International Scientific Colloquium
Modelling for Electromagnetic Processing
Hannover, September 16-19, 2014
Frequency dependence of an
alternating magnetic field driven flow
A. Cramer, V. Galindo
Abstract
The flow induced by a single-phase alternating magnetic field (AMF) is studied
mainly numerically, and by experiments at mains frequency. For validation, the well-known
dependence of the characteristic velocity υc on the magnetic induction B in the turbulent case
when the mean velocity scales with the shear velocity, υc ∝ B, is reproduced experimentally
and in the simulations. Ultrasonic flow mapping reveals that the eddies in the well believed
flow structure of two toroidal vortices one on top of another do not only simply oscillate, but
rather the topology of the flow changes slowly on a large scale. Besides such turbulence
characteristics, it will be shown that the global flow structure depends strongly also on the
frequency f of the AMF. An investigation of the change of υc with f suggests a quantitative
difference of flows in an AMF compared to rotating and travelling magnetic fields.
Introduction
The flow created by an imposition of an alternating magnetic field (AMF) on
electrically high conducting liquids receives attention in the field of metallurgy since quite a
long time. In 1976, Tarapore & Evans studied the flow in induction furnaces [1]. They
conducted fluid velocity measurements in an inductively stirred mercury pool of radius
R = 0.1445 m at a fixed frequency f of 3 kHz. The shielding parameter S = µσωR2, which
measures the crowding out of an electromagnetic field by its own eddy currents from the
interior of an electrical conductor, was 514. In the definition of S, ω = 2πf, µ is the
permeability, and σ the electrical conductivity. For such a high S » 1, the field does not
penetrate the conductor; it is expelled to a thin skin at the surface. The measurements in [1]
served the validation of the numerical code, the flow structure was calculated. It consists of
two toroidal vortices one on top of another, is symmetric in the axial direction, and the rolling
direction of the vortices is from the rim to the centre at half the height of the fluid cylinder.
Taberlet & Fautrelle (1985) varied the frequency in a similar configuration in such a
way that S was 3.9, 30, 166, and 372 [2]. Except for S = 30, where a small third counterrotating toroidal vortex appeared in the upper corner around the rim, the two vortices
described in [1] were observed. Opposing to there, they were not of equal size. For S = 3.9
and 30 the lower one was much larger than the upper one, and vice versa for S = 166 and 372.
While stating that the flow structure is determined by the particular distribution of the Lorentz
force, which distribution significantly changes with frequency, a perspicuous explanation for
the intermediate appearance of a third vortex and the reversal of the size of the eddies was not
provided. Regarding the dependence of the characteristic velocity υc on S, the maximum
υcmax(S) was found for S = 30. υc for S = 3.9 is much smaller than those for S = 166 and 372,
i.e., the decrease towards higher S is significantly flatter than the increase for smaller S.
Recently, studying flows in inductively processed melts gained stimulus from the field
of solar silicon block casting. A new crystallisation furnace is proposed in [3–5], in which the
traditional heating by resistance is replaced by induction heating. It is not only that the flow
driven by a single-phase alternating magnetic field obviously influences the shape of the
solidification front, thereby leading to higher efficiency photovoltaic material. Based on the
furnace described in [3–5], a project within the Framework Programme 7 of the European
Commission was yet launched to re-use silicon saw dust accruing during slicing the block
casts to wafers [6]. The challenge in this project is cleaning the silicon melt from polluting
silicon oxide and silicon carbide particles. Means for such cleaning is an electromagnetic
separation process (ESP), in which the non-conducting particles are moved to the vertical
crucible wall by Leenov-Kolin force (LKF) where they are accumulated in a dirty skin during
solidification. As the LKF is applied via an AMF, one is once more concerned with the flow
driven by this type of magnetic field.
The influence of flow on the ESP is discussed controversially in the literature; a too
strong flow inhibits ESP, whereas some flow is needed to move the particles from the bulk of
the melt to the crucible walls where they can be captured by the LKF. It is however not only
the vigour, but also the topology and the turbulence characteristics of the flow affecting the
effectiveness of the ESP. Two successful model experiments on ESP are reported in [7, 8].
Despite the high frequency mentioned in the title, S was in a moderate range owing to the
small scale of the experiments. Besides reporting on turbulence characteristics, which were
not subject in [7, 8], the present paper revisits the parameter range of the experiment in [8]
done on silicon.
The manuscript is organised as follows. Sections 1. and 2. describe the experiment and
the numerical procedure, respectively. Findings are reported and discussed in Section 3.
Besides a summarising interpretation, the Conclusions outline future work planned on flows
driven by single-phase alternating magnetic fields.
1. Experimental setup
All measurements have been carried out in
the cylindrical container of 20 mm in radius and 60
mm in height depicted to the left in Fig. 1.
