Flow measurement in liquid metals using Lo- rentz

Flow measurement in liquid metals using Lorentz force velocimetry: laboratory experiments
and numerical simulations
Dissertation
zur Erlangung des akademischen Grades
Doktoringenieur
(Dr.-Ing.)
vorgelegt der
Fakultät für Maschinenbau der
Technischen Universität Ilmenau
von Frau
Dipl.-Ing. Dandan Jian
geboren am 13.08.1978
in LiaoNing/China
Acknowledgements
I would like to express my immense pleasure to thank my supervisor Dr. Christian Karcher for the guidance and supporting throughout the research during the
past years. He was always ready to discuss my work, but let me free to pursue
my own goals in my own way. It has been a great pleasure to have him as a
mentor. Thanks for his valuable suggestions, constructive criticism, and his
great patience during the correction phase of this dissertation. His continued
support led my future self-development to the right path. Also I would like to remark upon his recognition, trust, and career advancement to me.
I would also like to express my sincere gratitude to Prof. Peter Ehrhard, not only
for taking his enthusiasm for my work, but also for organizing a very interesting
scientific talk on my studies in his institute. Moreover I appreciated the many
discussions with him, in which he was always full of ideas. Furthermore, it is a
pleasure to thank Prof. Klaus Zimmermann for his kind willingness to referee
this thesis and to take part in the examination committee. Thank him also for
providing constructive suggestions for presentation of research results.
Last but not the least, I would like to express my sincere gratitude to Prof. Andre
Thess, director of the institute of Thermodynamics and Fluid Mechanics, for giving me the opportunity to start research in the topic Time-of-Flight Lorentz Force
Velocimetry and for expanding my horizon in science and business. The support
provided by Dr. Christian Resagk is also acknowledged for his moral aid, his
motivation and technical advice over the past years.
I benefited immensely from the collaboration with Jörg Schumacher and Steffen
Badtke. It is in my opinion a rare privilege to discuss ones ideas at the time they
are being formed with someone able of grasping them in full and providing
feedback. With both I could discuss all my work in depth and they contributed
numerous useful ideas. Furthermore I gratefully acknowledge the discussions
with my students Alexander Schäfer, Sebastian Schädel that contributed to the
research reported in this thesis. Also, I thank them for their assistance in the
laboratory experiments.
Finally, special thanks goes to my husband Jörg Schumacher for understanding
and encouragement, my sister Weiwei Jian, who has always stand by me and
my parents, who are proud of me and I express my deepest thank for their unlimited patience.
The author gratefully acknowledges the financial support from the Deutsche
Forschungsgemeinschaft (DFG) in the framework of the Research Training
Group “Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing”
(grant GRK 1567) as well as by Bundesministerium für Bildung und Forschung
(BMBF) within the ForMaT2 program.
Ilmenau, Nov. 2012
Dandan Jian
Abstract
Non-contact flow measurement in hot and aggressive metallic melts is a big
challenge in metallurgic processes including continuous casting of steel and
production of secondary aluminum, among others. Due to aggressiveness of
metal melt, only non-contact measurement methods can be applied. Lorentz
Force Velocimetry (LFV) is an electromagnetic measurement technique to meet
these challenges. LFV is based on the principle of magnetofluiddynamics
(MFD). A respective flow meter which exploits these principles consists of a
magnet system with an attached force sensor that records the accelerating
force produces by the flow. This electromagnetically induced force is proportional to the flow rate or the local velocity in the vicinity of the magnet system.
A drawback of LFV is that the measured force depends on both the electrical
conductivity of the fluid and the strength of the externally applied magnetic field.
Therefore, an elegant method to circumvent this deficit is the so-called Time-ofFlight technique. During this technique two identical Lorentz force flow meters
are arranged one behind the other separated by certain distance. Time-of-Flight
Lorentz Force Velocimetry (ToF LFV) senses the passage of the triggered vortices at LFV 1 and measuring the time that the vortices need to travel to LFV 2. In
this case the flow rate may be purely determined by cross-correlating the force
signals provided by the two flow meters. The objective of the present investigation is to demonstrate experimentally that ToF LFV could be a promising tool for
measuring the local velocities and flow rates in liquid metal flows.
This thesis presents a series of experiments; first of all, the flow rate measurements in the turbulent liquid metal flow were performed on experimental facility
EFCO (Electromagnetic Flow Control Channel). Furthermore, a special measuring device termed Meniscus Velocity Sensor (MVS) relying on ToF LFV technique has been designed to record local surface velocities in metallic melts. The
method has been successfully tested using both solid body movement and liquid metal Ga68%In20%Sn12% on annulus channel LiMeSCo (Liquid Metal Surface
Velocity Correlation Measurement). In more detail, test measurements under
industry-relevant conditions using both metallic melts Sn32%Pb52%Bi at about
210°C and steel at about 1700°C were to be reported. Finally, to each experiment we present results of numerical simulations in order to get a deeper insight
in the physical principles of the magnetofluiddynamics.
Zusammenfassung
Die Durchflussmessung von heißen und aggressiven Metallschmelzen ist eine
große Herausforderung bei metallurgischen Prozessen, Beispiele hierfür sind
das Stranggießen von Stahl und Produktion von Sekundär-Aluminium. Aufgrund
der Aggressivität von Metallschmelzen kommen vorrangig berührungslose
Messmethoden in Betracht. Die Lorentzkraft-Anemometrie (LKA) ist ein vielversprechendes Messverfahren, um diesen Herausforderungen gerecht zu werden. Das Verfahren beruht auf den physikalischen Prinzipien der Magnetofluiddynamik (MFD). Die für diese Arbeit genutzte und auf den MFD Prinzipien basierende Strömungsmesser, sogenannte Lorentzkraft-Anemometer bestehen
aus einem Magnetsystem und einem Kraftsensor. Der Sensor misst die durch
die Bewegung der elektrisch leitfähigen Metallschmelze durch das Magnetfeld
induzierte elektromagnetische Lorentzkraft. Diese ist proportional zum Durchfluss oder der lokalen Geschwindigkeit in der Nähe des Magnetsystems.
Ein Nachteil der Messanordnung besteht darin, dass die Lorentzkraft abhängig
von der elektrischen Leitfähigkeit des Fluides und der Stärke des von außen
angelegten Magnetfeldes ist. Eine Methode, dieses Defizit zu umgehen, ist die
sogenannte Time-of-Flight Technik (ToF-LKA). Hierbei sind zwei Anemometer
hintereinander mit einem bestimmten Abstand angeordnet. Diese detektieren
den Durchgang einer Struktur oder einer Störung in der Strömung und misst die
Zeit, die die Wirbel zum Durchlaufen des Abstandes benötigen. Aus dieser
Laufzeit kann auf die Strömungsgeschwindigkeit geschlossen werden. Die
Laufzeit wird durch die Bildung der Kreuzkorrelation der Kraftsignale der beiden
Anemometer bestimmt. Das Ziel der vorliegenden Arbeit ist es, experimentell
nachzuweisen, dass die ToF-LKA ein vielversprechender Ansatz zur berührungsfreien der Strömungsmessung in Metallschmelzen ist.
Diese Arbeit stellt eine Reihe von experimentellen Untersuchungen vor. Zu Beginn wird die Durchflussmessung in einer turbulenten FlüssigmetallKanalströmung durchgeführt. Im Weiteren werden die Untersuchungen auf
Flüssigmetallströmung an der Oberfläche ausgeweitet. Hier ist die lokale Geschwindigkeitsmessung an der Oberfläche von Interesse. In diesem Experiment
wird das niedrigschmelzende Flüssigmetall GaInSn verwendet. Basierend auf
diesen Messungen wird ein spezielles Messgerät entwickelt, um lokale Oberflächengeschwindigkeiten in heißen Metallschmelzen zu erfassen. Mit diesem
Sensor werden, Testmessungen unter industrienahen Bedingungen mit der Metallschmelze (SnPbBi) bei ca. 210°C und mit geschmolzenem Stahl bei ca.
1700°C durchgeführt. Zu jedem Experiment wird das Ergebnis numerischer Simulationen präsentiert, um ein besseres Verständnis der Magnetofluiddynamik
herzustellen.
List of abbreviations
LFF
LFV
LKA
ToF LFV
EFCO
LiMeSCo
MVS
EMP
CCS
MFD
SEN
CFD
FVM
LES
RANS
DNS
WALE
GaInSn
UDV
US
Lorentz force flow meter
Lorentz Force Velocimetry
Lorentzkraft-Anemometrie
Time-of-Flight Lorentz Force Velocimetry
Electromagnetic Flow Control Channel
Liquid Metal Surface velocity Correlation measurements
Meniscus Velocity Sensor
Electromagentic pump
Continuous Casting of Steel
Magnetofluiddynamics
Submerged Entry Nozzle
Computational Fluid Dynamics
Finite Volume Method
Large Eddy Simulation
Reynolds Averaged Navier-Stokes
Direct Numerical Simulation
Wall-Adapting Local Eddy Viscosity
Galinstan-gallium, indium and tin
Ultrasonic Doppler Velocimetry
Ultrasound
Nomenclatures
B
B0
b
E 
j
FL
fL
U
UA
V
Vvortex
Re Recrit Red Rem Rec Ha N MaM
Cf x
Magnetic field [T]
Imposed magnetic field [T]
Induced magnetic field [T]
Electric field [V/m]
Electric potential [V]
Electric current density [A/m2]
Lorentz force [N]
Body force / Lorentz force density [N/m3]
Averaged velocity [m/s]
Alfven velocity [m/s]
Flow velocity [m/s]
Vortex velocity [m/s]
Reynolds number
Critical Reynolds number
Reynolds number of duct flow
Magnetic Reynolds number
Cylinder Reynolds number
Hartmann number
Interaction parameter
Magnetic Mach number
Drag force coefficient
Cf y
Lift force coefficient
C jy
Coefficient of eddy current density
µ0
υ
ρ
m
M
σ dH L0
L H
S
h
A
H
D
a β 
f
Q
Magnetic field constant [H/m]
Kinematic viscosity [m2/s]
Density [kg/m3]
Magnetic dipole moment [A/m2]
Magnetic diffusivity [m2/s]
Electrical conductivity s / m
Hydraulic diameter of duct flow [m]
Characteristic length of free surface flow [m]
Characteristic length [m]
Thickness of Hartmann layers [mm]
Thickness of Shercliff layers [mm]
Deformation of the free surface [mm]
Edge length of the cubic magnet [mm]
Height [mm]
Distance [mm]
Diameter of the submerged cylinder [mm]
Blockage ratio of the cylinder
Angle of the submerged cylinder [°]
Rotation frequency of the pump [Hz]
Volumetric flow rate [m3/s]
Contents
Acknowledgements....................................................................................................................... I Abstract ......................................................................................................................................... II Zusammenfassung...................................................................................................................... III List of abbreviations .................................................................................................................. IV Nomenclatures ............................................................................................................................ V 1 Introduction......................................................................................................................... 1 1.1 A qualitative overview ................................................................................................. 1 1.2 Problem and motivation .............................................................................................. 3 1.3 State of the art in the flow rate and local velocity measurement ................................ 6 1.4 Lorentz Force Velocimetry and Time-of-Flight Lorentz Force Velocimetry .............. 10 1.5 Objective and scope ................................................................................................. 13 2 Governing equations of magnetofluiddynamics ........................................................... 15 2.1 A brief review of electrodynamics ............................................................................. 15 2.2 A brief review of fluid mechanics .............................................................................. 18 2.3 Dimensionless parameters ....................................................................................... 19 2.4 Dimensionless equations.......................................................................................... 23 3 Electromagnetic flow rate measurement in turbulent liquid metal duct flow ............ 25 3.1 Introduction ............................................................................................................... 25 3.2 Using Lorentz Force Velocimetry ............................................................................. 26 3.2.1 Experimental results .................................................................................. 26 3.2.2 Numerical simulations ............................................................................... 27 3.3 Using Time-of-Flight Lorentz Force Velocimetry ...................................................... 31 3.3.1 Experimental results .................................................................................. 32 3.3.2 Numerical simulations ............................................................................... 37 4 Electromagnetic free surface velocity measurement in annulus flow ........................ 44 4.1 Introduction ............................................................................................................... 44 4.2 Metallic body experiments ........................................................................................ 45 4.2.1 Experimental results .................................................................................. 45 4.2.2 Numerical simulations ............................................................................... 55 4.3 Liquid metal free surface experiments ..................................................................... 57 4.3.1 Experimental results .................................................................................. 57 4.3.2 Numerical simulations ............................................................................... 61 5 Application of ToF LFV to free surface velocity measurement in metallic melt ........ 72 5.1 Introduction ............................................................................................................... 72 5.2 State of the art of surface velocity sensor for melt flow ............................................ 74 5.3 A sensor for high-temperature surface velocity measurement................................. 77 5.4 Preliminary test measurement under industry-relevant condition ............................ 84 5.4.1 Open channel flow measurement using SnPbBi at 210°C ........................ 84 5.4.2 Open channel flow measurement using steel at 1700°C .......................... 87 5.5 Sub-summary ........................................................................................................... 91 6 Conclusion and outlook .................................................................................................. 93 List of figures.............................................................................................................................. VI Bibliography ............................................................................................................................. XIII Appendix .................................................................................................................................. XXII A1 Physical properties of liquid metals ................................................................................ XXII A2 Experimental matrix for open channel flow measurements ........................................ XXIII A3 Strain gauge force sensor ...............................................................................................XXIV A4 Error estimation of experiments .....................................................................................XXVI 1
1 Introduction
1.1 A qualitative overview
Non-contact flow control and flow measurement in hot and aggressive metal
melts are big challenges in metallurgic processes including continuous casting
of steel [1] [55] and production of secondary aluminum [2] [56] [71], among others. Another example is the production of float glass [3]. During this process,
flow control in the tin bath, upon which the glass is solidifying, is crucial for the
quality of the final product. In most cases, non-invasive control methods are
favorable since at high temperatures metal melts are chemically very aggressive. Electromagnetic methods are promising tools since metal melts are both
non-magnetic and excellent electrical conductors. One of these electromagnetic
methods is Lorentz Force Velocimetry (LFV) [4] [5] [6] [7] [77]. LFV is a noncontact electromagnetic measurement technique and suitable for velocity and
flow rate measurement in high temperature metallurgical processes. This
measurement technique exploits the principles of magnetofluiddynamics (MFD)
[8] [9], as it is sketched in Fig.1. When an electrically conducting fluid like a
metal melt crosses the field lines of a primary magnetic field B0 produced by an
external magnet system, eddy currents j are generated within the melt. The eddy currents mainly flow in the direction that is perpendicular to both the magnetic field and the melt velocity. The eddy currents induce their own magnetic field,
the so-called secondary magnet field b that adds to the applied primary one.
Physically speaking, the magnetic field lines are slightly bent by the flow, see
Fig. 1. The interactions of the eddy currents with the magnetic field give rise to
the generation of Lorentz forces FL within the melt. The Lorentz force is proportional to both velocity or flow rate and electrical conductivity of the melt. The
force tends to brake the flow. A well-known industrial application of these effects
is the electromagnetic brake (EMBR) [10] [75] used during continuous casting of
steel. LFV is based on measuring the counterforce of this braking Lorentz force
which acts on the magnet system in the streamwise direction. This force is
called Kelvin force. In a physical sense, this force results from the flow-induced
