# Американский Научный Журнал EVALUATION OF DEPTH OF PENETRATION OF ELECTROMAGNETIC WAVE IN MELTED ALUMINUM

Abstract
Induction machines of small size are used in the modules for pumping molten metal, called electromagnetic
trays. Such devices are created according to the technical task of the metallurgical enterprise using engineering
methods of calculation. To improve the traction characteristics, a flat or cylindrical magnetic circuit is used. However, for induction machines with an open magnetic core, the asymmetry of the electromagnetic regime is characteristic, which is difficult to take into account in the preliminary calculation. Therefore, to clarify the regime parameters and preliminary evaluation of the efficiency of longitudinal field inducers, engineers perform numerical
simulation of the magnetic, hydrodynamic and thermal field of the tray on a computer. One of the most important
characteristics of the inductor can be considered the depth of penetration of the traveling magnetic field into the
molten metal. Скачать в формате PDF

44 American Scientific Journal № ( 21 ) / 201 8

ЭНЕРГЕТИКА

EVALUATION OF DEPTH OF PENETRATION

OF ELECTROMAGNETIC WA VE IN MELTED ALUMINUM

Tyapin A.A.

Postgraduate student, Siberian Federal University,

Svobodny prospect, 79, 660041, Krasnoyarsk, Russia

Andryushchenko V.Y. Postgraduate student,

Siberian Federal University, Krasnoyarsk, Russia

Avdulova Y.S. Assistant of the Department of Electrical

Engineering and Electrotechnology, Siberian Federal University,

Krasnoyarsk, Russia

Goremykin V.A. Ph.D., Associate Professor,

Siberian Federal University, Krasnoyarsk, Russia

О Ц Е Н К А Г Л У Б И Н Ы П Р О Н И К Н О В Е Н И Я Э Л Е К Т Р О М А Г Н И Т Н О Й В О Л Н Ы

В Р А С П Л А В Л Е Н Н Ы Й А Л Ю М И Н И Й

Тяпин Aлексей Андреевич

Аспирант, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект, 79, 660041, Красноярск, Россия

Адрющенко Вадим Юрьевич

Аспирант, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект, 79, 660041, Красноярск, Россия

Авдулова Юлия Сергеевна

Ассистент кафедры Электротехника и Электротехнологии

ФГАОУ ВО Сибирский Федеральный Университет,

Горемыкин Виталий Андреевич

К.т.н., доцент, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект , 79, 660041, Красноярск , Россия

Abstract

Induction machines of small size are used in the modules for pumping molten metal, called electromagnetic

trays. Such devices are created according to the technical task of the metall urgical enterprise using engineering

methods of calculation. To improve the traction characteristics, a flat or cylindrical magnetic circuit is used. How-

ever, for induction machines with an open magnetic core, the asymmetry of the electromagnetic regime is charac-

teristic, which is difficult to take into account in the preliminary calculation. Therefore, to clarify the regime pa-

rameters and preliminary evaluation of the efficiency of longitudinal field inducers, engineers perform numerical

simulation of the magnetic, hydrodynamic and thermal field of the tray on a computer. One of the most important

characteristics of the inductor can be considered the depth of penetration of the traveling magnetic field into the

molten metal.

Keywords

The running magnetic fi eld, electromagnetic tray, longitudinal magnetic field inductor, plane electromagnetic

wave, wave penetration depth into metal, hydrodynamics of aluminum melt, electromagnetic pump, mathematical

simulation.

Аннотация

Индукционные машины малого габарита при меняют в составе модулей для перекачивания расплав-

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kdhfmaZ^Zgbxf_lZeemj]bq_kdh]hij_^ijbylbyk ijbf_g_gb_fbg`_g_jguof_lh^bdjZkqzlZ>eymemq

r_gbyly]h\uooZ jZdl_jbklbdbkihevamxliehkdbcbebpbebg^jbq_kdbcfZ]gblhijh\h^H^gZdh^eybg

^mdpbhgguo fZrbg k jZahfdgmluf fZ]gblhijh\h^hf oZjZdl_jgZ g_kbff_ljby we_dljhfZ]gblgh]h

j_`bfZdhlhjmxljm^ghmq_klv\ij_^\Zjbl_evghfjZkqzl_Ihwlhfm^eymlhqg_gbyj_`bfguoiZ jZf_l

jh\bij_^\Zjbl_evghchp_gdbwnn_dlb\ghklbbg^mdlhjh\ijh^hevgh]hihey\uihegyxlqbke_ggh_fh^_

ebjh\Zgb_fZ]gblgh]h]b^jh^bgZfbq_kdh]hbl_ieh\h]hiheyehldZgZW<FH^ghcba\Z`g_crbooZjZd

l_jbklbdbg^mdlhjZ fh`ghkqblZlv ]em[bgm ijhgbdgh\_gby[_]ms _]hfZ]gblgh]hihey\jZkieZ\e_gguc

f_lZee

Ключевые слова

Бегущее магнитное поле, электромагнитный лоток, индуктор продольного магнитного поля, плоская

we_dljhfZ]gblgZy\hegZ]em[bgZijhgbdgh\_gby\heg\f_lZee]b^jh^bgZfbdZZexfbgb_\h]hjZkieZ\Z

we_dl jhfZ]gblgucgZkhkfZl_fZlbq_kdh_fh^_ebjh\Zgb_

American Scientific Journal № (21 ) / 201 8 45

Formulation of the problem . The use of electro-

magnetic devices for pumping molten aluminum can

overcome the shortcomings of mechanical pumps [1,

p.24]. Unlike mechanical devices, electromagnetic in-

ductors of the longitudinal magnetic field exclude di-

rect contact with the melt, provide high accuracy and

flexibility of control [2, p.45]. One of the most suitable

devices for moving melts is the electromagnetic tray.

However, the effic iency of the tray largely depends on

the depth of penetration of the magnetic field into the

aluminum. In the study of metallurgical equipment, nu-

merical modeling tools are used. However, the prelim-

inary evaluation is carried out according to analytical

expressions [3, p.28]. For induction machines of small

dimensions, known analytical expressions give large

errors [4, p.122]. Using mathematical modeling tools,

the formula for determining the depth of penetration of

an electromagnetic wave of a running magn etic field

can be clarified.

