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
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

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
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.
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
Индукционные машины малого габарита при меняют в составе модулей для перекачивания расплав-
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Ключевые слова
Бегущее магнитное поле, электромагнитный лоток, индуктор продольного магнитного поля, плоская
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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,
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.
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-

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.
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-
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-
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-
. (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-
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

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.

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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