secondary magnetic field that pulls at the magnet system. A respective flow
sensor that operates according to this principle is called Lorentz force flow meter (LFF). It basically consists of a magnet system and an attached force sensor.
2
Fig. 1: Principle of Lorentz Force Velocimetry. It is based on measuring the Kelvin force
that pulls at an externally arranged magnet system. The Kelvin force is the counteracting
force to the Lorentz force that is generated in the melt due to the movement of the electrically conducting melt through the magnetic field.
However, in metallurgic processes, the conductivity is often unknown or hard to
evaluate as it strongly depends on both temperature and composition of the
liquid metal alloy. To overcome this problem a modification of LFV has been
elaborated, namely, Time-of-Flight Lorentz Force Velocimetry (ToF LFV) [26]
[27]. Using this technique, two Lorentz force flow meters are arranged in a row
and separated by a defined distance D. This Time-of-Flight technique can be
employed for the flow measurement by purely cross-correlating the two force
signals recorded by the two flow meters, see Fig. 2. Therefore, the measurement becomes independent of any fluid properties and magnetic field parameters.
Fig. 2: Principle of Time-of-Flight Lorentz Force Velocimetry. Two Lorentz force flow meters are arranged one behind the other. This flow measurement technique is based on
just cross-correlating the force signals recorded by the two flow meters. The method is
independent of melt properties and magnetic field parameters.
The main goal of this thesis is to check experimentally the potential of LFV and
ToF LFV for flow measurement in metallurgical application. However, industrial
3
scale experiments with hot metallic melts are expensive and difficult to perform.
To eliminate this problem at first we propose small-scale model experiments
using the low-melting liquid metal GaInSn. This model melt is ideally suited for
laboratory experiments as it is liquid at room temperature and non-toxic. Here
we employ LFV and ToF LFV for flow rate measurement in turbulent channel
flow carried out in a closed liquid metal loop. As a flow driving device we use a
frequency controlled electromagnetic pump. Both Vives probe [15] [60] [61] and
Ultrasonic Doppler Velocimetry (UDV) are utilized [17] [62] [63] [64] as reference flow measurement techniques. Moreover, using ToF LFV we perform
measurements of local velocities in liquid metal free-surface flow. The respective test facility consists of an annulus gap filled with liquid metal GaInSn and
put into a controlled rotation by an electrical motor. Finally, the model experiments result in the development of a prototype of an electromagnetic surface
velocity sensor based on ToF LFV and capable of measuring locally freesurface velocities in high-temperature metallic melts. This prototype has been
tested in free-surface open channel flows of both PbBiSn at 210°C and steel at
1700°C. The prototype is shown to detect motion of liquid metal flow from a few
cm s-1 up to 0.65 m s-1. All the experimental investigations are supported by accompanying theoretical studies that are based on numerical simulations using
commercial codes.
1.2 Problem and motivation
A considerable part (about 80%) of the aluminum demand in the world is satisfied through secondary aluminum production [2] [56]. For the secondary production of aluminum, materials containing aluminum such as scrap, machining turnings, and dross are prepared, smelted and refined. A sketch of the production
line is shown in Fig. 3. Rotary drum and melt furnaces are used to melt down
aluminum scrap and materials containing aluminum. Typical melting temperatures are around 700 to 750 °C. After tagging, the primary melt, driven by gravity, is transferred via open-channel flow into converters for further treatment like
refining, mixing, and alloying with silicium (Si) or other metals such as copper
(Cu), magnesium (Mg), manganese (Mn) and zinc (Zn). The final aluminum alloy is also transferred in open channels -the so-called launders- to the casting
machine where it is subsequently cast into bars or directly processed in molten
form in the foundry. Nowadays, the production process is controlled only by
weighing the scrap at the beginning and the final product by the end of the process. Until now, however, no in-situ mass flux or flow rate measurement technique is available for high-temperature liquid metal melt.
4
Fig. 3: Secondary aluminum production process. The aluminium melt flux is indicated by
blue arrows. Scrap is melted in furnaces. The primary melt is delivered to converters
within which the final melt is prepared. The figure is taken from Ref. [71] (in German).
The information about the scrap performance as well as the amount of the alloying elements to be put into the converter cannot be evaluated exactly. The goal
of the present study is to demonstrate of the feasibility of the non-contact electromagnetic measurement techniques of LFV and ToF LFV to measure the flow
rate in such channel flows. In an aluminium recycling plant LFFs may be installed at the launders to register the flow rate during the transfer of the melt
from the furnaces to the converter and further to the tundish and the casting
machine.
The second technical application which we deal with is to measure the molten
steel flow in continuous casting of steel production. Demand for high-quality
steel product is continuously increasing. In continuous casting process [1] [55],
the molten steel is poured in to the mold through the submerged entry nozzle
(SEN) which is set in the center plane of the mold, see Fig. 4. As thermal insulator mold powder is put on the free surface (meniscus), serving as lubricator to
the mold wall and an adsorbent of impurities. Argon gas is injected into the nozzle to prevent clogging. Controlling the meniscus molten steel flow in the mold is
one of the keys to achieve good quality of the slab, because the impurities are
sometimes trapped from the free surface. The molten steel flow in the mold is
classified into two patterns. One is a double roll flow pattern and the other is a
single roll flow pattern, see the right picture in Fig. 4.
5
Fig. 4: Meniscus flow in Continuous Casting of Steel (left). Double and single roll flow
pattern (right). The figure is taken from Ref. [55].
In the double roll flow pattern, the molten steel flows from the nozzle to the narrow side of the mold. Then, some flows upward and some flow downward. At
the meniscus the molten steel moves from the narrow side of the mold to the
nozzle. In general, the double roll flow pattern appears when the molten steel
throughput is high or the mold width is narrow. In this case, mold powder is
trapped by the strong meniscus flow or meniscus level fluctuations generating
surface defects. On the other hand, when the molten steel throughput is low or
the mold width is wide, the molten steel occurs with ascending argon gas near
the nozzle and the molten steel moves from the nozzle to the narrow side of the
mold at the meniscus. It is called a single roll flow pattern. When the flow pattern is a single roll structure, molten steel flows from the nozzle does not reach
the narrow side of the mold and the temperature fall down at the mold corner of
the meniscus. Then, the slab corner might be cracked or the impurity carried
into the molten steel, which leads to the surface defects. In order to decrease
the defects on the slab and control the molten steel flow, various techniques
imposing of magnetic field system have been developed. In the present thesis
we report on the design of a non-contact electromagnetic meniscus velocity
sensor based on ToF LFV that allows detecting both the magnitude and direction of the flow near the free surface. Thus, the sensor may register the flow
pattern in the entire mold. In Fig. 4 we show schematically such a sensor as it is
place in some distance above the melt surface.
6
1.3 State of the art in the flow rate and local velocity
measurement
Precise and reliable velocity measurement is required during casting in metallurgy process [1] [2]. A number of flow measurements methods in electrically
conducting fluids have been developed over the past decades. These methods
can be divided into (integral) flow rate measurement methods and (local) velocity measurements.
Flow rate measurement
Flow rate measurements in liquid metals are required in various technological
processes ranging from the cooling of nuclear reactors to the dosing and casting of molten metals. A variety of electromagnetic flow meters have been developed starting from the late 1940s and described by Shercliff [23]. Commercially
available electromagnetic flow meters typically use a pair of electrodes in direct
contact to the liquid to measure the potential difference induced by the electrically conducting fluid moving through a static magnetic field. This approach is
now well developed and works reliably for common liquids like water etc., but
not so for liquid metals. Major problem in molten metals, especially at higher
temperatures, is that electrodes are not resistant against the chemically aggressive liquid metals.
Alternatively, the liquid metal flow can be determined in a contactless way by
eddy-current flow meters, which determine the flow rate by sensing the flowinduced perturbation in an applied magnetic field. The magnetic fly wheel is
such a flow meter which was invented by Shercliff [23] and employed by Bucenieks [11] [57] [58]. It consists of permanent magnets, which are mounted on
an axle and placed close to a tube carrying the liquid metal flow; see the left
picture in Fig. 5. The disk rotating rate is proportional to the flow rate.
Fig. 5: Magnetic fly wheel (left). Single-magnet rotary flow meter (right). The figure is taken from Ref. [11] and [12].
Priede [12] [59] suggested an alternative design of such a flow meter, which
uses just a single magnet mounted on the axle and which can freely rotate
7
around and the axis perpendicular to the direction of the magnetization, see the
right picture in Fig. 5. In contrast to fly wheel, which is driven by the electromagnetic force acting on separate magnets, the single magnet is set into rotation
only by the electromagnetically induced torque. This driving torque is due to the
eddy currents induced by the flow across the magnetic field. As the magnet
starts to rotate, additional eddy currents are induced, which break the rotation.
An equilibrium rotation rate is attained when the braking torque balances the
driving one, and this rate depends only on the flow velocity and the flow meter
arrangement, whereas it is independent of the electromagnetic torque itself as
well as the electrical conductivity of the melt. The rotation rate of the magnet
was measured by an inductive magnetic proximity sensor. The main problem
with this type of flow meters is the weak magnetic field perturbation which may
be caused not only by the flow.
Recently a new flow rate measurement method was developed which show that
the flow-induced phase shift of alternating magnetic field (AC) [13] is more reliable for flow rate measurements. The phase disturbance is found to be more robust than that of the amplitude perturbation used in conventional eddy-current
flow meters. The flow rate is determined by applying a weak AC magnetic field
to a liquid metal flow and measuring the flow-induced phase disturbance in the
external electromagnetic field, see Fig. 6. This flow meter employs the fact that
the flow of a conducting liquid disturbs not only the amplitude but also the phase
distribution of an applied AC magnetic field. The asymmetry of the phase distribution caused by the fluid motion can be used to determine the velocity. However, similar to standard eddy-current flow meters, phase-shift sensor is still
sensitive to the electrical conductivity, thus, to the temperature of the liquid.
Fig. 6: Contactless electromagnetic phase-shift flow meter for liquid metals. The figure is
taken from Ref. [13].
8
Local velocity measurement
Information about local velocity in conducting fluids was obtained by various
types of local probes. For instance, hot-wire and hot-film sensors by Hill and
Sleicher [83] in mercury and by Platnieks and Uhlmann [84] in sodium. Permanent magnet probes were developed by Ricou and Vives [60] and Von Weissenfluh [85]. However, the described technology above reveals considerable limitations. For instance, it is intrusive but not absolutely contactless. Moreover, the
short lifetime of the sensor probe limits the application potential of this method
[14].
The principle of the Vives probe measurement [15] [60] [61] is governed by
Ohm’s law. The probe is immersed in the molten metal at the location where the
velocity is to be measured. The permanent magnet creates a magnetic field and
the movement of electrically conducting liquid through this field generates an
electric field, which is measured by the four electrodes lying on alongside the
magnet, see the left picture in Fig. 7. The potential difference between diagonally placed sensing electrodes, which is measured by a nano-voltmeter, is proportional to the flow velocity. Vives Probe is not to be applied in high-temperature
metallic melt because of his limitations of Curie temperature (570 K neodymiumiron-boron magnets to 1120 K Alnico magnets).
Fig. 7: Design of Vives Probe (left) and Potential Probe (right). The figure (right) is taken
from Ref. [54].
The technical principle of the potential probe [16] [51] [52] [53] [54] is governed
also by the Ohm´s law in moving fluids. The probe consists of four copper wire
electrodes insulated by a varnish except the sensitive tips which are in direct
contact with the liquid metal. The probe provides the electric potential differences between the measuring electrodes in spanwise and streamwise directions see the right picture in Fig. 7.
In recent years, ultrasound technique has drawn much attention as a nonintrusive and non-invasive measurement method due to its attractive advantages over the conventional techniques. The principle of the Ultrasound
9
Doppler Velocimetry (UDV) [17] [62] [63] [64] method is to use the pulsed echo
signal of ultrasound wave and to detect the same signal reflected by the moving
particles suspended in the fluid flow. The delay time between emission of ultrasound pulse and reception of the corresponding echo signal provides the distance of the particle, while the corresponding Doppler shift provides its velocity,
see Fig. 8. The limitations of the application UDV in liquid metal flow are such
as multiply reflection at the wall. Another problem is the wetting of the sensor by
the liquid metal for achieving a sufficient acoustic coupling and the allocation of
suitable tracer particles.
Fig. 8: The US transducer is submerged at an angle of 60° into the liquid metal duct EFCO (left). Multiple reflections of US wave may result in imaginary velocity values outside
the region of the liquid flow (right). The right figure is taken from Ref. [17].
The typical problem for UDV applications is multiple reflections of the Ultrasonic
wave. As shown the right picture in Fig. 8, the US beam is reflected by the opposite channel wall in point B and transforms this interface in the transmitter.
Consequently, a particle contained in the liquid moving along the dashed line
may backscatter Doppler energy more than once in the direction of the transducer (at points A and C). The depth associated with the reflection at point C is
located outside the flow region. Imaginary velocity component are added to the
real velocity profile. The velocity measurement near the wall is affected by this
phenomenon. Therefore, it is common to obtain a non-zero velocity at the far
wall.
10
1.4 Lorentz Force Velocimetry and Time-of-Flight
Lorentz Force Velocimetry
Recently, we suggested a new electromagnetic non-contact technique, termed
Lorentz Force Velocimetry (LFV), for measuring local surface velocity and flow
rates in electrically conducting fluids, as well as in high-temperature metallic
melts. LFV is based on the principles of magnetofluiddynamics [8] [9]. A Lorentz
force flow meter measures the electromagnetically induced force acting on a
magnet system that produces the magnetic field with which the flow field interacts. Hence, such a LFV basically consists of a magnet system -preferably built
up by an arrangement of permanent magnets- and an attached force sensor.
The force recorded by the force sensor is proportional to the product of the electrical conductivity of the fluid σ, the square of the typical applied magnetic field
strength B0, and the flow rate Q or the typical flow velocity U. The scaling law
between force F and flow rate Q is given by the relations
FL  QB 02L or FL  VB 02L3 ,
(1), (2)
where L denotes a characteristic electromagnetic interaction length. See chapter 2.1 for the theoretical derivation of these equations. An enlarged view of the
magnetic field within the Lorentz force flow meter is shown in Fig. 9.
Fig. 9: Principle of Lorentz Force Velocimetry for the measurement of flow rate (left) and
local surface velocities (right).
The difference between flow rate and local velocity measurement using LFV
techniques depends on the variety of alignment of the magnet system. In case
for measuring the flow rate we use big-size magnet systems of which the magnetic field penetrates the entire cross-section of the flow, see left picture in Fig.
9. In this case we have a relatively large characteristic length L. On the other
hand, for local velocity measurement we may use small-size cubic permanent
magnets of which the magnetic field penetrates only a small flow volume adjacent to the surface, see the right picture in Fig. 9. This case corresponds to a
relatively small value of L.
11
However, in the latter case we expect that another characteristic length enters
into the problem: the height H, at which the magnet is arranged above the surface, as it mainly influences the magnitude of B0, see right picture in Fig. 9. To
analyze this dependence in some more detail, we consider a cubic small permanent magnet with edge length A and dipole moment m. The magnet is located at distance H above a semi-infinite fluid moving with uniform velocity u parallel to its free surface.
3
H
m
B
The magnetic field, which we refer to as the primary field B0, is of the order