Metallurgical enterprises are equipped with melt-

ing and casting units, which operate on a two -stage or

single -stage process diagrams. Melting and casting

units can differ in the types of furnaces, arrangement of

equipment, feat ures of the technological cycle [1, p.12].

A common feature is the need to pump the melt from

the furnace to the furnace or from the furnace to the

crystallizer, and various devices are used for this. Strict

requirements for pumping melt remain unchanged,

therefore mechanical pumps are considered morally ob-

solete. Among many machines for pumping aluminum

between furnaces, one can distinguish a group of flat

and cylindrical inductors of a longitudinal magnetic

field [2, p.80]. As a rule, they are used in tra ys where it

is not necessary to create powerful devices with a large

depth of electromagnetic wave penetration. This condi-

tion corresponds to shortened induction machines with

a ferromagnetic secondary element. A feature of multi-

phase machines can be consi dered their short -pole and

low -frequency structure [5, p.62].

A sketch of the construction of a flat electromag-

netic tray is shown in Figure 1, a. The tray consists of a

channel (1), a magnetic circuit (2), in whose slots a

multiphase winding (3) is locate d. The induction ma-

chine is placed under the molten metal (4). The side of

the inductor, directly below the channel, is called the

working one. The distance from the surface of the in-

ductor to the melt is usually called the working gap. The

larger the valu e of the working gap, the less the mag-

netic field penetrates into the melt. Conditionally be-

lieve that the inductor is an analog of the deployed sta-

tor of the induction motor (Figure 1, b), on the surface

of which a running magnetic field is created [4, p. 37].

The amplitude of magnetic induction Bz over the

active zone of the inductor varies exponentially (Fig. 1,

c). The intensity of the damping depends on the magni-

tude of the pole pitch, by which is meant the distance

occupied by each of the inductor pole s. The distance

along the inductor, at which the phase changes by 2 π,

is the period of the spatial structure of the winding. At

this distance 2 τ there are two magnetic poles. As the

step of pole increases, the length of the phase zones in-

creases proportion ally. In addition to flat inductors, in

practice, multiphase induction machines with a cylin-

drical magnetic core and disk coils placed around the

core are used [5, p.64].

а b c

Figure 1

The magnetic flux lines crossing the melt induce

eddy currents in it (Fig. 2, b). The interaction of the

traveling magnetic field and eddy currents creates vol-

umetric electromagnetic forces that cause the melt to

move (Figure 2, a). With an increase in any of the pa-

rameters ( , , f) , the increase in force goes to the

maximum region. But further it is seen that the force

decreases and tends to zero [3, p.70] . The diagram of

the distribution of tractive effort is shown in Figure 2,

c.

а b c

Figure 2

The improvement of numerical methods and anal-

ysis tools led to a radical change in the situation when

using numerical models of complex technical systems.

Numerical methods have obtained decisive advantages

due to the evolut ion of computer technology, software

and rapid increase in processing power. The input lan-

guages of modern software are highly developed, the

user interface is improved. This made it possible to sim-

plify the construction of numerical models, up to semi-

auto matic algorithms for their generation. On the basis

of models, an iterative numerical solution of the field

equations is organized [1, p.111] . Software tools for

modeling induction machine regimes are used at the

stage of analysis of the first technical s olutions ob-

tained, often in comparison with the results of a physi-

cal experiment.

Two -dimensional numerical models give a satis-

factory agreement with experiment for axisymmetric

systems with a traveling magnetic field. For planar sys-

tems, the quantitative errors in the calculation of two -

46 American Scientific Journal № ( 21 ) / 201 8

dimensional problems can be excessive, and often lead

to a qualitatively different result. This is due to difficul-

ties in describing in the two -dimensional model the spe-

cific elements of the system, for example, the outer

parts of the windings and their mutual arrangement.

Accounting in the two -dimensional formulation of edge

effects in the melt is also problematic.

The induction machine for the electromagnetic

tray is a complex technical system built by solving in-

terrelated problems in electrical circuits, electromag-

netic, magnetohydrodynamic and heat fields. The study

of heterogeneous objects of high complexity should be

carried out in a complex manner, taking into account

the most significant factors and their interrelatio nships.

In the "inductor -channel" system, this is the mutual in-

fluence of magnetic and hydrodynamic fields on each

other. Their ratio forms the dynamics of the displace-

ment of the aluminum melt in the channel and the dis-

tribution of the resultant electroma gnetic field of the in-

ductor. The results of application of both 2D and 3D

models are presented in the literature [1, p.123].

As a rule, the results of the calculation in a three -

dimensional formulation better coincide with the re-

sults of the experiment. H owever, it was not possible to

find a system assessment of the question of the quality

of docking of two -dimensional and three -dimensional

models in metallurgical problems. Apparently, there is

no quantitative assessment of the reliability and com-

parison o f the accuracy of such calculations for large -

dimensional problems. Therefore, it was not possible to

establish recommendations on the limits of the use of

models. A sketch of the computational domain in the

model intended for the analysis of the electroma gnetic

field in the "inductor -channel" system is shown in Fig.

3.

The working area of the "inductor" of the electro-

magnetic tray is called a flat electromagnetic system.

And in the investigated object it is necessary to accu-

rately estimate the nature of th e moving magnetic field.

It is known that truncated induction machines are char-

acterized by the presence of edge effects (longitudinal,

transverse, input and output). Therefore, it is important

to compare three -dimensional and twodimensional

models and ass ess the degree of coincidence of the re-

sults.

The problem is that when taking into account the

design features of the inductor, the calculation time can

be excessive. Therefore, in the construction of the

model, a number of Figure 3 assumptions and limita-

tithe assumptions made should ons are introduced, but

not distort the real picture of the field. It should be

noted that

three -dimensional problems require substantial

computing resources. Duration of calculations on clus-

ter computers can be hundreds of hours. At the same

time, sometimes, for making a decision, it is necessary

to quickly evaluate the effect of design paramet ers on

the device mode. The question of comparing models is

relevant, since the system under investigation can have

several design options. Therefore, to make a decision,

it is necessary to ensure minimum requirements for

computational resources.