Although the magnetic system allows for a height
covering more than that of the melt volume as it
was the case in [1, 2, 7, 8], only the central part of
the segmented coil was used. Since limitations
were only imposed by the maximum available
power of the current source, which could be fed to
that one segment, the choice of the small height of
the coil provided the maximum possible vigour of
stirring. This is due to the particularity of an AMF
that the magnitude of the rotational part of the
Lorentz force depends not only on the strength of
Fig. 1. Drawn to true scale sketch of the
the magnetic field but on its gradient as well.
fluid container, the magnetic system, and
The velocity measurements were done by
the measuring unit.
ultrasound Doppler velocimetry (UDV). A transducer installed at the top with the ultrasonic beam directed downward measures the vertical
velocity component υz. Traversing the transducer allowed mapping mean properties of the
flow field. For similar measurements with a description of the principles of UDV see [9].
2. Numerical procedure
For the numerical simulations of the induced electric current j and magnetic induction
B in electrically conducting regions, the finite element code OPERA 3d (Cobham plc., [10])
was used. The computational grid was refined near the walls in order to resolve the skin layers
and has a total number of 2 million finite elements.
The flow in the volume containing the melt was simulated numerically by means of
the open source library OpenFOAM [11] solving the Navier-Stokes equation together with the
incompressibility condition ∇ ⋅ υ = 0 and including an electromagnetic force density term
averaged over one period T = 1/f:
ρ ⋅
ρ, p, and η are the density, the
pressure, and the dynamic viscosity. The
boundary conditions for the flow field is the
no-slip condition υ = 0 at the solid container
wall. For the melt surface, either υ = 0 or the
conditions for a stress-free non-deformable
surface υn = 0 and ∂υt / ∂z = 0 (the subscripts
n and t denote the normal and the tangential
component, respectively) are applied
depending on whether the melt flow is to be
calculated in an open or an enclosed
container. A computational grid with 650000
volume elements was used. The discretisation scheme for the convective term
was of second order.
η
Fig. 2. Dependence of velocity on induction.
3. Results and discussion
3.1. Characteristic velocity vs. induction
Numerical calculation and measurement of the dependence of υ
on B was done mainly for validation purposes. Since the experimental
setup allowed measuring υz over the height of the container, the
maximum υzmax(z) along the ultrasonic beam was taken as a representative for υc. The computations were done with a stationary solver
where possible. Higher values of B required transient calculation and
using a turbulence model, which was the simple k–ε one.
Fig. 2 shows that the well-known linear relation for υc continues
to be true for υz measured on the cylinder axis: υzmax(z) ∝ B. The
numerically obtained results are consistent between the stationary and
transient calculations. Agreement can be stated between the
computations and the experiments.
3.2. Turbulence
It is reported in the numerical work [12] that the time Fig. 3. Relative (top)
dependence of an AMF driven flow in configurations similar to the and absolute stanpresent one consists in an oscillation of the mean flow eddies (c.f. Fig. 1 dard deviation of υ.
there). This means, however, that the mean flow structure of two vortices one on top of
another is preserved. Further findings were that these oscillations show a sharp peak in the
spectrum at quite low frequencies, and that the flow is most unstable in between the vortices
adjacent to the rim. The low frequency peak was reproduced in the present work at about 0.1
Hz; a spectrum is not shown due to space limitations. As the turbulence degree, in its strict
definition, is meaningless in a recirculating flow, Fig. 3 plots a modification thereof calculated
from the local standard deviation sd and the local time averaged velocity (upper panel). The
lower panel shows sd for comparison—note that only a quarter of a central section through
the cell is plotted. A free-hand drawn rough course of streamlines should serve as a guide to
the eye. Both mappings of statistical properties largely support the findings in [12], except
that the region of strong fluctuations extends farther in both directions, vertically towards the
eye of the vortex and radially towards half the radius.
Fig. 4. Equidistant in time series of vertical sections of υz(z) at r = 15 mm. For pictorially
better comprehendible presentation, abscissas and ordinates are exchanged.
The series of υz(z) in Fig. 4 evinces that one is not concerned with mere oscillations of
the mean flow eddies. If it were so, the shape should be an “S” as, e.g., in the second panel,
with changing amplitudes and the root moving up and down. Whereas, the third panel exhibits
only an upper vortex and the existence of two or even three roots (panels 1 and 5) makes
evident that eddies are created and destroyed. That is to say, the well believed flow structure
exists only as a long-term average and is probably never realised at any instant in time.
3. 3. Dependence of velocity on frequency
It is well known that, keeping B
constant, a maximum of υc is observed for an
f where S is in the range from 5 to 10 in the
case of rotating (RMF) and travelling magnetic fields (TMF). Among their coarsely
spaced S, Taberlet & Fautrelle found υcmax(S)
at S = 30 [2]. This is noticeable since
shielding, i.e. expelling an AC field from a
conductor thereby diminishing the Lorentz
force, depends on f, only.