at the surface of the fluid. The order of magnitude of the induced

3
H
m
u
B
u
j
0
0
0
0

eddy currents is given by  
. The eddy currents are con 
centrated below the surface, and interact with the primary field to produce a Lorentz force density of the order fL  jB 0   02 um 2H6 . Since the force acts on a
volume H3, the total Lorentz force is FL f LH3 and we have
FL 
 02 um 2
.
H3
(3)
From this formula we can draw the following conclusions [5]:
(i) F is proportional to the product of velocity and electrical conductivity.
(ii) F increases with the second power of the magnetization.
(iii) F decreases with third power of distance.
A drawback of LFV is that the measured force depends on various other quantities like electrical conductivity, geometry parameters L, H and magnetic parameters, see Eqs. (1)-(3) listed above. Therefore, intense calibration of Lorentz
force flow meters is necessary to find the device constants under various flow
conditions, magnetic properties and geometry parameters. An elegant method
to circumvent this deficit is the so-called Time-of-Flight LFV [26] [27]. ToF LFV
rely on the fundamental measurements of time and distance and do not require
calibrating for individual applications. This feature contrasts strongly with another commercial electromagnetic flow meters which have to be individually calibrated for the specific pipe size in use. Here, we just measure the flow by using
two LFFs that are arranged one behind the other and that sense the passage of
the any vortex structures that are transported by the flow. In more detail, ToF
LFV measures the flow by sensing the passage of the triggered vortices at LFV
1 and measuring the time that the vortices need to travel to LFV 2 with the typical velocity Vvortex. In Fig. 10 we have sketched such an arrangement for the
case of flow rate measurement (left picture) and the case of local surface velocity measurement (right picture).
12
Fig. 10: Principle of Time-of-Flight Lorentz Force Velocimetry for the measurement of
flow rate (left) and local surface velocities (right). The transit time  of a tagging signal
flowing through the two flow meters is determined by cross-correlating the voltage data
provided by the two force sensors.
With the separation distance D between the LFVs and the measured transit
time at hand, the vortex Velocity Vvortex, correlating with the typical flow velocity
V, can easily be calculated by Equations (4) and (5).
Vvortex  D  ,
(4)
V  k 2  Vvortex ,
(5)
Q  k1  A  Vvortex .
(6)
The flow rate can be determined by Equations (4) and (6). Here, k1 and k2 are
calibration constants and A denotes cross-section of the flow. Mathematically,
the transit time  is obtained by evaluation the cross-correlation function R12()
[73] using the raw voltage signals U1(t) and U2(t) provide by the two force sensors. The cross-correlation function is defined by the Equation (7)
R12 ( ) 
1 T
U1t   U2 ( t  ) dt .
T 0
(7)
Cross-correlation techniques [28] [66] [67] [68] are based on measuring the
transit time  of tagging signals (turbulence, clumps of particles, etc.) in the flow
between two axially separated sensors. Such techniques are well known from
the use of hot-wire anemometry for the measurement of two-point velocity correlations [29]. Cross correlation is more directly used where measurements are
the knowledge of time delays required, such as velocity measurement of steel
[82].
13
1.5 Objective and scope
The thesis aims to summarize the fundamental as well as application-oriented
research on global and local flow measurement in model experiments applying
LFV, see Fig. 11 for an overview. In a first series of global flow rate measurements, employing the closed-loop test facility EFCO (Electromagnetic Flow
Control channel) and utilizing both LFV and ToF LFV, we determine the volumetric flow rate Q by evaluating equations (4) and (6) and using magnet systems of which the magnetic field penetrates the entire cross-section of the flow.
Vortex structures are triggered by submerging a cylinder in the flow. Flow control may be managed by adjusting the power of the respective flow-driving unit
represented by an electromagnetic pump (EMP). The experimental set-up is
shown in the left picture in Fig. 11.
In a second series of local velocity measurements, exploiting the test facility
LiMeSCo (Liquid Metal Surface velocity Correlation measurements), we apply
as well the ToF LFV to determine free surface velocities. Here, we evaluate
equations (4) and (5) and use small-size cubic permanent magnets of which the
magnetic field penetrates only a small flow volume adjacent to the surface. To
check the potential of the method, in a first step, we apply the technique to solid
metal bodies that are put into controlled rotation. In this case, the calibration
constant k2 is equal to unity, cf. Eq. (5). In a second step, we conduct freesurface liquid metal experiments using low-melting eutectic alloy GaInSn. Vortex structures are generated intrinsically as the used magnets themselves act
as a magnetic obstacle [39] [70]. In this case, we expect that the calibration
constants k2 will considerably deviate from unity. Local control shall be achieved
by coupling the force measurements with electromagnetic actuators that manipulate the local flow field in a favorable manner. The experimental set-up is
shown in the middle picture of Fig. 11.
Moreover, a special prototype of a respective measuring device, called Meniscus Velocity Sensor (MVS), has been developed to measure local free-surface
velocity in high-temperature metallic melts. In Fig. 11 the prototype is shown in
the right picture. To check the functionality of the ToF LFV, in a first step we
measure surface velocities during rotation of metallic substances. We observe
that small fluctuations in the signals that are due to the natural inhomogeneity of
the solid body can be used to determine surface velocities. In a second step we
apply the technique to liquid metal free-surface flow. Here we use the liquid
metal GaInSn as model liquid. In this case we observe that the crosscorrelations between the signals are much rarer and much weaker than in the
solid-body cases. In more detail, we present test measurements under industryrelevant conditions using both SnPbBi at about 210°C and molten steel at about
1700°C. These experiments were conducted at Key Laboratories on EPM at
North Eastern University. The evaluation of the data shows that our prototype of
MVS works well in producing signals of which surface velocity can be determined. Nevertheless, there is still a big gap between the research development
and the industrial application in real production process [81].
14
Fig. 11: Sketches of the experimental facility EFCO (left) for flow rate measurements,
LiMeSCo for surface velocity measurement (middle), and MVS for measurement of surface velocity in high-temperature metallic melt (right).
The main goal of the present study is to demonstrate experimentally that ToF
LFV is suited for the non-contact measurement of both flow rates and local surface velocities in liquid metal flow. In order to support the experimental observations, at the end of each chapter, the predictions of corresponding numerical
simulations using commercial codes and simplifying assumptions were presented.
This thesis is organized as follows. The principle of LFV and ToF LFV are introduced in details in chapter 1.4. In chapter 2, we will provide the theory of fundamental magnetofluiddynamics. The test facility EFCO and the results of respective flow rate measurements are described in chapter 3. In chapter 4 we
are concerned with free-surface liquid metal measurement on test facility
LiMeSCo and discuss the results of local velocity experiments. Studies concerning the electromagnetic surface velocity measurement for high-temperature metallic melt are reported in chapter 5. Main conclusions and perspectives are
drawn in chapter 6.
It should be mentioned that due to the significance to metallurgic application
most of the results shown in this thesis have been pre-published in peer-review
scientific journals. These articles are given in Refs. [27] [30] [31] [32] [33] [78]
[79]. Therefore, some parts of these papers have been taken into this thesis.
15
2 Governing equations of magnetofluiddynamics
In this chapter we would like to present the theoretical background of this thesis.
Magnetofluiddynamics (MFD) is a science that covers phenomena and interactions resulting from the coupling of the velocity field u of on electrically conducting fluid and a magnetic field B. Hence, we have to consider the governing
equations of both fluid mechanics and electrodynamics, i.e. the Navier-Stokes
equations [42] and the Maxwell equations [46], respectively. The aim of this
chapter is to provide the reader with the physical and mathematical information
that serves to explain the experimental findings and that represents the basis of
the numerical simulations shown in sub-chapters 3.2.2, 3.3.2, 4.2.2 and 4.3.2.
The governing equations of MFD are well known since the pioneering works of
the Danish physicist Julius Hartmann on experiments in liquid metal channel
flow affected by a uniform magnetic field [47] [48] and of former Swedish Nobel
Prize winner Hannes Alfven [43] on the existence of electromagnetichydrodynamic waves (later on referred to as Alfven waves, cf. chapter 2.4).
More recently, a number of famous textbooks on this subject have been published including the standard books of Shercliff [23] [44], Roberts [49], Moreau
[8], Davidson [24], and Asai [45], among others. Therefore, in this chapter we
only give a brief review of the fundamental MFD relations that are already well
documented in the literature listed above. Our focus shall be to point out the
application of these relations towards LFV.
2.1 A brief review of electrodynamics
In the following we consider the general equations of the electrodynamics of
continuous conducting media under the assumption that electrical displacement
currents generated by a time-dependent electrical field are much smaller than
any current related to the flux of electrical charge carriers. Moreover, we assume that all fluid velocities are small compared to the speed of light. Both assumptions are well met in the present case of liquid metal flow. Within these
assumptions the Maxwell equations can be written in the following form.