The probl em of analyzing the field must be solved

in the system of Maxwell's equations. For simplicity,

the known method is used and a universal variable is

introduced, the vector potential of the magnetic field Ā.

7KH XVH RI– H[FOXGHVXQNQRZQV IURPWKH V\VWHPDO

lowing one to obtain an equation of one variable.

At the initial stage, the currents of the induction

unit are assumed to be sinusoidal. Therefore, the field

American Scientific Journal № (21 ) / 201 8 47

in the working region can be considered harmonic in

accordance with (1), and the regime parameters are rep-

resented in a complex form. By the equations of a qua-

sistationary field, one proceeds to a simplified, flat

twodimensional or three -dimensional formulation. The

magnetic induction vector is consider ed to be located in

the plane of the tray, while the vector of the electric

current density J and the vector magnetic potential A

are orthogonal to it. In the twodimensional formulation,

the components Jy and Ay are nonzero, and in the

threedimensional for mulation there are all components.

The solution of the field equations (2, 3) is a

boundary value problem. For its correct description,

boundary conditions are applied and the equations are

solved together with the boundary conditions. Next, the

traction characteristics of the electromagnetic field are

calculated. For this, engineers usually use specialized

software environments [5, p.66].

Calculation of the electromagnetic field in the

working region is pe rformed by the finite element

method. For this, the software Ansys Multiphysics is

used. When building models, the programming lan-

guage APDL was used. Conversion of input and output

information flows is performed in ASCII code, which

is beneficial from the point of view of the formation of

hybrid models. At the same time, integration of third -

party calculation models with external software mod-

ules is performed. In addition, the ASCII code is con-

venient for processing the calculation results. The mod-

els desc ribed here are formalized in the format of inter-

nal program code and additional modules - macros.

Macros act as docking nodes for heterogeneous tasks,

and also carry service options to support the execution

of non -standard functions [1, p.55].

At the first stage, to accept the decision on the

model, the launch regime was investigated. The elec-

tromagnetic force in the starting mode determines the

starting pulse necessary to start the melt movement.

The intensity of metal movement depends on a large

number of criteria. These include linear current load,

frequency, working gap, pole division, melt parame-

ters, magnetic circuit characteristics, etc. Approxi-

mately 50 criteria are adopted. Identifying the most sig-

nificant dependencies and their formalization requir es

a large number of calculations and leads to optimization

problems. Consequently, already at this stage the ques-

tion of the applicability and adequacy of two -dimen-

sional or three -dimensional models becomes relevant.

In the study, two parametric models of the "induc-

tor -channel" system were developed and tested: in two -

dimensional and three -dimensional formulations. To

compare the calculation results, the geometry parame-

ters and the power supply mode of the model are set to

the same. Some results are shown in Fig. 4.

The regulation of the supply voltage frequency has

shown that the tangential force F τ has a characteristic

optimum. And for the two -dimensional case (curve 1),

the optimum is located at a frequency of 3 Hz, and for

a three -dimensional (curve 2) at 17 Hz. The difference

in effort values is 53 %. In addition, the behavior of the

curves is significantly different. The nature of the re-

gion of optimal values along curve 1 looks relatively

high -quality.

Curve 2, on the other hand, shows a low selective

extremum. At a frequency of 23 Hz, the curves inter-

sect, and further curve 1 is located below curve 2 (~ 7

%). Thus, the results of calculating the electromagnetic

force by the twodimensional and three -dimensional

models are different.

Further studies sho wed a different character of the

distribution of the amplitude value of the magnetic in-

duction in the melt at frequencies below 12 Hz. The in-

duction

Figure 4

48 American Scientific Journal № ( 21 ) / 201 8

distribution for the three -dimensional model is

more uniform along the length of the channel of

the melt and has no characteristic dips. The two -

dimensional model is characterized by too high induc-

tion values (up to 76 %). This explains the higher cal-

culated values of the forces in Fig. 4, but does n ot ex-

plain the change in the nature of their distribution.

The presence of previously developed prototypes

of induction machines facilitates the task of comparing

the results of numerical simulation. Despite the fact that

the samples of longitudinal field inducers were manu-

factured without the use of optimization algorithms, the

main regularities of the force effect for the threedimen-

sional models were basically confirmed. To power

physical samples, a transistor IGBT converter with a

capacity of 50 kVA was used, and the devices them-

selves were mounted on an experimental bench.

The two -dimensional model allows to take into ac-

count the input and output edge effects (x -axis). But

does not take into account the transverse effect associ-

ated with a change in the direction of the vortex current

vector by 90 °, along the lateral surface of the channel.

In addition, due to the presence of frontal parts of wind-

ing, there is a distortion of the distribution of the mag-

netic field within the active part of the inductor ( y axis).

Therefore, two -dimensional numerical models for the

analysis inductors of short -pole linear induction ma-

chines in such a formulation should not be used. For the

adoption of technical and economic solutions, it is nec-

essary to form a full -fledged t hree -dimensional numer-

ical model at the stage of preliminary calculations and

to calculate the mode parameters, taking into account

the most important factors determining the device's ef-

ficiency.

To quantify the limit values of the starting angles

of the e lectromagnetic tray, a thorough study of the set

of criteria, the construction of a detailed model, and the

refinement of the research algorithm are required. In

addition, a large amount of computation and a system

analysis of the results are required. The refore, the study

of the launch regime is allocated to a separate project.

In the theory of induction machines, the concept

of the depth of penetration of the magnetic field into

the melt is used. If the thickness of the metal is much

greater than , then in the most remote layers of the melt,

backflows can occur and condition

(1) will not be fulfilled. Therefore, take the condi-

tion hk = 1.41 , at which the optimum mode of operation

of the inductor takes place. According to the literature,

the penetration d epth is determined by the expres-

sion:

. (4)

The calculation results from expression (4)

showed a significant difference from the values de-

scribed above. For 1.41 = 50 mm, the frequency is f

= 40 Hz. But with a decrease in thickness, the depend-

ence is shifted to the range of kilohertz. Consequently,

the use of classical expressions does not accurately de-

scribe the electromagnetic mode of the induction ma-

chine.

