1. Introduction
Currently, due to the increased requirements for the energy efficiency of electrometallurgical plants, there is a steady trend towards an increase in their capacity. In connection with this circumstance, in practice, one has to deal with the enhancement of the influence of various magnetohydrodynamic (MHD) effects and, in particular, the generation of so-called electrovortex flows (EVF) in a current-carrying liquid—a metal melt [
1]. The presence of EVF, which are formed as a result of the interaction of an electric current passed through an electrically conductive liquid, with the intrinsic magnetic field (MF) of this current, leads to a radical restructuring of the hydrodynamic structure in the volume of the melting metal. This circumstance significantly affects energy consumption and the quality of products, which determines the paramount importance and relevance of the study of electrovortex flows. The experimental study of electrovortex flows in industrial conditions is very difficult due to the complexity of measuring the velocity in a high-temperature and aggressive liquid metal environment, and the computational methods for studying EVF require additional verification.
Below we give a brief overview of the methods for measuring the velocity in liquid metals (LM) with the justification of their application under certain conditions. All methods for measuring velocity in a LM can be divided into several categories: probe and non-contact, methods that allow to measure average values or pulsation characteristics, and methods that measure average values in space or allow measurements in a small volume of finite size. At the same time, if in laboratory conditions the problem of measuring the velocity in a liquid metal can still be solved relatively successfully due to the use of various model metals and alloys that retain a liquid state at room or low temperatures (mercury, sodium, tin, and various gallium alloys), then in industrial conditions, at temperatures reaching 1000–2000 °C, few workable methods are known.
The Pitot tube is the oldest method for measuring the velocity of liquid media. This method is ineffective due to the high density of liquid metals (except sodium), it allows you to obtain only average values of the velocity and, like any probe method, obstructs the flow; hot-wire anemometer—ineffective in non-isothermal environments and environments with high thermal conductivity, which include liquid metals (nevertheless, there are attempts to use this method in relation to LM [
2]; the “Lorentz Force Velocimetry” method [
3] based on measuring the force acting with the side of a moving conductive liquid on a permanent magnet, allows you to measure only the space-averaged values of velocity (flow rate); the ultrasonic Doppler anemometer [
4] combines the advantages of contactlessness and the ability to measure at a point (measurement occurs along the beam), but, firstly, there are limitations associated with the orientation beam on an installation or industrial unit, and, secondly, this method is extremely demanding on the composition of the alloy—particles (usually oxides) from which ultrasound is reflected, for this reason, UDA is usually used to measure the velocity in gallium alloys, which oxidize well in air and oxide particles fall into the volume of LM; and fiber-optic method [
5]—a probe method that allows performing fine measurements (including velocity pulsations) in a small measuring volume (40 × 40 × 500 microns), but having temperature limitations (up to 100 °C).
In our work, we use the thermocorrelation method [
6,
7] to measure the velocity in an electric vortex flow in a hemispherical container. Apparently, this is the first work where this method is used to measure velocity in a current-carrying fluid. This problem, about the study of the hydrodynamic structure of the flow caused by the spreading of electric current from a point source into a hemispherical volume filled with liquid metal, occupies a special place among the model studies of EVF. This geometry of the working bath has a number of important practical advantages and features, which make it possible to study EVF in its most general form. These features include, first, the existence under these conditions of analytical dependences for the current density and magnetic field. Secondly, this geometry is typical for industrial tasks associated with metal remelting (electroslag welding, and electric arc and electroslag metal remelting). Third, an interesting effect of swirling of an axisymmetric electric vortex flow is observed on the hemispherical model.
The thermal correlation method is a probe method and allows you to measure only the average value of the velocity during a certain specified time interval (in our conditions, the duration is 30–120 s). At the same time, the probe is simple in design and can be used in industrial installations. The use of high-temperature, tungsten–rhenium or platinum–rhodium thermocouples in the probe makes it possible to measure the velocity of liquid metal at temperatures up to 2500 °C. We use the thermocorrelation method in a laboratory setup.
Despite a significant number of studies of EVF, analytical [
8], numerical [
9], and experimental [
10], many questions remain insufficiently studied, which is associated with both the complexity of the object itself and the laboriousness of the measurements.
2. Mathematical Description of Processes
As mentioned above, EVF are formed as a result of the interaction of an inhomogeneous electric current of density
J with its own magnetic field
B (see
Figure 1). The Navier–Stokes equation describing the hydrodynamics of an electric vortex flow has the form
where
U—velocity,
t—time,
ρ—density,
ν—kinematic viscosity,
F =
J ×
B is the electromagnetic force that causes the movement of an electrically conductive fluid.
For a hemispherical geometry, in which the current density
Jr depending on the radius
r (in the spherical coordinates), changes as
from Maxwell’s equation
the expression for the distribution of the magnetic field can be easily obtained
Then the expression for the electromagnetic force in a liquid metal has the form
The system can be characterized by the following dimensionless parameters
and this parameter determines the intensity and flow regimes.
At currents > 30 kA (as shown in [
11]), or with an external axial field of more than 0.0025 T [
12], we must also take into account the currents induced by the motion of the liquid, and then the expression for the current density will have the form
here
μ0—magnetic constant,
φ and
θ azimuthal and polar angles in the spherical coordinates, σ is the conductivity of the medium,
E is the electric field, and, in the general case, it is also necessary to take into account the magnetic fields of the induced currents and to solve the complete magnetohydrodynamic problem including equations for the magnetic field.
A schematic diagram of the formation of the EVF in a hemispherical container is shown in
Figure 1. In the absence of external magnetic fields, the magnetic field of an axisymmetric system has only one component of the magnetic field (azimuthal) and, accordingly, one component of the force (poloidal). In a hemisphere, such a force creates a toroidal flow directed downward on the axis.
External magnetic fields interacting with an electric current passing through a liquid metal also create electromagnetic forces, then B = Bown + Bext, where Bown is the own magnetic field of the current, Bext is an external magnetic field, for example, the magnetic field of the Earth or current leads.
At the latitude of Moscow, the Earth’s magnetic field has an inclination of 19° and a magnitude of ~52.175 μT (Data of The Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences). Thus, the vertical component of the magnetic field is ~49 μT. The vertical component of the magnetic field
Bz interacting with the radial component of the current density creates an azimuthal component of the electromagnetic force
For clarity, the expression is given in cylindrical coordinates (here
JR is a radial component of current density), in spherical coordinates, the expression for the force is
In a converging flow, this component creates a significant force and the study of the influence of the Earth’s magnetic field on the EVF swirl is discussed in [
13,
14,
15].
In experiments, the Earth’s magnetic field can only be compensated by an oppositely directed magnetic field.
Author Contributions
Conceptualization, methodology, validation, formal analysis I.T. and D.V.; software, D.V.; writing—original draft preparation, I.T.; writing—review and editing, supervision, Y.I. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (State Assignment No. 075-00460-21-00). The authors are grateful to K. Kubrikov and P. Polyakov for their help in preparing the experimental setup and preparing the article.
Conflicts of Interest
The authors declare no conflict of interest.
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