Btj
0
.
0
B B

E



 ,
 ,

(8)
(9)
(10)
Here, Eq. (8) represents Faraday’s law. It denotes that due to a time-dependent
magnetic field B, a vortical electrical field E is induced. This law is also referred
to as the induction law. Furthermore, Eq. (9) describes Ampere’s law. It denotes
that an electrical current density j generates a vortical magnetic field. In Eq. (9)
µ0 is the magnetic field constant given by µ0 = 4 x 10-7 H/m. Finally, Eq. (10) is
16
the so-called Gauss’ law. It states that the magnetic field is divergence-free, i.e.
that any magnetic flux entering a volume is equal to the flux exiting the volume.
Taking the divergence of Eq. (9) it easily follows that the electrical current density is also a solenoidal field, i.e.
.
0
j
 
(11)
This set of Maxwell equations is accompanied by a constitutive law –called
Ohm’s law– that expresses the capability of an electrically conducting substance to transport electric charge. Under the assumptions that the material is
isotropic and that the transport is dominated by conduction rather than convection, Ohm’s law writes as follows.
j   E  U  B .
(12)
Here,  denotes the electrical conductivity of the material. Physically Eq. (12)
reveals that in a conductor an electrical current density -the so-called eddy current density- is driven by an electrical field E and/or is induced due to the
movement of the conductor through a magnetic field whenever the vectors u
and B are non-parallel. Concerning LFV where we face liquid metal channel
flow or free-surface flow affected by superimposed spatially localized magnet
fields, from Eqs. (11), (12) we can conclude that in bulk flow regions within
which the imposed magnetic field is present, eddy currents are mainly generated by the u x B term. On the other hand, in the vicinity of non-conducting rigid
walls, where velocity is significantly reduced, or in regions within which the
magnetic field is absent, the electrical field E is the dominant driving mechanism
for electrical currents. It ensures that the current path lines are closed. As we
shall see later on, the eddy current distribution in the melt mainly controls the
distribution of the induced electromagnetic forces, cf. Eq. (15). Unfortunately, in
our test experiments on LFV we cannot measure j. Therefore, to get a better
understanding in the physical principles we also perform simulation to determine j numerically.
Using some elementary rules of vector calculus, Eqs. (8)-(12) can be combined
to yield the following magnetic field transport equation, also known as the induction equation, which is an expression relating magnetic field B to velocity u.
B
t

1 2
 (U   ) B 
 B  (B   ) U


 



0
Advection of


 Stretching of
Time dependence
of magnetic field
magnetic field
Diffusion of
magnetic field
(13)
magnetic field
Magnetic field transport equation describes the change in the magnetic field due
to the fluid flow. Here, the left hand side represents the total rate of change of B
due to the effects of time-dependence of the field (first term) and its advection
by the flow field (second term). Moreover, the first term on the right denotes the
diffusion of B within the conducting material while the second term describes
the generation of B in a shear flow. In Eq. (13), the pre-factor 1/( 0 ) = M is
the material property called magnetic diffusivity. As all diffusion coefficients, its
dimension is given in [m2/s]. A typical value for liquid metals -that shall exclu-
17
sively be considered in this thesis- is M = 1 m2/s. Hence, in liquid metal MFD
the magnetic fields are typically highly diffusive. As a physical consequence, the
magnitude of any magnetic field perturbation induced by the flow remains small
as it is rapidly diffused. That means that the usually strong imposed magnetic
field is only weakly influenced by the flow. In this case the coupling of the u field
to the B field remains also small. We will see later on that this weak coupling is
due the small values of the magnetic Reynolds number. Please note that in Eq.
(13) B denotes the total magnetic flux density. For the application of this equation to LFV it is instructive to split the magnetic field in the primary externally
applied static magnetic field, B0 say, and in the secondary field, b say, that is
induced by flow due to the generation of eddy currents. Thus we write
B
 x , y , z; t 
 B0  x , y , z   b  x , y , z; t  .
(14)
Finally we finish this sub-chapter by introducing another very important constitutive relation that expresses a fundamental effect of MFD being the physical basis of the LFV measuring technique investigated in this thesis: The movement of
an electrical conductor (liquid metal flow) through a magnetic field generates a
force density within the conductor. This force density, resulting from the interaction of the induced eddy current density j and the magnetic field B, is called Lorentz force density fL and can be calculated using the relation
fL  j  B .
(15)
As seen from Eq. (15) this body force is acting in the direction which is perpendicular to both j and B. If we plug in Ohm’s law, i.e. Eq. (12) into the above formula we can draw three important conclusions for the application of this interaction to LFV:
(i)
The magnitude of the Lorentz force density is scaling according to the
relation
.
2
B
U
fL

(16)
Here, U is a characteristic velocity of the flow.
(ii) In the bulk region the Lorentz force density is pointing in the negative
flow direction. Hence, the induced Lorentz forces tend to brake the flow
as we have
e fL   eu
.
(17)
where e denotes the respective unit vector.
(iii) In regions adjacent to non-conducting walls or a free surface, the eddy
currents have to change their direction in order to form closed loops. By
that, also the Lorentz force changes its direction. In these regions we ex-
18
pect a strong component pointing into the normal direction of the wall or
the surface as well as a weak component pointing into the positive flow
direction, i.e. pushing the flow.
LFV is based on measuring the counter force to the Lorentz forces generated in
the liquid metal. This force -the so-called Kelvin force- acts on the magnet system that produces the primary magnetic field superimposed to the flow. As obvious from Eq. (16), upon measuring this force we are able to get information
about the flow velocity U and/or the volumetric flow rate Q = UA. Moreover, the
measured quantity, i.e. the force, is directly proportional to the unknown quantity
to be determined, i.e. U and/or Q. From a point of view of measurement technology, this is an ideal constellation. Furthermore, Eq. (16) reveals, that the
magnitude of the measured force is proportional to the electrical conductivity of
the melt and the square of the magnetic field strength.
2.2 A brief review of fluid mechanics
The flow field u of a Newtonian simple and isothermal fluid is governed by the
Navier-Stokes equations. These equations comprise balances of both mass and
momentum. In case of an incompressible fluid with constant density these
equations write as follows.
  U  0,
(18)
U
t

(19)
Unsteady
acceleration
1
1
2
 (U  ) U   P  

U  (jB )







friction
Convective
 Viscous


force
acceleration
Pr esseure
gradient
Density of
Lorentz force
Here ρ is the fluid density,  denotes its kinematic viscosity, and p is fluid pressure, respectively. Eq. (18) is the so-called continuity equation that expresses
the physical fact that the flow field is divergence-free. The physical meanings of
the various terms within the momentum balance, i.e. Eq. (19), are explicitly described. In contrast to ordinary fluid mechanics, in the case of MFD, we have to
take into account the Lorentz force density which presents the Lorentz force per
unit volume as an additional source term on the right hand side of the momentum balance. During LFV, where strong magnetic fields are of interest, we expect that the generated electromagnetic forces are relatively strong. Hence,
LFV is characterized by strong coupling of the u field to the B field. This means
that the electromagnetic forces are able to significantly modify or change the
flow structure. In the next sub-chapter we shall see that the strength of this coupling can be quantified using the dimensionless groups of Hartmann number
and interaction parameter that measure the magnitude of the electromagnetic
force in units of the viscous friction force (2nd term on the right of Eq. (19)) and
the inertia force, respectively. The latter force on the left of Eq. (19) is denoted
by its synonym, i.e. convective acceleration.
19
As we have already mentioned before, both the magnitude and the direction of
the induced electromagnetic force are controlled by the eddy current distribution. Using the arguments (i)-(iii) listed in the sub-chapter 2.1 we conclude that
applying LFV we may expect the following modification to an ordinary flow profile:
(i)
The bulk velocity will be significantly reduced due to the braking action
of strong electromagnetic forces.
(ii) In turn, due to continuity, cf. Eq. (18), the velocity in the wall regions
must significantly increase.
(iii) This formation of boundary layers near the walls is supported by the
electromagnetic force acting in these regions as they tend both to push
fluid towards the wall and to accelerate the flow.
A more detailed study of these effects is given by Müller & Bühler [25].
2.3 Dimensionless parameters
Based on the governing MFD equations derived above we can deduce the dimensionless parameters that control LFV and reflect all the physical phenomena described so far. Formally, these parameters can be set up by evaluating
and comparing the order of magnitudes of the various terms occurring in the
governing equations. In a first step, we introduce characteristic scales for
length, velocity, time, pressure, and magnetic flux density. We choose
x, y, z   L,
(u, v, w )  U, t  L / U, p  U2 , (B x , B y , B z )  B0 .
(20)-(24)
By that the scales for the electrical current density and the electrical field are
also fixed. We obtain
j  UBo , E  UB0 .
(25), (26)
We now introduce these scales into the induction equation (13) governing the
magnetic field and the Navier-Stokes equation (19) governing the flow field.
Physically, Rem is the ratio of the time scales of diffusion of the magnetic field
d  L2o due to the fluid motion across the magnetic field lines and the time
scales of advection of the magnetic field a  L U . This parameter is defined by

.M
/
L
U
L
U
0
eM
R
 

(27)
20
Having in mind that the total magnetic field consists of a primary and a secondary field, cf. Eq. (14), we also can write
.
B0
/
b
eM
R

(28)
This definition shows that the magnetic Reynolds number can be interpreted as
the ratio of the strengths of the magnetic field induced by the flow and the externally applied magnetic field. In liquid metal applications, including LFV, ReM
is small due to the high magnetic diffusivity (M  1 m2/s) and limitations in both
U and L. Typically we have ReM  10-2. Therefore, upon calculating the Lorentz
force density according to Eq. (15) it is physically justified to neglect the contribution of the induced field. In this case -referred to as the quasi-static approximation- the governing MFD equations can be considerable simplified, see next
sub-chapter 2.4. This is in contrast to geophysical and astrophysical MFD phenomena where the typical length scales and therefore the respective magnetic
Reynolds numbers are extremely high [50]. In this case there exists a very
strong coupling of the B field to the u field.
We now turn to the Navier-Stokes equation (19) and take the ratio of the inertia
force and the friction force. This ratio defines the Reynolds number Re given by
.
/
L
U
e
R


(29)
Physically, the Reynolds number characterizes different flow regimes, i.e. laminar or turbulent flow. When its value is less than a certain threshold, i.e. Re <
Recrit, the flow is laminar characterized by smooth and steady fluid motion due
to the relative dominance of momentum diffusion. On the other hand, when Re
˃ Recrit, the flow is called turbulent flow and is dominated by advection of momentum which tends to trigger random eddies, vortices, and other flow instabilities. In the present case of liquid metal flow applied to LFV, we are mostly in the
turbulent regime. In our laboratory experiments we will typically have Re  1.6 x
104. Here, the large Re values are due to the small kinematic viscosity of liquid
metals, typically being of the order   10-7 m2/s. Large Reynolds numbers indicate that friction forces are only significant in thin layers adjacent to rigid walls.
Within this so-called boundary layer the flow velocity has to drop from the value
in the inviscid bulk to zero right at the wall. In turbulent flow the typical thickness
 of such a boundary layer is of the order /L  Re-1/5.
.M
/
rM
P
Upon taking the ratio of the parameters ReM and Re we obtain the magnetic
Prandtl number PrM. We find
 
(30)
Hence, the magnetic Prandtl number represents the ratio of two diffusion coefficient of the melt, i.e. the kinematic viscosity being responsible for diffusion of
momentum and the magnetic diffusivity. Using the typical values for  and M of
a liquid metal, we obtain PrM  10-7. This indicates that the diffusive length and
time scales in liquid metals differ extremely. Up to now in numerical simulations
it is not possible to simultaneously resolve both diffusive fields in a physical
21
way. In liquid metal flow we can elegantly circumvent this problem by applying
the quasi-static approximation within which the induced magnetic field must not
be calculated.
We consider again the Navier-Stokes equation (19) and take the ratio of two
typical time scales. The first one, t say, is set by the friction force and is given
by t  (L3/(U))1/2. The second one, t say, is related to and the electromagnetic
force and can be written as t  (L/(UB02)1/2. The ratio of these time scales
defines the Hartmann number Ha. We obtain


.
L
B0
a
H

(31)
More physically, Ha2 denotes the relative strength of the electromagnetic force
and the viscous force. In our experiments with liquid metals we typically have
Ha2  104. This relatively high value indicates that the electromagnetic forces
clearly dominate viscous friction. This is clearly the case in the inviscid bulk region of the flow. However, as stated before, near solid walls friction becomes
important. Hence, adjacent to walls we expect thin electromagnetic boundary
layers within which there is a balance of both forces. Such layers are referred to
as Hartmann layers of thickness H at walls perpendicular to the applied magnet
field and Shercliff layers of thickness S at walls parallel to the applied magnetic
field. The respective scaling of these layers is given by the relations H/L  Ha-1
and S/L  Ha-1/2.
Finally, within Eq. (19) we take the ratio of the electromagnetic force and the
inertia force. By that we obtain the electromagnetic interaction parameter N,
sometimes also called the Stuart number. This parameter is defined as