These authors published the dependence of υc on S in a joint work with a
numerical group [13], based on the experiments in [2], one year later. υcmax(S) was
Fig. 5. Dependence of the maximum velocity
found at S about 80 in the calculations and,
on f. The upper abscissa translates the
surprisingly, a second local maximum at
frequency to the shielding parameter S.
higher S appeared. The interest in the present
work is in the range of S for which successful ESP experiments were done. Figure 5 plots the
numerically obtained maxima of |υ| for the conditions in [8]. A Levenberg-Marquardt fit
suggests that |υ|max(S) > 40. The difference for 10 and 11 kHz is noticeable; both solutions,
albeit the velocity fluctuates distinctly, are converged with respect to the total kinetic energy.
Such a significant difference may be attributed to a change of the global flow structure. This
aspect will be returned to in the next section, where it is discussed in conjunction with the
obvious result that υcmax(S) for an AMF is distinctly higher than those for an RMF and a TMF.
3.4. Lorentz force and flow structure
It is well known that the flow structure evoked by an RMF neither depends significantly on f, nor on the axial distribution of the Lorentz force FL. The movement of an
electrically conducting fluid follows the azimuthal FL, accompanied by a secondary poloidal
flow due to Ekman pumping. Shielding affects essentially the vigour of the flow. From own
experience, it turns out that the same
holds for the axial forcing in a TMF.
That is to say, the more complex
spatially-temporarily
alternating
fields drive a comparably simple
flow. To imagine why a simple singlephase, i.e. not moving in space, AMF
may lead to rather complex flow
patterns,
an
axially
infinite
configuration is considered. The fluid
stays at rest because ∇×FL = 0. Said
Fig. 6. Lorentz force distribution for 50 Hz (left) and in words, the AMF produces only
for 11 kHz (right). Plotted is the Lorentz force density pressure. In any finite system, it is the
divided by the density, so the dimension of F is m/s2. spatial variation of FL which determines the global flow structure.
Iso-plots of |FL| for the system investigated in [8] are shown in Fig. 6. 50 Hz and 11
kHz correspond to S = 0.19 and 42.7, respectively. The change of |FL| with S is remarkable,
albeit not in the focus here. The interest is in the topology of the flow, which is to be seen in
Fig. 7. Also at mains frequency, |FL| is maximal in the corners. Despite this, a considerable
central portion of the height where |FL| is sufficiently high drives the fluid inwards since FL is
made mainly of the radial component. The
fluid consequently moves opposing to the
Lorentz force at the top and at the bottom, the
well-believed double-torus is observed.
Shielding obviously does not only expel
B simply radially outwards. The magnetic field
is also crowded out in the axial direction, with
the consequence that FL loses dominance in
the same central-axial region that determined
the flow at mains frequency. Driven all but by
the strong force in the corners, four vortices
one on top of another establish. That the flow
Fig. 7. Plot of velocity vectors for the same appears asymmetric in the vertical direction is
a matter of the plot depicting the flow at a
frequencies as in Fig. 6.
certain instance in time; also the lower of the
two central vortices is the larger one at times. Loss of axial symmetry is to be seen at the
different height of the eyes of the vortices along the circumference. This may explain the
significant difference in |υ|max(S) between 10 and 11 kHz. The flow is still almost axisymmetric
at 10 kHz. While increasing f merely by 10 %, the asymmetry with respect to the cylinder
axis becomes significant.
Conclusion and perspective
Being subject of investigation for decades, the flow driven by an AMF is still far from
being completely understood. Even restricting to the case where the axial extension of the coil
is much larger than that of the fluid volume, which quasi generic configuration will hardly be
realisable for silicon recycling, both dependences of the characteristic velocity and flow
topology on f are not precisely know. Besides more parametric numerical studies on different
geometries aiming at suitable ones for recycling purposes via separation, experiments are
badly needed, the latter of which should be capable to visualise the flow structure. Mapping
of turbulent flow by means of ultrasonic Doppler anemometry is readily available [14].
Acknowledgement
This work was financially supported by the European Commission in the Framework
Programme 7 ENV-2013 under the grant agreement number 603-718.
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Authors
Dr. Cramer, Andreas
Institute of Fluid Dynamics
Helmholtz-Zentrum Dresden-Rossendorf
Bautzner Landstraße 400
D-01328 Dresden, Germany
E-mail: a.cramer@hzdr
Dr. Galindo, Vladimir
Institute of Fluid Dynamics
Helmholtz-Zentrum Dresden-Rossendorf
Bautzner Landstraße 400
D-01328 Dresden, Germany
E-mail: v.galindo@hzdr.de