The study [1, p.76] shows an expression that clar-

ifies this discrepancy. The justification is the indication

of the evaluation of the results for a plane electromag-

netic wave (3). The conce pt of a plane wave assumes

that . This assumption is inapplicable for a real short -

pole inductor. Therefore, the calculated values of pen-

etration depth differ significantly from those measured.

For their correct definition, the following expression is

used:

(5)

The structure of expression (5) indicates that the

depth of penetration depe nds not only on the conduc-

tivity , the frequency f of the supply voltage, but also

on the magnitude of the pole division . However, anal-

ysis of the results of numerical simulation showed that

the calculated values of the frequencies also signifi-

cantly exce ed the results obtained from expression (4).

A detailed quantitative evaluation shows the unsuitabil-

ity of the analytical expression (5) for determining the

penetration depth of the wave developed by the induc-

tor of the electromagnetic tray. Studies have s hown that

taking into account the non -magnetic gap Δ leads to a

sharp change in the location of the optimal regime char-

acteristics, whose behavior is shown in Fig. 6. This

shows the necessity of taking into account the influence

of Δ in expression (5). To refine the analytical expres-

sion (5), a series of approximation procedures is per-

formed, using splines. As a result, we obtained a refined

expression for estimating the penetration depth of an

electromagnetic wave, which makes it possible to reli-

ably descr ibe the result.

(6)

where ξ π / τ, is the correction factor of the

penetration efficiency of the running electromagnetic

wave through the non -magnetic gap; k = 1 ÷ 2 - the co-

efficient of influence of the secondary element on the

penetration depth of the electromagnetic wave.

The obtained analytical expression (5) shows the

corrected regularity of the distribution of values of the

penetration depth of the wave, for the considered class

of electromagnetic trays. The formula mak es it possible

to calculate the penetration depth at a given frequency,

American Scientific Journal № (21 ) / 201 8 49

taking into account the joint influence of the essential

technological parameters of the "inductor -channel"

system. The product of the newly introduced coeffi-

cients ξ and k in the form ula (5) integrally takes into

account the effect of the physical parameters of the de-

vice and the geometric characteristics of the short -pole

machine. For the considered design designs, with the

secondary element and without it, on the basis of mul-

tivariat e numerical studies the values of the coefficient

k = 1 ,2 and 1,8, respectively, were adopted.

Comparison of calculation results by analytical

expression and results of numerical simulation con-

firmed the high degree of coincidence of not only dis-

crete val ues, but also accurate reproduction of the mo-

notonous regularity of distribution of optimal values of

fopt and hк for the system under study. In addition, a

comparison of the results of the calculation of the re-

gime characteristics with the results of a ph ysical ex-

periment on the trays from a number of designed prod-

ucts. Comparison showed an acceptable quality of co-

incidence. Therefore, the need for preliminary numeri-

cal simulation with long and resource -intensive itera-

tive calculations has been exhausted.

The dependence of the optimal tangential force Fτ

on the melt height hk for the case with the secondary

element (2) and without it (1) is shown in Fig. 5. De-

pendences of the optimal frequencies fopt on the melt

height hk for the case with the secondary ele ment (2)

and without it (1) is shown in Figure 6. The behavior of

the regime parameters of the induction machine is im-

portant to evaluate when searching for the optimum fre-

quency. Automated search of optimal values of fre-

quency fopt, allowed to conclude th at all dependences

are essentially nonlinear (Figure 5). An increase in hk

leads to an exponential decrease in the frequency. And

for both constructive options the behavior is the same.

The displacement of the curves is 26 % in practically

the entire range .

Figure 5 Figure 6

Based on the results of the study, engineers can

formulate generalized recommendations for construct-

ing an effective tray design. At the same time, it is pos-

sible to take into account a combination of technologi-

cal, energy, constructive and cost factors. In particular,

an inductor with a magnetic core of circular cross sec-

tion is proposed. It makes it possible to simplify the de-

sign, facilitate the device, improve cooling condit ions,

and also solve the problem of parametric synthesis of

operating modes in a simplified formulation.

For an inductor with an ascending channel, the

first criterion of the efficiency of work is the minimum

starting level of the melt. The determining fac tor is the

traction force F in the channel. The value determined

from the condition for balancing the hydrostatic pres-

sure force. It should be noted that, for pressure electro-

magnetic trays, the starting level of the melt is critical

only at the launch sta ge. For non -pressure trays, it is

important throughout the entire process. The inductor

of the electromagnetic tray has an uneven density of

electromagnetic forces along the length of the channel.

The lowest density is characteristic for the zones of the

beginning and end of the inductor, and also beyond its

limits. In the course of a numerical experiment, the in-

tegral power characteristics were estimated on the in-

duction machine model for the tray. A characteristic

feature of the constructed models is thei r greater dimen-

sionality and high complexity, since the real geometry

of the tray and the electrophysical characteristics of the

induction machine are taken into account.

The study used a comparison of the two models of

the electromagnetic tray, created fo r the pressure and

non -pressure versions. The steady state is studied at dif-

ferent frequencies. The melt level in the channel is con-

sidered in the range from 0 ( idle) to 1.2 m. The numer-

ical experiment carried out confirmed the hypothesis of

increased eff iciency of the proposed inductor design of

the electromagnetic tray with the ferromagnetic sec-

ondary element.

Conclusions. As a result of the development and

research of parametric numerical models in the ANSYS

software environment, it was possible to eval uate the

differential characteristics of the electromagnetic field

and to study the regularities of the occurrence of phys-

ical processes in the electromagnetic system of tray. In

addition, in the study was received:

1. The integral characteristics of the fiel d made it

possible to evaluate the magnitude of the electromag-

netic force and the energy efficiency of various induc-

tor modifications in comparison with the proposed

short -pole design.

2. Dependencies of the mode parameters of the

low -pole inductor in the tas k of optimizing the tray

modes, as well as practical recommendations for the

operation of the device.

50 American Scientific Journal № ( 21 ) / 201 8

3. Modified dependence of the penetration depth

of the magnetic field in the metal melt, taking into ac-

count additional factors: the magnitude of the non -mag-

netic gap, pole division and the secondary element.