.
L
20
U
B
2
e
a
HR
N



(32)
In our experiments we shall typically have N  1, i.e. both forces are of the
same order of magnitude. Physically this means that the induced electromagnetic forces are strong enough to re-shape the flow structure significantly. We
expect that this flow shaping effect is more pronounced while the flow Reynolds
number is decreased.
In this thesis we investigate experimentally and numerically turbulent liquid
metal channel flow and turbulent liquid metal free-surface flow affected by
strong localized magnet fields that are produced by the magnet systems of various Lorentz force flow meters. According to the relative values of the MFD parameters defined above we expect the following configurations.
(i)
The case of channel flow: in regions in front of the magnetic field we
shall have a turbulent hydrodynamic flow profile characterized by an almost constant velocity in the bulk and thin viscous boundary layers near
the walls. Upon entering the magnetic field this flow profile will be signifi-
22
cantly re-shaped due to the high Hartmann number and an interaction
parameter of order unity. Due to the braking Lorentz forces the bulk velocity will be considerable reduced. Fluid is pushed towards the walls.
Here we expect jet-like flow structures. Upon exiting the magnet field the
re-shaped remains nearly unchanged due to the high-Reynolds number.
(ii) The case of free-surface flow: the flow will tend to by-pass the region
within which the magnetic field is present. This is the so-called magnetic
obstacle effect that have been intensively studied both experimentally
and numerically, see [38] [39] for reference. Hence the surface flow velocity underneath the magnet will be considerably reduced. In turn, jetlike flow is expected aside of the magnet. Again, due to the high Reynolds number the electromagnetic wake that is formed by the magnet will
persist far in the downstream direction. Moreover, we will expect to detect large deformations of the free surface and the formation of surface
waves due to the high Hartmann number and an interaction parameter of
order unity.
In the following table we summarize the main dimensionless parameters discussed above.
Parameter Symbol Definition
Physical
Significance
Typical
Values
of
experiments
Magnetic
Reynolds Rem
number
Interaction
N
Parameter
d
a
UL
advection of B0
diffusion of B0
10-2
Ha 2
Re
B02 L
U
Lorentz force
inertia
1
Hartmann
number



B0 L

 Lorentz force 


 viscous force 
inertia
viscous force
Ha
12

UL
Reynolds
Re
number
u

Magmetic
viscous diffusion

Re m
Prandtl
Prm
magnetic diffusion
M
Re
number
Table 1: Essential dimensionless parameters in MFD.
102
1.6 x 104
10-7
23
2.4 Dimensionless equations
By using the scalings given in Eqs. (20)-(26) we can transform the governing
equations into a dimensionless form. In the following we would like to give two
representations. The first one refers to the full MFD case characterized by a
small but infinite magnetic Reynolds number. In this case the full magnetic induction equation is considered. The second representation refers to the quasistatic approximation. Here, upon taking the limit ReM << 1, the induced magnet
field must not be calculated. Instead, a Poisson equation for the scalar electric
potential has to be considered. Both representations of the sets of equations
are implemented in the commercial program package FLUENT which will be
used later on for conducting the numerical simulations.
The set of the full MFD equations is given by the following relations.
  U  0,   B  0,
(18) (10)
U
1 2
Ha
  B   B ,
 U    U    p 
 U
t
Re
Re  Re M
B
1
 (U   ) B  (B   ) U 
 2B .
t
ReM
2
(33)
(34)
Here we have used Ampere’s law (cf. Eq. (9)) to eliminate the electric current
density in the Lorentz force term, i.e. the last term in Eq. (33). Interestingly, in
this term the parameter group Ha2/(ReReM) appears. Using the definitions of
we
can
rewrite
this
group
to
obtain
Ha,
Re,
and
ReM
2
2
2
2
2
2
Ha Re ReM   N ReM  B0 0 U  UA U  MaM , where UA = B0(µ0)-1/2 is
the Alfven velocity representing the propagation velocity of electromagnetichydrodynamic waves and MaM  U UA is the so-called magnetic Mach number.
In liquid metal MFD we typically find that MaM is much less than unity. Physically this means that the magnetic field lines are very stiff and can only slightly be
bent by the flow.


