The method of claims 1 - 6, wherein frequencies of amplitude modulation w_a and the frequency modulation w _f are equal to each other (

_{S a} frequency of

^J w₀

modulation).

8. The method according to claims 3 and 4, wherein the transition from the resonant mode of the liquid core to the vibration of the mold wall with the required duty cycle is performed.

9. The method according to claims 5 and 6 wherein the transition from the resonant mode of the liquid core to the vibration of the mold wall with the required duty cycle is performed.