References

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fh^_ebjh\Zgb_ we_dljhfZ]gblgh]h ehldZ ^ey

ljZgkihjlbjh\db jZkieZ\h\ Zexfbgby ^bk «

dZg^l_oggZmd

05.09.01. – Красноярск: ИПЦ СФУ, 2015. –

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2. Верт е, Л. А. Электромагнитная разливка и

h[jZ[hldZ`b^dh]hf_lZeeZ ± М.:

Металлургия, 1967. – 207 с.

3. Гельфгат, Ю. М. Жидкий металл под дей-

kl\b_fwe_dljhfZ]gblguokbe

Ю. М. Гельфгат, О. А. Лиелаусис, Э. В. Щер-

[bgbg ± Рига: Зинатне, 1976. – 251с.

4. Полищук, В. П. Магнитодинамические

gZkhku^ey`b^dbof_lZeeh\ / В. П. Полищук,

FJPbgb^j ± Киев: Наукова думка, 1989. – 257

с.

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Center LP, Lubljana, №19 -1, 2018. – p. 64 -67.

ЭНЕРГЕТИКА

EVALUATION OF DEPTH OF PENETRATION

OF ELECTROMAGNETIC WA VE IN MELTED ALUMINUM

Tyapin A.A.

Postgraduate student, Siberian Federal University,

Svobodny prospect, 79, 660041, Krasnoyarsk, Russia

Andryushchenko V.Y. Postgraduate student,

Siberian Federal University, Krasnoyarsk, Russia

Avdulova Y.S. Assistant of the Department of Electrical

Engineering and Electrotechnology, Siberian Federal University,

Krasnoyarsk, Russia

Goremykin V.A. Ph.D., Associate Professor,

Siberian Federal University, Krasnoyarsk, Russia

О Ц Е Н К А Г Л У Б И Н Ы П Р О Н И К Н О В Е Н И Я Э Л Е К Т Р О М А Г Н И Т Н О Й В О Л Н Ы

В Р А С П Л А В Л Е Н Н Ы Й А Л Ю М И Н И Й

Тяпин Aлексей Андреевич

Аспирант, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект, 79, 660041, Красноярск, Россия

Адрющенко Вадим Юрьевич

Аспирант, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект, 79, 660041, Красноярск, Россия

Авдулова Юлия Сергеевна

Ассистент кафедры Электротехника и Электротехнологии

ФГАОУ ВО Сибирский Федеральный Университет,

Горемыкин Виталий Андреевич

К.т.н., доцент, ФГАОУ ВО Сибирский Федеральный Университет,

Свободный проспект , 79, 660041, Красноярск , Россия

Abstract

Induction machines of small size are used in the modules for pumping molten metal, called electromagnetic

trays. Such devices are created according to the technical task of the metall urgical enterprise using engineering

methods of calculation. To improve the traction characteristics, a flat or cylindrical magnetic circuit is used. How-

ever, for induction machines with an open magnetic core, the asymmetry of the electromagnetic regime is charac-

teristic, which is difficult to take into account in the preliminary calculation. Therefore, to clarify the regime pa-

rameters and preliminary evaluation of the efficiency of longitudinal field inducers, engineers perform numerical

simulation of the magnetic, hydrodynamic and thermal field of the tray on a computer. One of the most important

characteristics of the inductor can be considered the depth of penetration of the traveling magnetic field into the

molten metal.

Keywords

The running magnetic fi eld, electromagnetic tray, longitudinal magnetic field inductor, plane electromagnetic

wave, wave penetration depth into metal, hydrodynamics of aluminum melt, electromagnetic pump, mathematical

simulation.

Аннотация

Индукционные машины малого габарита при меняют в составе модулей для перекачивания расплав-

e_ggh]hf_lZeeZgZau\Z_fuowe_dljhfZ]gblgufbehldZfbLZdb_mkljhckl\Zkha^Zxlkh]eZkghl_ogbq_

kdhfmaZ^Zgbxf_lZeemj]bq_kdh]hij_^ijbylbyk ijbf_g_gb_fbg`_g_jguof_lh^bdjZkqzlZ>eymemq

r_gbyly]h\uooZ jZdl_jbklbdbkihevamxliehkdbcbebpbebg^jbq_kdbcfZ]gblhijh\h^H^gZdh^eybg

^mdpbhgguo fZrbg k jZahfdgmluf fZ]gblhijh\h^hf oZjZdl_jgZ g_kbff_ljby we_dljhfZ]gblgh]h

j_`bfZdhlhjmxljm^ghmq_klv\ij_^\Zjbl_evghfjZkqzl_Ihwlhfm^eymlhqg_gbyj_`bfguoiZ jZf_l

jh\bij_^\Zjbl_evghchp_gdbwnn_dlb\ghklbbg^mdlhjh\ijh^hevgh]hihey\uihegyxlqbke_ggh_fh^_

ebjh\Zgb_fZ]gblgh]h]b^jh^bgZfbq_kdh]hbl_ieh\h]hiheyehldZgZW<FH^ghcba\Z`g_crbooZjZd

l_jbklbdbg^mdlhjZ fh`ghkqblZlv ]em[bgm ijhgbdgh\_gby[_]ms _]hfZ]gblgh]hihey\jZkieZ\e_gguc

f_lZee

Ключевые слова

Бегущее магнитное поле, электромагнитный лоток, индуктор продольного магнитного поля, плоская

we_dljhfZ]gblgZy\hegZ]em[bgZijhgbdgh\_gby\heg\f_lZee]b^jh^bgZfbdZZexfbgb_\h]hjZkieZ\Z

we_dl jhfZ]gblgucgZkhkfZl_fZlbq_kdh_fh^_ebjh\Zgb_

American Scientific Journal № (21 ) / 201 8 45

Formulation of the problem . The use of electro-

magnetic devices for pumping molten aluminum can

overcome the shortcomings of mechanical pumps [1,

p.24]. Unlike mechanical devices, electromagnetic in-

ductors of the longitudinal magnetic field exclude di-

rect contact with the melt, provide high accuracy and

flexibility of control [2, p.45]. One of the most suitable

devices for moving melts is the electromagnetic tray.