The problem of the above representation is that we have to equip Eq. (34) with
proper boundary condition for the induced magnetic field b, cf. Eq. (14). The
physically correct one would be the far-field condition b  0 as x  . This
means that the induced magnetic field vanishes far away from the location
where it is created, i.e. far away from the fluid region within which the superimposed primary magnetic field is present. However, in numerical simulations the
computational domain must be kept infinite. Therefore, in the used program
package Fluent a default boundary condition is implemented that sets the tangential components of b to zero at the fluid boundaries. By that only the fluid
domain has to be considered. Less problematic are the boundary conditions for
the velocity field u. Along with velocity inlet and a pressure outlet condition we
choose the no-slip condition at rigid walls and the stress-free condition at the
free surface.
24
Within the quasi-static approximation the governing MFD equations read as
  U  0,
(18)
U
1 2
 U    U    p 
 U  N    U  B 0   B o ,
t
Re
(35)
 2     U  B 0  
(36)
Here B0 denotes a given primary magnetic field distribution normalized by a
characteristic value B0. Moreover, we have introduced the electric potential  as
within the quasi-static approximation the electric field is curl-free and can be
represented by the gradient of a scalar, i.e. E = -. The electric potential is
governed by a Poisson equation, cf. Eq. (36). Physically, this equation expresses that the electric current density field is divergence-free, cf. Eq. (11). Here, the
gradient of the electromotive force u  B0 acts as a source term. Hence, a
strongly vortical applied magnetic field or a flow with a high vorticity is needed to
feed the potential. During LFV both conditions are met. In this representation,
along with the same hydrodynamic boundary condition as above, now we have
to formulate boundary condition for . In case of an electrically non-conducting
wall or a free-surface the electric current cannot penetrate into the wall or cross
the surface. Hence normal derivative of  is zero. At the inlet and the outlet 
can be set to be zero.
As it can be seen from Eqs. (35)-(36), by applying the quasi-static approximation we have derived a considerable simplification compared to the full MFD
case. Instead of solving the unknown vector field b we just have solve the scalar field  determined by a Poisson equation. Moreover, we have got rid of the
extremely different diffusive length and time scales set by  and . Finally, without any restrictions the computational domain is given by the fluid region as
physically correct electromagnetic boundary condition can be applied at all fluid
boundaries.
25
3 Electromagnetic flow rate measurement in turbulent liquid metal duct flow
3.1 Introduction
Electromagnetic flow rate measurements are carried out in the test facility EFCO. The experimental set-up is shown in Fig. 12. The facility consists of a
closed channel with rectangular cross-section of height × width = 80 × 10mm2
corresponding to a hydraulic diameters of dH = 18mm. Each Lorentz force flow
meter consists of two block-type permanent magnets of height × width × thickness = 100 × 30 × 20mm3 that produce a magnetic induction about 300mT at
their inner surfaces corresponding a Hartmann number of 140. The magnets
are connected by an iron yoke that guides the magnetic flux density. A force
sensor is mounted on the yoke to record the force that the fluid exerts on the
magnet. Fluid flow is driven by a frequency-controlled electromagnetic pump
(EMP) based on rotating permanent magnets. Moreover, the facility is equipped
with a Vives probe to measure the local velocity in the mid plane of the channel
and an Ultrasonic Doppler Velocimetry to measure velocity profile across the
height of the channel. The low-melting liquid metal GaInSn was applied as
working medium. Its temperature is measured by a submerged thermocouple
and regulated at 20°C by a water-cooled heat exchanger.
Fig. 12: Set-up of test facility EFCO for non-contact flow rate measurement in turbulent
liquid metal flow using single LFV.
26
3.2 Using Lorentz Force Velocimetry
3.2.1 Experimental results
In this chapter a summary of the experimental results of non-contact flow rate
measurements using single Lorentz Force Velocimetry was provided. The
channel is equipped with a single LFV consisting of two permanent magnets
that generate a localized spanwise magnetic field. Attached to the magnetic
system is a force sensor that records the Kelvin force acting in the streamwise
direction. In Fig. 13, the measured Lorentz force is plotted as a function of the
volumetric flow rate. We observe that at low and moderate flow rates there is a
linear relation between the measured force and the flow rate. This finding corresponds exactly to the scaling law according to Eq. (1-2). However, at higher
flow rate values, we find a deviation from the linear behaviour: upon increasing
volumetric flow rate Q, there is only a slight increase of the corresponding Lorentz force. We attribute this experimental result mainly to fact that in this regime, saturation of the EPM may take place.
Fig. 13: Lorentz force FL as a function of volumetric flow rate Q.
Fig. 14 shows results of UDV measurements at rotation frequency of EMP at
25Hz corresponding to a Reynolds number of 1.6 ×104. Here, the streamwise
velocity is plotted versus the depth of the channel. When the EPM operates in
the counterclockwise mode, we detect a purely hydrodynamic turbulent flow
profile characterized by a nearly constant bulk velocity and sharp gradients near
the top and bottom of the channel. On the other hand, in the case of clockwise
EPM operation mode, a typical M-shaped MFD profile is registered. Here, the
bulk velocity is reduced due to the braking Lorentz forces. However, according
to the principle of mass flux conservation, fluid is pushed aside. Within this socalled side layers (at top and bottom of the channel) the induced electric eddy
currents turn to loop back. The generated Lorentz forces basically act in the
27
side layers supporting the pushing. This is a well-known finding is MFD flow
with electrically insulating side walls.
Fig. 14: Velocity profiles at the mid plane of the channel measured by UDV. Counterclockwise mode: turbulent hydrodynamic profile. Clockwise mode: M-shaped MFD profile.
The reason why we do not find the velocity going to zero at the bottom of channel wall are considered to be occurrence of multiple reflections of ultrasonic
beam on the channel walls. The detail explanation about this effect was described in chapter 1.2.
3.2.2 Numerical simulations
To support the experimental findings we perform numerical simulations using
the commercial code ANSYS/FLUENT. We consider turbulent liquid metal
channel flow affected by a localized constant magnetic field that is pointing in
the spanwise direction (positive z-direction), see Fig.15. The geometry of the
channel for the numerical simulation is identical to the experimental set-up.
Here, we simultaneously solve the Navier-Stokes equations and the magnetic
induction equations. These equations are fully coupled via the Lorentz force
term in the Navier-Stokes equations (19) and the effects of advection-stretching
of magnetic field lines by the flow in the induction equation (13). As appropriate
hydrodynamic boundary conditions we use no-slip conditions at rigid channel
walls, a turbulent purely hydrodynamic flow profile as an inlet condition, and a
zero-pressure as outlet condition. Furthermore, as electrodynamic boundary
conditions, we choose electrically insulating channel walls, as well as perfectly
28
conducting interfaces at the entrance and exit planes of the magnetic field. It is
to be noticed, at these planes the tangential component of the induced magnetic field is set to be zero.
Fig. 15: Numerical simulation for Re = 1.54 x 106 and Ha = 140. Top graph: geometry of
the channel wall. Bottom graph: velocity profiles in entrance, inside, and at the exit of the
magnet field.
We observe that the initially hydrodynamic profile is reshaped under the influence of the localized magnetic field. In accordance with the measurements we
find a considerable reduction of the bulk velocity and the pushing of fluid into
the side layers, see Fig.15. At the walls perpendicular to the magnetic field, socalled Hartmann layers [25] [65] are formed, the thickness Ha of which scaling
according to the relation
Ha/dH  Ha-1.
(37)
In the Hartmann layers the velocity sharply increases from zero at the wall to
the almost constant value in the core. As we shall see later on, in these layers
strong eddy currents are induced that are flowing in the opposite direction than
in the core in order to form closed loops. Hence, the Lorentz forces act as a flow
driving mechanism contributing to the creation of such thin Hartmann layers.
The transversal electric current of maximal intensity is generated in the central
area of the gap between the magnetic poles due to the interaction of the horizontal component of magnetic field with the streamwise component of the flow
velocity. So far as the electrical current must close outside the region of magnetic field, it changes its direction from spanwise to streamwise near the side
walls. By virtue of this effect, the streamwise component of the braking Lorentz
force is higher in the mid plane of the magnet gap compared to the value in the
29
vicinity of the sidewalls. It is characterized by low values of velocity in the central part of the channel and two strong maxima near the sidewalls. Such kind of
velocity profile is usually referred to as an M-shaped profile. [25]
The flow profile deformations described above can physically be understood by
analyzing the flow paths of the induced electric eddy currents. The results are
shown in Fig. 16. Here, the magnitude of eddy current density is plotted in the xy plane at position z = 0, i.e. at mid plane of the channel. The flow comes from
positive x-direction. The region, within which the localized magnetic field is present, is marked by red lines. The magnetic field points out of the paper plane.
Fig. 16: Induced eddy currents in the region of the localized magnetic magnetic field.
We observe that inside the magnetic field region strong eddy current are flowing
in the negative y-direction. However, in the side layer the eddy currents turn
their direction. Here, the currents basically flow in the streamwise direction (xdirection) and loop back outside the magnetic field region. Hence, also the Lorentz forces shall change direction, acting basically in the y-direction. The simulations results correspond well to the experimental investigation using UDV, see
Fig.14.
A three-dimensional picture of the eddy currents paths at half-height of the
channel is shown in Fig. 17. As stated before, we observe that near the Hartmann walls which are perpendicular to the magnetic field, strong eddy currents
are flowing in the positive y-direction. The interaction of these currents with the
magnetic field causes the formation of thin Hartmann layers while fluid is accelerated by the generated Lorentz forces.
30
Fig. 17: Eddy currents in the region of the localized magnetic field at half-height of the
channel.
Fig. 18 shows the Lorentz forces density generated within the region of the localized magnetic field at half-height of the channel. As discussed before, we
observe that in the core region the Lorentz forces act as a braking force while in
the Hartman layers they act as a flow driving force.
Fig. 18: Lorentz force density in the region of the localized magnetic field at half-height of
the channel.
31
Fig. 19: Lorentz force density in the region of the localized magnetic field at the mid
plane of the channel.
Finally, Fig. 19 shows the Lorentz force density within the region of the localized
magnetic field at the mid plane of the channel. We observe the braking effect of
the Lorentz forces within the core region of the flow. Moreover, we observe that
near the top and bottom of the channel the Lorentz forces act towards the walls
contributing to the formation of the M-shaped velocity profile within these side
layers.
The numerical results demonstrate that the main contributions to the integral
Lorentz force measured by the flow meter come from the entrance and exit regions of the localized magnetic field. In this region, the strongest braking forces
are induced. In the core region of the magnetic field, the braking Lorentz forces
are much weaker and may completely be compensated by the accelerating
forces in the Hartmann layers.
3.3 Using Time-of-Flight Lorentz Force Velocimetry
In the second step we extend the experimental investigations to ToF LFV, using
the same test facility EFCO. Here, two identical Lorentz force flow meters are
arranged in a row and separated by a certain distance D, see Fig. 20. The
measurement principle is described in chapter 1.3. In the experiments we vary
the rotation frequency of EMP and the separation distance of the two flow meters. We determine the volumetric flow rate Q by using magnet systems of
which the magnetic field penetrates the entire cross-section of the flow. The
magnet has a flux density B of 200 mT on its surface. Vortex structures are triggered by submerging a cylinder in the flow. The measured velocity range 0 ≤ u
≤ 32 cm/s. Therefore, the corresponding ranges of Reynolds and Hartmann
numbers are Re = [0, 1.6 x104] and Ha = 140, respectively.
32
3.3.1 Experimental results
The present study compares the measured correlation time  (transit time) with
the mean convective time which the ratio of distance D between the two flow
meters and the velocity V of the flow. Most of results about ToF Lorentz force
flow meter in this chapter have been published in the scientific journal “Measurement Science and Technology”, see Ref. [27].
Fig. 20: Set-up of the test facility EFCO for non-contact flow rate measurement in turbulent liquid metal flow using Time-of-Flight Lorentz Force Velocimetry.
Fig. 21 shows velocity profiles across the depth of the channel measured by
UDV. Here, the rotating frequency of EMP is fixed at f = 25Hz. The distance
between two flow meters keeps at D = 220mm. This corresponds to a mean
velocity of about 32cm/s and a duct flow Reynolds number of Red = 1.64 × 104.
Hence, we are in the regime of fully turbulent liquid metal channel flow.
33
Fig. 21: Velocity profiles at the mid plane of the channel measured by UDV. The green
curve gives the profile when both flow meters are present. The blue curve gives the profile when LFV 2 was removed.
When the second flow meter is removed, the single LFV results [32] [33] are
retained, cf. blue curve. In this case, a typical M-shaped MFD profile is registered. Due to the second flow meter, the M-shape profile becomes even more
pronounced, cf. green curve in Fig. 21. This indicates that a turbulent flow
profile that is already shaped by MFD effects is very sensitive to the presence of
a second localized magnetic field. For instance, due to the second magnetic
field, the core velocity decreases by a factor of 2 while the peak velocity in the
side layer increases by a factor of 3. Moreover, the side layer becomes thinner.
One may expect that the profiles are symmetric with respect to the half-height of
the channel at 40mm (total height of the channel 80mm). Unfortunately the used
ultrasound transducer cannot resolve properly the region close to the bottom of
the channel due to undesired reflections on the bottom wall. Details of the UDV
measurements are given in [34].
Our experiment procedure results in evaluating the travelling speed Vvortex = D/,
cf. Eq. (4), of any vortex structure that are present in the flow. To increase the
rate of such vortex structures and likewise the rate of usable signals, cf. Eq. (7),
a cylindrical obstacle is submerged into the flow. Such cylindrical obstacles are
commonly used in vortex flow meter devices [35] [69]. The used cylinder has a
diameter of a = 8mm and is submerged at an angle of  = 60° to the horizontal
direction. This corresponds to a submerged length of 100mm. Here the angle of
60° is experimentally pre-fixed for the UDV measurement; see the left picture in
Fig. 8. Due to the restriction of experimental set-up, measurements using both
34
UDV and submerged cylinder cannot be performed simultaneously. In the present experiments the cylinder serves to trigger vortex structures that may be
first registered by LFF 1 and then by LFF 2.
Fig. 22: Flow velocity (in m/s) in the horizontal plane at y = 2.5mm around the cylinder
submerged at an angle of  = 60° into the flow. Parameters are Red = 1.6 x 104, ReC = 7.42
x 103, and the blockage ratio of the cylinder of  = 0.8.
To illustrate the effect of the cylinder on the flow we perform purely hydrodynamic 3D numerical simulations using the commercial CFD code ANSYS with
LES turbulence modelling. A total of 3 million elements and an extra fine meshing of the cylinder region have been used. As an example, Fig. 22 shows the
contours of the velocity magnitude in the cylinder wake at the position
y = 2.5mm in the horizontal plane parallel to the bottom of the channel. The flow
Reynolds number Red and the cylinder Reynolds number ReC, defined by
Re d  V  dH / , Re C  V  a / ,
(38)(39)
Respectively, are fixed at Red = 1.65 x 104 and ReC = 7.42 x 103. Here,
dH = 18mm is the hydraulic diameter of the channel. In the present case the
blockage ratio of the submerged cylinder is  = 0.8 as the channel width is
10mm. Here, a fully developed turbulent flow profile is used as an inlet condition. At the outlet a fixed pressure condition is applied. We find that due to the
presence of the side walls, the formation of a classical Karman vortex street [36]
is suppressed. Instead, generated vortex structures travel in the near-wall jets.
A parametric numerical study on this vortex shedding problem with variation of
the parameters Red, , and  is given in [37].
Figs. 23 and 24 illustrate our procedure to evaluate the transit time. In Fig. 23
we present the raw data of the voltages U1(t) and U2(t) delivered by the strain
gauge of LFF1 and LFF 2. Here, signals are shown within a typical time period
of 10s for an experimental run for which the rotation frequency of the EMP and
the separation distance were fixed at f = 23Hz and D = 170mm. Each run lasts
35
320s. In Fig. 23 the red curve gives the voltage data registered by LFF 1 while
the green curve refers to LFF 2. Due to the limitation of the data acquisition system is the sampling frequency in our experiments restricted to 25s-1.
60000
Original signals of LFV1 and LFV2
LFV1
LFV2
58000
56000
Voltage, µV
54000
52000
50000
48000
46000
44000
0,00
0,40
0,80
1,20
1,60
2,00
2,40
2,80
3,20
3,60
4,00
4,40
4,80
5,20
5,60
6,00
6,40
6,80
7,20
7,60
8,00
8,40
8,80
9,20
9,60
10,00
42000
Run time [s]
Fig. 23: Raw voltage signals U1(t) and U2(t) delivered by the strain gauge of flow meters
LFF 1 and LFF 2 over a period of 10s.
Fig. 24: Auto- and cross-correlation functions calculated according to Eq. (7) from the
voltage signals U1(t) and U2(t) shown in Fig. 23 above. Experimental parameters are fixed
at f = 23Hz and D = 170mm.
36
Fig. 24 shows the auto- and cross-correlation functions obtained by evaluating
Eq. (7) and using the voltage data given in Fig. 23. In the respective graphs, the
value on the abscissa of the first peak corresponds to the desired transit time  .
We observe that there is a clear peak at correlation time  = 0.24s, see red
curve in Fig. 24.
The two figures (25) and (26) below summarize the experimental findings.
These graphs show the measured interrelation of the flow velocity V and the
convective velocity of vortex Vvortex at the two different separation distances
D = 170mm (Fig. 25) and D = 220mm (Fig. 26). The data points represent values that have been double-averaged over 20 individual time periods of 10s of
each run and a total of 12 experimental runs for each pump rotating frequency.
The typical standard deviation is about 10%. As expected, the diagrams indicate a linear behavior between the flow velocity V and Vvortex. Moreover, we observe that when the distance between the two LFFs is increased, the transit
time for the vortex structures passing through is extended. However, we find
that for each separation distance, the vortex velocity is considerably higher than
the velocity of the flow, i.e. Vvortex > V. Furthermore, the slope of the curves is
directly representing the calibration constant k2, cf. Eq. (5).
We find the values k2 = 1.90 for D = 170mm and k2 = 0.95 for D = 220mm.
Hence, the calibration constant decreases with increasing separation distance.
We attribute these findings to the facts that due to the submerged cylinder nearwall jets are created that are mainly carrying the vortex structure detected by
the flow meters, cf. Fig. 22, and that under the influence of the localized magnetic fields the flow profile is deformed into an M-type shape, cf. Fig. 14 and Fig.
21. At large separation distance the jets may have decayed so that calibration
constant approaches unity.
Fig. 25: Averaged relation between the vortex velocity Vvortex measured by ToF LFF and
the velocity of the flow V measured by UDV and Vives probe. The separation distance is
fixed at D = 170mm. The slope of the linear fitting curve is k2 = 1.90.
37
As mentioned before, in these test experiments, the sampling period of the data
acquisition system is restricted to 40ms. This may explain the somewhat stepwise distribution of the data points shown in Fig. 26. Moreover, we observe that
at larger distances, both the rate and the reproducibility of usable data decreases. We attribute this finding to the fact that any vortex structure generated by
the obstacle is strongly re-shaped by the first magnetic field. Therefore, it shall
hardly be re-detected by the second LFF. On the other hand, at smaller distances, the two localized magnetic field may overlap; see Fig. 51 in sub-chapter
4.2.2.
Fig. 26: Averaged relation between the vortex velocity Vvortex measured by ToF LFF and
the velocity of the flow V measured by UDV and Vives probe. The separation distance is
fixed at D = 220mm. The slope of the linear fitting curve is k2 = 0.95.
3.3.2 Numerical simulations
To support the experimental findings we perform numerical simulations using
the commercial code FLUENT. We consider turbulent liquid metal channel flow
affected by two localized constant magnetic fields that are pointing in the
spanwise direction (positive z-direction). The geometry of the channel is identical to the experimental set-up. Here, we simultaneously solve the Navier-Stokes
equations, Eq. (19) and the magnetic induction equations, Eq. (13).
Fig. 27 shows the computational domain and the used mesh. A total of 3 million
elements have been used. Here, within the volumes Fluid 2 and Fluid 4, a constant transverse magnetic field B0 is applied. Fluid volumes 1 and 5 represent
the inlet and the outlet regions, while Fluid 3 is the volume between the two
LFFs assumed to be field-free. As appropriate hydrodynamic boundary conditions we use no-slip conditions at rigid channel walls, a turbulent purely hydrodynamic flow profile as an inlet condition, and a zero-pressure outlet condition.
Moreover, as electrodynamic boundary conditions, we choose electrically insulating channel walls, as well as perfectly conducting interfaces at the entrance
38
and exit planes of the magnetic field. Furthermore, at these planes the tangential component of the induced magnetic field is set to be zero. In the simulations
we apply the WALE-LES turbulence model.
Fig. 27: Computational domains and meshes used in the numerical simulations. Within
the volumes Fluid 2 and Fluid 4 a constant spanwise magnetic field is applied.
In the following we show to the graphical presentation of flow profiles in the x-yplane (at the mid plane of the channel). Fig. 28 and Fig. 29 illustrate the effect
of the flow Reynolds number at a fixed value of the Hartmann number Ha =140.
The Reynolds numbers are fixed at Red = 2.0×104 (Fig. 28) and Red = 3.0×104
(Fig. 29).
Fig. 28: Flow velocities (in m/s) in the mid plane of the channel at Red = 2.0×104
(V = 0.4m/s) and Ha = 140.
39
The flow is from the left to the right. Black vertical lines indicate the regions
within which the constant magnetic field is applied. The field points out of the
plane. We observe that due to the influence of the first magnetic field, M-shaped
flow profiles are formed. Therefore, MFD side layers appear at top and bottom
of the channel.
Fig. 29: Flow velocities (in m/s) in the mid plane of the channel at Red = 3.0×104
(V = 0.6m/s) and Ha = 140.
In the region between the magnetic fields, these profiles remain unchanged.
However, upon entering the second localized magnet field, the bulk flow is once
more slow and more fluid is pushed into the side layers. By that, the M-shape
profile becomes more pronounced. This finding corresponds well with UDV
measurements, see Fig. 21. At higher Reynolds number, the braking effect is
increased resulting in higher velocity gradients across the height of channel.
Fig. 30 shows the effect of the Hartmann number. Here, the parameters are
fixed at Red = 3.0 x 104 and Ha = 205. We observe that upon increasing the
strength of the magnetic field, the MFD effects increase likewise. The difference
between the bulk velocity and velocity in the side layers strongly increases and
the thickness of the side layers decreases.
40
Fig. 30: Flow velocities (in m/s) in the mid plane of the channel at Red = 3.0×104
(V = 0.6m/s) and Ha = 205.
Finally, Fig. 31 shows some of the calculated profiles at various positions in the
streamwise direction. Parameters are fixed at Red = 4.8 x 103 (V = 0.1m/s) and
Ha = 205 (B0 = 0.3T). The blue line refers to the turbulent purely hydrodynamic
entrance profile. As the flow has passed LFF 1, due to the action of the Lorentz
forces it has been transformed into an M-shape profile, see line 2 (green curve).
As already stated above, in the field-free region between LFF 1 and LFF 2 this
profile is mainly unchanged. We observe a small increase of the bulk velocity,
since in this region braking Lorentz forces are absent, see line 3 (red curve).
After having passed LFF 2, the profile was once more re-shaped into an M-type
form and the bulk velocity has decreased again, see line 4 (black curve). Finally, at the outlet, the profile starts to transform back into a purely hydrodynamic
shape. The bulk velocity increases and the side layers start to dissipate. However, the peak velocity in the side layers has increases, see line 5 (orange
curve). This prediction is due to the fact that the so-called Hartmann layers rapidly breakdown. These layers are formed adjacent to the side walls of channel
that are perpendicular to the magnetic field. By conservation of mass, fluid is
pushed into the side layers. These numerical findings qualitatively correspond
with the UDV measurements shown in Fig. 21. However, the predicted reduction of the bulk velocity and the increase of the peak velocity in the side layer
due to the presence of the second magnetic field are considerably lower than
observed in the experiments. We attribute these quantitative deviations to both
the limited resolution of the UDV and the simplifications made in the simulations.
41
Fig. 31: Reduced velocity profiles at various positions in the streamwise direction: Line
1: inlet, Line 2: right behind LFF1, Line 3: between LFF1 and LFF2, Line 4: right behind
LFF2, Line 5: outlet. The separation distance of the flow meters is D = 150mm.
To get more physical insight into the formation of the M-shape velocity profile,
Figs. 32 and 33 show the distribution of the induced eddy current density and
the resulting distribution of the Lorentz force density in the mid plane of the
channel. The flow comes from right to left. Here, LFV 1 and LFV 2 denote the
regions within which the localized magnetic field is present. Parameters are
fixed at Red = 1.43 x 104 (V = 0.3m/s) and Ha = 68. The separation distance of
the flow meters is D = 100mm.
Fig. 32: Induced eddy current density (in A/m2) in the mid plane of the channel. The applied localized magnetic fields are pointing out of the plane.
42
According to Ohm´s law, cf. Eq. (12), strong eddy currents are induced within
the regions LFV 1 and LFV 2. In the bulk areas of these regions the eddy currents are mainly flowing in the positive vertical direction (positive y-direction), cf.
Fig. 32 and Eq. (12). According to Eq. (15), this eddy current distribution gives
rise to strong Lorentz forces pointing opposite the flow direction, cf. Fig. 33.
Hence, these forces tend to brake the flow and the bulk velocities are considerably reduced.
Fig. 33: Lorentz force density (in N/m3) in the mid plane of the channel. The applied localized magnetic fields are pointing out of the plane.
Fig. 34 illustrates 3D M-shape velocity profile in stream- (a) and spanwise direction (b). Moreover, due to the solenoidal constraint according to Eq. (11), in the
vicinity of the electrically insulation top and bottom wall, the eddy currents must
turn into the flow direction. Eventually they loop back inside the liquid within the
regions where the magnetic field is absent, cf. Fig. 32. Therefore, in these top
and bottom regions, the Lorentz forces change their direction likewise and are
mainly pointing into the top and bottom wall, see Fig. 33. This effect reinforces
the formation of the M-shape profile. A parametric numerical investigation on
the impact of magnetic field penetrating in spanwise direction in turbulent liquid
metal duct flow is given in [41].
43
Fig. 34: Velocity profiles in streamwise direction (a) and in spanwise direction (b).
44
4 Electromagnetic free surface velocity measurement in annulus flow
4.1 Introduction
In this chapter we describe experimental results of respective measurements in
the test facility LiMeSCo. To demonstrate the general feasibility of Time-ofFlight technique to sense local surface velocities employed miniaturized LFV, in
a first step we measure surface velocities during solid-body rotation of metallic
substances. We observe that small modulations in the signals that are due to
the natural unevenness of the metallic body can be used to determine surface
velocities. In a second step we apply the Time-of-Flight technique to liquid metal
free-surface flow. Here we use the low-melting eutectic alloy GaInSn as a model fluid. In this case we observe that the cross-correlations between the signals
are much rarer and much weaker than in the solid-body experiments. We conclude that the presence of the localized magnetic fields, produced by the permanent magnets of the miniaturized LFV, give rise to intense deformation of the
free surface and the creation of surface waves. The field acts as a magnetic
obstacle. We observe that the flow velocity is decreased right underneath the
magnet and is accelerated in the side regions.
A photograph of the experimental test facility LiMeSCo is shown in figure 35. It
consists of a ring channel that is put into controlled rotation by an electrical motor. The inner and outer radii of the channel are 17.5cm and 24.5cm, respectively. At a certain height H above the channel, we arrange two small cubic magnets attached with force sensors separated by a certain distance D. Each cubic
magnet is with edge length A = 20mm producing a magnetic induction of 270mT
at its bottom face. Moreover, the permanent magnets are mounted on strain
gauge sensors that record the Lorentz forces generated in the motion of electrically conducting liquid metal. To support these experimental findings we present
results of numerical simulations in sub-chapter 4.2.2 and 4.3.2 using the commercial code MAXWELL.
45
Fig. 35: Experimental set-up of test facility LiMeSCo.
4.2 Metallic body experiments
4.2.1 Experimental results
On top of the channel there are two solid aluminum sheets of thickness 1.5mm.
In one sheet there is a drilling hole of 1mm in diameter. At a certain height H
above the channel we arrange a small cubic permanent magnet attached with a
force sensor. A blow-up is shown in the left picture in Fig. 36. To check the
functionality of the measurement principle and for calibrating purposes, a thin
iron wire is tightened across the channel; see the right picture in Fig. 36.
Fig. 36: Single LFV with two rotating aluminum sheets, one of which with drilling hole
(left); The iron wire (circled in red) is used to trigger controlled signals for counting the
experimental runs (right).
46
As an example, Fig. 37 shows the force signals produced whenever the iron
wire (signal 1) and the aluminum sheets (2 with hole and 3 without hole) are
passing underneath the miniaturized LFV. The iron wire produces two sharp
force peaks corresponding to entering (downward peaks) and exiting (upward
peaks) the localized magnetic field. The iron wire signal (4) demonstrates the
beginning with the 2nd rotation run of the channel. On the other hand, due to
their finite extend the aluminum sheets produce more smooth signals. Here signal (2) is slightly modulated due to the presence of the small hole on the aluminium sheet. We conclude that LFV can sense both surface velocities and small
perturbations caused by sudden changes of the electrical conductivity of the
moving metallic bodies.
Fig. 37: Voltage signals of the iron wire (1); the Aluminum sheets with hole (2) and without hole (3) and iron wire (4) as function of time.
Fig. 38 summarizes the entire series of experiments. It shows the measured
Lorentz forces FL [mN] as a function of the height H [mm] above the channel
and the moving velocity V [cm/s]. As expected, the Lorentz force increases upon increasing velocity and by decreasing height.
47
Fig. 38: Lorentz force FL vs. driving velocity V and height H.
Fig. 39 shows more details about the measurements in a dimensionless representation. Here, the measured coefficient of the Lorentz force CL (Lorentz force
divided by hydrodynamic pressure force) eq. (40) is plotted against the magnetic Reynolds number eq. (27) for various heights H.
CL  FL (1 2  V 2 A )
(40)
We observe that all experiments we are within the so-called quasi-static MFD
approximation characterized by Rem<< 1, i.e. the magnetic field induced by the
eddy currents remain small compare to the externally applied magnetic field. As
expected, the dimensionless Lorentz force CL decreases upon increasing both
ReM and H. This finding reflects the fact that the Lorentz force is linearly increasing with Velocity V and decreases with height according to H-3, Eq. (3).
48
H= 0.10 cm
H= 0.20 cm
H= 0.32 cm
H= 0.44 cm
H= 0.80 cm
H= 1.30 cm
H= 1.90 cm
H= 2.90 cm
0.025
F/1/2v2A
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
Re
0.4
0.5
0.6
m
Fig. 39: Coefficient of Lorentz force CL as a function of magnetic Reynolds number Rem
and height H.
In the starting experiment, we show that using a single LFV, precise measurement of surface velocities in controlled solid body rotation is possible. Furthermore, we extend the rotary metallic bodies experiments by applying the ToF
LFV to determine local velocities. A photograph of the experimental set-up is
shown in Fig. 40. A thin iron wire is tightened across the channel (not shown in
Fig. 40). On top of the channel there is a solid aluminum disk. At a certain
height H above the channel we arrange two small cubic magnets attached with
force sensors separated by a certain distance D. Here, we record the voltage
signal delivered by the two LFVs as a function of time. We evaluate the time
shift  between the two voltage signals and recalculate the velocity V according
to the relation V = D/.
Fig. 40: Set-up of solid-body rotation experiments using Time-of-Flight LFV.
49
An example is given in Fig. 41. Here, large-amplitude peak signals are due to
the motion of the iron wire through the two localized magnetic fields. Small
peaks and signal fluctuations are due to the natural inhomogeneity of the aluminum disk. We observe that there is a clear time shift between the raw voltage
signals. We detect that not only the sharp peaks produced by the iron wire but
also the small fluctuations of the signals can be used to determine surface velocities.
Fig. 41: Raw voltage data U1(t) (blue curve) and U2(t) (green curve) recorded by the force
sensors LFV 1 and LFV 2. The velocity is fixed at V = 30cm/s.
In means of standard signal processing methods, we additionally calculate the
transit time using the averaged auto- and cross-correlation functions Eq. (7)
denoted by Rxx and Rxy, respectively.
50
Fig. 42: Auto- and cross-correlation functions calculated according to Eq. (7) and using
the raw voltage signals U1(t) and U2(t) shown in Fig. 41.
On the top of the channel there is a solid aluminum sheet of thickness 1.5mm,
see the left picture in Fig. 43. In a second series of experiments we replace the
sheet by a ring-type aluminum plate, see the right picture in Fig. 43. At a certain
height H above the aluminum plates we arrange the prototype of a Meniscus
Velocity Sensor (MVS) based on the Time-of-Flight technique. A detailed description of the MVS is to be found in chapter 5.3. The two magnets are separated by the distance D = 65mm. The edge length of both cubic magnets is