However, the effic iency of the tray largely depends on

the depth of penetration of the magnetic field into the

aluminum. In the study of metallurgical equipment, nu-

merical modeling tools are used. However, the prelim-

inary evaluation is carried out according to analytical

expressions [3, p.28]. For induction machines of small

dimensions, known analytical expressions give large

errors [4, p.122]. Using mathematical modeling tools,

the formula for determining the depth of penetration of

an electromagnetic wave of a running magn etic field

can be clarified.

Metallurgical enterprises are equipped with melt-

ing and casting units, which operate on a two -stage or

single -stage process diagrams. Melting and casting

units can differ in the types of furnaces, arrangement of

equipment, feat ures of the technological cycle [1, p.12].

A common feature is the need to pump the melt from

the furnace to the furnace or from the furnace to the

crystallizer, and various devices are used for this. Strict

requirements for pumping melt remain unchanged,

therefore mechanical pumps are considered morally ob-

solete. Among many machines for pumping aluminum

between furnaces, one can distinguish a group of flat

and cylindrical inductors of a longitudinal magnetic

field [2, p.80]. As a rule, they are used in tra ys where it

is not necessary to create powerful devices with a large

depth of electromagnetic wave penetration. This condi-

tion corresponds to shortened induction machines with

a ferromagnetic secondary element. A feature of multi-

phase machines can be consi dered their short -pole and

low -frequency structure [5, p.62].

A sketch of the construction of a flat electromag-

netic tray is shown in Figure 1, a. The tray consists of a

channel (1), a magnetic circuit (2), in whose slots a

multiphase winding (3) is locate d. The induction ma-

chine is placed under the molten metal (4). The side of

the inductor, directly below the channel, is called the

working one. The distance from the surface of the in-

ductor to the melt is usually called the working gap. The

larger the valu e of the working gap, the less the mag-

netic field penetrates into the melt. Conditionally be-

lieve that the inductor is an analog of the deployed sta-

tor of the induction motor (Figure 1, b), on the surface

of which a running magnetic field is created [4, p. 37].

The amplitude of magnetic induction Bz over the

active zone of the inductor varies exponentially (Fig. 1,

c). The intensity of the damping depends on the magni-

tude of the pole pitch, by which is meant the distance

occupied by each of the inductor pole s. The distance

along the inductor, at which the phase changes by 2 π,

is the period of the spatial structure of the winding. At

this distance 2 τ there are two magnetic poles. As the

step of pole increases, the length of the phase zones in-

creases proportion ally. In addition to flat inductors, in

practice, multiphase induction machines with a cylin-

drical magnetic core and disk coils placed around the

core are used [5, p.64].

а b c

Figure 1

The magnetic flux lines crossing the melt induce

eddy currents in it (Fig. 2, b). The interaction of the

traveling magnetic field and eddy currents creates vol-

umetric electromagnetic forces that cause the melt to

move (Figure 2, a). With an increase in any of the pa-

rameters ( , , f) , the increase in force goes to the

maximum region. But further it is seen that the force

decreases and tends to zero [3, p.70] . The diagram of

the distribution of tractive effort is shown in Figure 2,

c.

а b c

Figure 2

The improvement of numerical methods and anal-

ysis tools led to a radical change in the situation when

using numerical models of complex technical systems.

Numerical methods have obtained decisive advantages

due to the evolut ion of computer technology, software

and rapid increase in processing power. The input lan-

guages of modern software are highly developed, the

user interface is improved. This made it possible to sim-

plify the construction of numerical models, up to semi-

auto matic algorithms for their generation. On the basis

of models, an iterative numerical solution of the field

equations is organized [1, p.111] . Software tools for

modeling induction machine regimes are used at the

stage of analysis of the first technical s olutions ob-

tained, often in comparison with the results of a physi-

cal experiment.

Two -dimensional numerical models give a satis-

factory agreement with experiment for axisymmetric

systems with a traveling magnetic field. For planar sys-

tems, the quantitative errors in the calculation of two -

46 American Scientific Journal № ( 21 ) / 201 8

dimensional problems can be excessive, and often lead

to a qualitatively different result. This is due to difficul-

ties in describing in the two -dimensional model the spe-

cific elements of the system, for example, the outer

parts of the windings and their mutual arrangement.

Accounting in the two -dimensional formulation of edge

effects in the melt is also problematic.

The induction machine for the electromagnetic

tray is a complex technical system built by solving in-

terrelated problems in electrical circuits, electromag-

netic, magnetohydrodynamic and heat fields. The study

of heterogeneous objects of high complexity should be

carried out in a complex manner, taking into account

the most significant factors and their interrelatio nships.

In the "inductor -channel" system, this is the mutual in-

fluence of magnetic and hydrodynamic fields on each

other. Their ratio forms the dynamics of the displace-

ment of the aluminum melt in the channel and the dis-

tribution of the resultant electroma gnetic field of the in-

ductor. The results of application of both 2D and 3D

models are presented in the literature [1, p.123].

As a rule, the results of the calculation in a three -

dimensional formulation better coincide with the re-

sults of the experiment. H owever, it was not possible to

find a system assessment of the question of the quality

of docking of two -dimensional and three -dimensional

models in metallurgical problems. Apparently, there is

no quantitative assessment of the reliability and com-

parison o f the accuracy of such calculations for large -

dimensional problems. Therefore, it was not possible to

establish recommendations on the limits of the use of

models. A sketch of the computational domain in the

model intended for the analysis of the electroma gnetic

field in the "inductor -channel" system is shown in Fig.

3.

The working area of the "inductor" of the electro-

magnetic tray is called a flat electromagnetic system.

And in the investigated object it is necessary to accu-

rately estimate the nature of th e moving magnetic field.

It is known that truncated induction machines are char-

acterized by the presence of edge effects (longitudinal,

transverse, input and output). Therefore, it is important

to compare three -dimensional and twodimensional

models and ass ess the degree of coincidence of the re-

sults.

The problem is that when taking into account the

design features of the inductor, the calculation time can

be excessive. Therefore, in the construction of the

model, a number of Figure 3 assumptions and limita-

tithe assumptions made should ons are introduced, but

not distort the real picture of the field. It should be

noted that

three -dimensional problems require substantial

computing resources. Duration of calculations on clus-

ter computers can be hundreds of hours. At the same

time, sometimes, for making a decision, it is necessary

to quickly evaluate the effect of design paramet ers on

the device mode. The question of comparing models is

relevant, since the system under investigation can have

several design options. Therefore, to make a decision,

it is necessary to ensure minimum requirements for

computational resources.

The probl em of analyzing the field must be solved

in the system of Maxwell's equations. For simplicity,

the known method is used and a universal variable is

introduced, the vector potential of the magnetic field Ā.

7KH XVH RI– H[FOXGHVXQNQRZQV IURPWKH V\VWHPDO

lowing one to obtain an equation of one variable.

At the initial stage, the currents of the induction

unit are assumed to be sinusoidal. Therefore, the field

American Scientific Journal № (21 ) / 201 8 47

in the working region can be considered harmonic in

accordance with (1), and the regime parameters are rep-

resented in a complex form. By the equations of a qua-

sistationary field, one proceeds to a simplified, flat

twodimensional or three -dimensional formulation. The

magnetic induction vector is consider ed to be located in

the plane of the tray, while the vector of the electric

current density J and the vector magnetic potential A

are orthogonal to it. In the twodimensional formulation,

the components Jy and Ay are nonzero, and in the

threedimensional for mulation there are all components.

The solution of the field equations (2, 3) is a

boundary value problem. For its correct description,

boundary conditions are applied and the equations are

solved together with the boundary conditions. Next, the

traction characteristics of the electromagnetic field are

calculated. For this, engineers usually use specialized

software environments [5, p.66].

Calculation of the electromagnetic field in the

working region is pe rformed by the finite element

method. For this, the software Ansys Multiphysics is

used. When building models, the programming lan-

guage APDL was used. Conversion of input and output

information flows is performed in ASCII code, which

is beneficial from the point of view of the formation of

hybrid models. At the same time, integration of third -

party calculation models with external software mod-

ules is performed. In addition, the ASCII code is con-

venient for processing the calculation results. The mod-

els desc ribed here are formalized in the format of inter-

nal program code and additional modules - macros.

Macros act as docking nodes for heterogeneous tasks,

and also carry service options to support the execution

of non -standard functions [1, p.55].

At the first stage, to accept the decision on the

model, the launch regime was investigated. The elec-

tromagnetic force in the starting mode determines the

starting pulse necessary to start the melt movement.

The intensity of metal movement depends on a large

number of criteria. These include linear current load,

frequency, working gap, pole division, melt parame-

ters, magnetic circuit characteristics, etc. Approxi-

mately 50 criteria are adopted. Identifying the most sig-

nificant dependencies and their formalization requir es

a large number of calculations and leads to optimization

problems. Consequently, already at this stage the ques-

tion of the applicability and adequacy of two -dimen-

sional or three -dimensional models becomes relevant.

In the study, two parametric models of the "induc-

tor -channel" system were developed and tested: in two -

dimensional and three -dimensional formulations. To

compare the calculation results, the geometry parame-

ters and the power supply mode of the model are set to

the same. Some results are shown in Fig. 4.

The regulation of the supply voltage frequency has

shown that the tangential force F τ has a characteristic

optimum. And for the two -dimensional case (curve 1),

the optimum is located at a frequency of 3 Hz, and for

a three -dimensional (curve 2) at 17 Hz. The difference

in effort values is 53 %. In addition, the behavior of the

curves is significantly different. The nature of the re-

gion of optimal values along curve 1 looks relatively

high -quality.

Curve 2, on the other hand, shows a low selective

extremum. At a frequency of 23 Hz, the curves inter-

sect, and further curve 1 is located below curve 2 (~ 7

%). Thus, the results of calculating the electromagnetic

force by the twodimensional and three -dimensional

models are different.

Further studies sho wed a different character of the

distribution of the amplitude value of the magnetic in-

duction in the melt at frequencies below 12 Hz. The in-

duction

Figure 4

48 American Scientific Journal № ( 21 ) / 201 8

distribution for the three -dimensional model is

more uniform along the length of the channel of

the melt and has no characteristic dips. The two -

dimensional model is characterized by too high induc-

tion values (up to 76 %). This explains the higher cal-

culated values of the forces in Fig. 4, but does n ot ex-

plain the change in the nature of their distribution.

The presence of previously developed prototypes

of induction machines facilitates the task of comparing

the results of numerical simulation. Despite the fact that

the samples of longitudinal field inducers were manu-

factured without the use of optimization algorithms, the

main regularities of the force effect for the threedimen-

sional models were basically confirmed. To power

physical samples, a transistor IGBT converter with a

capacity of 50 kVA was used, and the devices them-

selves were mounted on an experimental bench.

The two -dimensional model allows to take into ac-

count the input and output edge effects (x -axis). But

does not take into account the transverse effect associ-

ated with a change in the direction of the vortex current

vector by 90 °, along the lateral surface of the channel.

In addition, due to the presence of frontal parts of wind-

ing, there is a distortion of the distribution of the mag-

netic field within the active part of the inductor ( y axis).

Therefore, two -dimensional numerical models for the

analysis inductors of short -pole linear induction ma-

chines in such a formulation should not be used. For the

adoption of technical and economic solutions, it is nec-

essary to form a full -fledged t hree -dimensional numer-

ical model at the stage of preliminary calculations and

to calculate the mode parameters, taking into account

the most important factors determining the device's ef-

ficiency.

To quantify the limit values of the starting angles

of the e lectromagnetic tray, a thorough study of the set

of criteria, the construction of a detailed model, and the

refinement of the research algorithm are required. In

addition, a large amount of computation and a system

analysis of the results are required. The refore, the study

of the launch regime is allocated to a separate project.

In the theory of induction machines, the concept

of the depth of penetration of the magnetic field into

the melt is used. If the thickness of the metal is much

greater than , then in the most remote layers of the melt,

backflows can occur and condition

(1) will not be fulfilled. Therefore, take the condi-

tion hk = 1.41 , at which the optimum mode of operation

of the inductor takes place. According to the literature,

the penetration d epth is determined by the expres-

sion:

. (4)

The calculation results from expression (4)

showed a significant difference from the values de-

scribed above. For 1.41 = 50 mm, the frequency is f

= 40 Hz. But with a decrease in thickness, the depend-

ence is shifted to the range of kilohertz. Consequently,

the use of classical expressions does not accurately de-

scribe the electromagnetic mode of the induction ma-

chine.

The study [1, p.76] shows an expression that clar-

ifies this discrepancy. The justification is the indication

of the evaluation of the results for a plane electromag-

netic wave (3). The conce pt of a plane wave assumes

that . This assumption is inapplicable for a real short -

pole inductor. Therefore, the calculated values of pen-

etration depth differ significantly from those measured.

For their correct definition, the following expression is

used:

(5)

The structure of expression (5) indicates that the

depth of penetration depe nds not only on the conduc-

tivity , the frequency f of the supply voltage, but also

on the magnitude of the pole division . However, anal-

ysis of the results of numerical simulation showed that

the calculated values of the frequencies also signifi-

cantly exce ed the results obtained from expression (4).

A detailed quantitative evaluation shows the unsuitabil-

ity of the analytical expression (5) for determining the

penetration depth of the wave developed by the induc-

tor of the electromagnetic tray. Studies have s hown that

taking into account the non -magnetic gap Δ leads to a

sharp change in the location of the optimal regime char-

acteristics, whose behavior is shown in Fig. 6. This

shows the necessity of taking into account the influence

of Δ in expression (5). To refine the analytical expres-

sion (5), a series of approximation procedures is per-

formed, using splines. As a result, we obtained a refined

expression for estimating the penetration depth of an

electromagnetic wave, which makes it possible to reli-

ably descr ibe the result.

(6)

where ξ π / τ, is the correction factor of the

penetration efficiency of the running electromagnetic

wave through the non -magnetic gap; k = 1 ÷ 2 - the co-

efficient of influence of the secondary element on the

penetration depth of the electromagnetic wave.

The obtained analytical expression (5) shows the

corrected regularity of the distribution of values of the

penetration depth of the wave, for the considered class

of electromagnetic trays. The formula mak es it possible

to calculate the penetration depth at a given frequency,

American Scientific Journal № (21 ) / 201 8 49

taking into account the joint influence of the essential

technological parameters of the "inductor -channel"

system. The product of the newly introduced coeffi-

cients ξ and k in the form ula (5) integrally takes into

account the effect of the physical parameters of the de-

vice and the geometric characteristics of the short -pole

machine. For the considered design designs, with the

secondary element and without it, on the basis of mul-

tivariat e numerical studies the values of the coefficient

k = 1 ,2 and 1,8, respectively, were adopted.

Comparison of calculation results by analytical

expression and results of numerical simulation con-

firmed the high degree of coincidence of not only dis-

crete val ues, but also accurate reproduction of the mo-

notonous regularity of distribution of optimal values of

fopt and hк for the system under study. In addition, a

comparison of the results of the calculation of the re-

gime characteristics with the results of a ph ysical ex-

periment on the trays from a number of designed prod-

ucts. Comparison showed an acceptable quality of co-

incidence. Therefore, the need for preliminary numeri-

cal simulation with long and resource -intensive itera-

tive calculations has been exhausted.

The dependence of the optimal tangential force Fτ

on the melt height hk for the case with the secondary

element (2) and without it (1) is shown in Fig. 5. De-

pendences of the optimal frequencies fopt on the melt

height hk for the case with the secondary ele ment (2)

and without it (1) is shown in Figure 6. The behavior of

the regime parameters of the induction machine is im-

portant to evaluate when searching for the optimum fre-

quency. Automated search of optimal values of fre-

quency fopt, allowed to conclude th at all dependences

are essentially nonlinear (Figure 5). An increase in hk

leads to an exponential decrease in the frequency. And

for both constructive options the behavior is the same.

The displacement of the curves is 26 % in practically

the entire range .

Figure 5 Figure 6

Based on the results of the study, engineers can

formulate generalized recommendations for construct-

ing an effective tray design. At the same time, it is pos-

sible to take into account a combination of technologi-

cal, energy, constructive and cost factors. In particular,

an inductor with a magnetic core of circular cross sec-

tion is proposed. It makes it possible to simplify the de-

sign, facilitate the device, improve cooling condit ions,

and also solve the problem of parametric synthesis of

operating modes in a simplified formulation.

For an inductor with an ascending channel, the

first criterion of the efficiency of work is the minimum

starting level of the melt. The determining fac tor is the

traction force F in the channel. The value determined

from the condition for balancing the hydrostatic pres-

sure force. It should be noted that, for pressure electro-

magnetic trays, the starting level of the melt is critical

only at the launch sta ge. For non -pressure trays, it is

important throughout the entire process. The inductor

of the electromagnetic tray has an uneven density of

electromagnetic forces along the length of the channel.

The lowest density is characteristic for the zones of the

beginning and end of the inductor, and also beyond its

limits. In the course of a numerical experiment, the in-

tegral power characteristics were estimated on the in-

duction machine model for the tray. A characteristic

feature of the constructed models is thei r greater dimen-

sionality and high complexity, since the real geometry

of the tray and the electrophysical characteristics of the

induction machine are taken into account.

The study used a comparison of the two models of

the electromagnetic tray, created fo r the pressure and

non -pressure versions. The steady state is studied at dif-

ferent frequencies. The melt level in the channel is con-

sidered in the range from 0 ( idle) to 1.2 m. The numer-

ical experiment carried out confirmed the hypothesis of

increased eff iciency of the proposed inductor design of

the electromagnetic tray with the ferromagnetic sec-

ondary element.

Conclusions. As a result of the development and

research of parametric numerical models in the ANSYS

software environment, it was possible to eval uate the

differential characteristics of the electromagnetic field

and to study the regularities of the occurrence of phys-

ical processes in the electromagnetic system of tray. In

addition, in the study was received:

1. The integral characteristics of the fiel d made it

possible to evaluate the magnitude of the electromag-

netic force and the energy efficiency of various induc-

tor modifications in comparison with the proposed

short -pole design.

2. Dependencies of the mode parameters of the

low -pole inductor in the tas k of optimizing the tray

modes, as well as practical recommendations for the

operation of the device.

50 American Scientific Journal № ( 21 ) / 201 8

3. Modified dependence of the penetration depth

of the magnetic field in the metal melt, taking into ac-

count additional factors: the magnitude of the non -mag-

netic gap, pole division and the secondary element.

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