20mm. For protection of the sensor against heat and dust in an assumed industrial condition, it is arranged inside a double-walled housing made of stainless
steel. In the experiments we vary the rotation speed V of the channel, the separation distance D between magnets and the height of the gap between MVS
sensor and the aluminum plate.
Fig. 43: Set-up of the solid-body rotation experiments using an aluminum sheet (left) and
an aluminum ring-type plate (right).
51
Fig. 44 represents the experimental results of the rotating sheet at V =
41.27cm/s and H = 28mm. The graph on the top of Fig. 44 shows the raw signals, i.e. the voltage recorded by the two force sensors. The graph below in Fig.
44 shows the corresponding cross-correlation functions Rxy and Ryx. We observe that there are clear peaks in the raw signals whenever the sheet passes
underneath the force sensor. Moreover, we observe a clear time shift in the raw
signals.
4
x 10
-1.5062
LKA1
-1.5064
Voltage, mV
-1.5066
-1.5068
-1.507
-1.5072
-1.5074
0
1
2
3
4
5
6
7
8
-1721
LKA2
-1722
Voltage, mV
-1723
-1724
-1725
-1726
-1727
-1728
-1729
0
1
2
3
4
Time, s
5
6
7
8
Averaged Cross Correlations for LFV1 and LFV2, T=1s
0.7
 mean=0.1575
0.6
<Rxy>
<Ryx>
Cross-Correlation
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time
Fig. 44: Raw signals (top) and cross-correlation function (below) for the case of a rotating
aluminum sheet at V = 41.27cm/s and H = 28mm.
1
52
4
Voltage, mV
-1.496
x 10
LKA1
-1.4965
-1.497
0
2
4
6
8
10
12
14
16
18
20
Voltage, mV
-1610
22
LKA2
-1612
-1614
-1616
-1618
0
2
4
6
8
10
12
Time, s
14
16
18
20
Averaged Cross Correlations for LFV1 and LFV2, T=1s
0.3
<Rxy>
<Ryx>
0.25
0.2
Cross-Correlation
22
 mean=0.2377
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.1
0.2
0.3
0.4
Time
0.5
0.6
0.7
0.8
Fig. 45: Raw signals (top) and cross-correlation function (below) for the case of a rotating
aluminum sheet at V = 27.35cm/s and H = 33mm.
Fig. 45 show the corresponding results for the cases V = 27.35cm/s, H = 33mm,
respectively. We observe that in these cases the peaks in both the raw signals
and the cross-correlation functions are considerably weaker. We attribute this
finding to the fact that a decrease in velocity and an increase in height both
cause a significant decrease in the magnitude of the Lorentz force, cf. Eq.(3).
Repeating the experiments for various rotation speed and height, we observe
that our prototype is able to produce reliable results at relatively low heights H <
33mm and at rotation speeds within the range of a few cm/s to 65cm/s.
53
To verify the results measured using the MVS, we installed a photoelectric sensor as reference measurement. For these comparison experiments we decrease the height in a range of 5-13mm. At first, we employed a force sensor
with a resolution of 0.1g (see Fig.72) and set the height of H = 10mm above the
channel, thereby the edge length of the applied magnets are A = 20mm. Fig. 46
shows the results if we let the channel rotate in the counter-clockwise direction.
It means that the aluminum sheet first passes underneath the LFV 1. Accordingly, Fig. 47 depicts the results if we let the channel rotate in the clockwise direction, as the aluminum sheet passes first underneath the LFV 2. In both cases
the measured rotating velocity with MVS is almost the same as the measured
with photoelectric sensor. However, it is remarkable that in the range of 4060cm/s the velocity is not detectable by MVS. Even though the rotating speed
increases, the MVS measures always the constant velocity of 53cm/s.
Fig. 46: Comparison of measured velocity using MVS with measured velocity using laser;
channel rotating counter clockwise.
54
Fig. 47: Comparison of measured velocity using MVS with measured velocity using laser;
channel rotating in clockwise.
One possible reason for this unexpected finding might be the limited resolution
of the force sensor used at that time. For further experiments we decided to
employ a more sophisticated force sensor which produced by company Velomat, see Appendix 3. With the new sensor we broaden the detectable measuring capacity from a few cm/s to 65 cm/s without significant deviations from the
measured reference velocity.
Fig. 48: Experimental set-up for measuring the rotating speed using an aluminium sheet
without housing.
55
Fig. 49: Comparison between the measured velocity using MVS with the one measured
using laser by varying height above the rotating channel.
Nevertheless, it can be found that by increasing the height, the sensitivity of the
MVS sensor decreases. The results given in Fig. 49 show that by increasing the
height from 5 to 13mm, the lowest detectable velocity will increase drastically.
With the height 13mm we are not able to measure the velocities lower than
20cm/s, see the graph at the bottom right of the Fig. 49.
4.2.2 Numerical simulations
For this experimental setup we perform numerical simulations using the commercial software MAXWELL. Here we solve a slightly different problem where
an aluminium sheet is linearly pulled at constant speed underneath two localized magnetic fields. Magnet dimensions and arrangements correspond exactly
to the experimental set-up. MAXWELL solves the three-dimensional transient
induction equation in the magnetic vector potential representation. By that, we
obtain the distributions of the magnetic field produced by the permanent magnets as well as the distribution of the induced eddy currents and Lorentz forces
within the moving metallic sheets.
Figs. 50 and 51 show the results of numerical simulations. Here, both the calculated Lorentz forces Fx [N] in the streamwise direction and induced eddy current
Jz [A/m2] in spanwise direction are plotted as a function of position in streamwise direction. Moreover, the y-component (vertical direction) of the magnetic
field By [mT] is added. The pulling speed is fixed at V = 30cm/s. The height is
fixed at H = 2cm.
56
Fig. 50: Graphs of the y-component By of the magnetic field (red curve), the z-component
jz of the induced eddy current density (blue curve), and the x-component Fx of the generated Lorentz force (purple curve). Parameters are fixed at V = 30cm/s, H = 2cm and D =
10mm.
Fig. 50 refers to a separation distance of D = 10mm corresponding to the halfedge length of the cubic magnets. We observe that in this case the two magnetic field overlap. Interestingly, there are localized regions in front of the first magnet field and at the end of the second magnetic field within which the Lorentz
force acts as an acceleration force on the solid body, i.e. pushing it in the
streamwise direction. This is due to the fact that within these regions the eddy
currents are flowing in the opposite direction to form closed loops. By that, also
the Lorentz force changes its direction.
57
Fig. 51: Graphs of the y-component By of the magnetic field (red curve), the z-component
jz of the induced eddy current density (blue curve), and the x-component Fx of the generated Lorentz force (purple curve). Parameters are fixed at V = 30cm/s, H = 2cm and D =
80mm.
Fig. 51 shows the numerical results for a separation distance of D = 80mm and
the same set of parameters as before. In this case the magnetic fields do not
overlap. Moreover, within the region between the two localized magnetic fields,
the Lorentz force acts in the positive streamwise direction. This indicates that
the eddy currents are using the entire gap between the magnets to loop back.
4.3 Liquid metal free surface experiments
4.3.1 Experimental results
Finally, we would like to present first experimental results for the case when the
ring channel is filled with GaInSn. Here, we want to demonstrate that ToF LFV
may also serve as a method to determine local velocities in free-surface flow. A
sketch of the experimental set-up for liquid metal free-surface experiment is
shown in Fig. 52. A ring channel filled with liquid metal GaInSn is put into controlled rotation. This case is closer to the metallurgic applications during continuous casting of steel and production of secondary aluminum.
58
Fig. 52: Set-up for measuring surface velocities in free-surface flow using ToF LFV.
The results of flow visualization using a single permanent magnet are shown in
Fig. 53. We observe that the presence of a localized magnetic field, produced
by the permanent magnet, gives rise to intense deformation of the free surface
and the creation of surface waves. The field acts as a magnetic obstacle which
has already been
Fig. 53: Visualization of liquid metal free-surface flow influenced by a localized magnet
field that is produced by a small permanent magnet located above the surface. Surface
deformation and surface waves are created due to the magnetic obstacle effect.
predicted by numerical simulations [38] and confirmed in respective experiments [39] [70]. We observe that the flow velocity is decreased right underneath
the magnet and is accelerated in the side regions.
Due to the obstacle effect of the magnet on the liquid metal flow the main part of
the liquid flows next to the wall boundaries. Therefore, any flow pertubation
triggered by the first LFV does not reach the zone of influence of the second
LFV. For this reason we equiped the first magnet with an edge length of 10mm
and the second magnet with an edge length of 15mm. The experimental set-up
is shown in Fig. 53. With the liquid metal in rotation and the height of the MVS
kept at 5mm above the channel, we obtain the results shown in Fig. 54. The
blue points are the measurable velocites. The red points represent the linear
59
fitting and the black line is the curve fitted to the blue points. Unfortunaly, very
weak linear behaviour in the results has been recognised.
Fig. 54: Comparison between the measured velocity using MVS and measured velocity
using laser; magnet in LFV 1 with edge length 10mm and magnet in LFV 2 with edge
length 15mm.
Afterwards, we decided to immerse a piece of styrofoam with diameter 40mm
as disturbance body in the liquid metal GaInSn, see Fig. 55, and repeat the
experiment.
60
Fig. 55: Submerged styrofoam as a disturbing body in liquid metal free-surface flow
measurement.
With the aid of this submerged body, we can drastically increase the measurable velocities, see Fig. 56. Moreover, using the styrofoam even higher velocity
up to 65cm/s were detectable.
Fig. 56: Comparison of measured velocity using MVS with measured velocity using laser;
magnet of LFV 1 with edge length 10mm and magnet of LFV 2 with edge length 15mm.
61
Finally, two magnets with the same edge length of 15mm are used. By that, we
achieve better results, i.e. less discrepancy from the linear relationship and a
wider measurement range of velocity, see Fig. 57.
Fig. 57: Comparison of measured velocity using MVS with measured velocity using laser;
both magnets in LFV 1 and LFV 2 are with edge length 15mm.
We observe that the raw signals are very noisy and the cross-correlation is
weak. Repeating the experiments for different values of H and V reveals that so
far the rate of cross-correlations is clearly less than that in the solid-body experiments and that the reproducibility is modest. We attribute these findings to both
the magnetic obstacle effect and the lack of design of MVS. Improvements of
these deficits are a part of the future work.
4.3.2 Numerical simulations
Our numerical approach consists of two steps. In a first step we calculate the
applied magnetic field B0 (x, y, z) that is produced by the two small-size permanent magnets using the commercial finite-element solver MAXWELL. Basically,
MAXWELL solves the Laplace equation for the magnetic scalar potential  defined by
62
 2   0,
where
B 0   0   
(41)(42)
As input parameters we define the geometry of the cubic-type magnets (with
edge length A) and their separation distance D. Moreover, according to the data
sheet provided by the manufacturer of the NdFeB permanent magnets, we
specify both the magnetic remanence (Br = 1.32T) and the coercive field
strength (Hc = 12kA/m).
In a second step we calculate the velocity field u, the total magnetic field B = B0
+ b, where b is the magnetic field induced by the flow, and the pressure field p
in the flow domain using the commercial finite-volume solver FLUENT. Here, we
simultaneously solve the Navier-Stokes equations (19) and the magnetic induction equations (13). As appropriate boundary conditions we apply the no-slip
condition at rigid walls and a zero shear stress condition at the free surface
which is assumed to be non-deformable.
Fig. 58: Geometry of the problem. Two cubic-type permanent magnets of edge length A
are arranged in a row separated by a certain distance D and some height H above the
free surface. Numerical solutions will be given along the lines A (symmetry line in the
streamwise direction), lines B (spanwise direction in front of the first magnet), and lines
C (symmetry line in the spanwise direction).
Moreover, we apply a constant velocity profile u = u0ex as an inlet condition
and zero-pressure p = 0 as an outlet condition. All boundaries are taken to be
electrically insulating. In this case the normal component of the current density
is zero. ANSYS deals with this by setting the tangential component of the induced magnetic field identical to zero. Fig. 58 shows the geometry within which
the equations (13) and (19) are solved. For meshing we use about 6 x 106 elements. As an appropriate turbulence model we choose the Large Eddy Simulation (LES) where the sub-scale structures are modeled by the Wall-Adapting
Local Eddy Viscosity (WALE) method.
63
Applied magnetic field B0
Fig. 59 shows the magnetic field strength along line A for a fixed magnet edge
length of A = 10 mm and various values of the separation distance D and height
H above the free surface.
Fig. 59: Magnetic field strength B0 [T] in streamwise direction.
We observe that upon increasing H the magnetic field strength decreases considerably. Moreover, upon increasing the separation distance D the overlapping of the magnetic fields decreases.
Results of magnetohydrodynamic simulations
In the following we prefer to discuss the results using the dimensionless parameters of the problem. These are the Reynolds number Eq. (29) and the Hartmann number Eq. (31) as well as geometry parameters d, a, and h defined by
the following relations
d = D/L0, a = A/L0, h = H/L0.
(43)-(45)
Here, Bmax denotes the maximum magnetic field strength at the free surface
directly underneath the magnet and L0 is the characteristic length (L0 = 0.04 m).
Moreover, we use scaled values for the streamwise coordinate x, the streamwise velocity u, the Lorentz force density fL  j B Eq. (15) with drag force fx
64
and lift force fz, and the eddy current density j  1  0   B  Eq. (9) (spanwise
component jy) according to the relations
L̂  x L 0 ,Û  u u0 , cfx = fx L0/(½u02), cfz = fz L0/(½u02), cjy = jy/(Bmaxu0).
(46-50)
The Reynolds number represents the ratio of inertia forces to viscous forces.
Usually we have Re  1 indicating that the flow is turbulent. The Hartmann
number represents the ration of electromagnetic forces to viscous forces. Usually we have also Ha >> 1 indicating that Lorentz force clearly dominates over
friction. Please note that Ha varies with Bmax, i.e. with the height H at which the
magnets are positioned above the free surface. The Lorentz forces coefficients
cfx and cfz represent the electromagnetically induced drag and lift force per unit
area reduced by the dynamic pressure of the flow. Physically, they indicate how
strongly the flow will be reorganized under the action of the magnetic field.
Streamwise velocity
As an example Fig. 60 shows the reduced streamwise velocity Û at the free surface along line A for various values of the parameters Re and Ha. The geometry
parameters is fixed at d = 1.25 and a = 0.25. In these graphs vertical lines indicate the location of the permanent magnets. We observe that there is a slight
acceleration of the flow in front of magnet 1, i.e. Û > 1. As we shall see later on
this due to the distribution of the induced eddy current density that are looping
back in this region in the negative y-direction. According to the right-hand rule
this gives rise to Lorentz forces acting in the position x-direction that are pushing fluid in the streamwise direction. On contrast, right underneath the magnets,
the flow is strongly decelerated due to the strong braking Lorentz forces acting
in the negative x-direction. According to mass conservation fluid is pushed
aside in the spanwise direction surround the magnet region. This finding is wellknown as the magnetic obstacle effect. After having left the zone within which
the magnetic field is present, the flow is again accelerated due to the absence
of the braking Lorentz forces. Furthermore, from Fig. 60 we conclude that this
obstacle effect becomes more pronounced upon increasing Hartmann number
and decreasing Reynolds number. This behavior can be physically understood
in terms of the electromagnetic interaction parameter N = Ha2/Re, Eq. (32)
which represents the ratio of electromagnetic forces to inertia forces. In case of
N > 1, Lorentz forces dominate over inertia forces. Hence, the flow profile can
easily be deformed by the electromagnetic forces. On the other hand, in the
case N < 1, the relatively strong inertia forces are able to prevent strong deformations of the flow profile by the Lorentz forces.
65
Fig. 60: Reduced streamwise velocity along line A for various Reynolds number and the
Hartmann number. Vertical lines indicate the locations of the permanent magnets.
This magnetic obstacle effect becomes more obvious when analyzing the
streamwise velocity at the entire free-surface. Fig. 61 shows an example for two
different Reynolds numbers. Hartmann numbers are fixed at Ha = 460 and
d = 1.25, a = 0.25. As expected, for lower Reynolds number at constant Hartmann number, the magnetic obstacle is more pronounced. Moreover, Fig. 62
shows streamwise surface velocity profiles as a function of the spanwise coordinate at two different streamwise locations, i.e. along line B (left graph) and
along line C (right graph), and for two different sizes of magnets. Reynolds and
Hartmann numbers are fixed at Re = 21.000 and Ha = 460, and d = 1.25. As
expected, we find that for larger magnets, the obstacle effect is more pronounced. Results of more parametric studies are given in Ref. [40].
66
Fig. 61: Streamwise velocity distribution (in m/s) at the free surface. Flow direction is
from left to right. Squares illustrate locations of the permanent magnets. Parameters are
Re = 21.000 (top graph) and Re = 63.000 (below graph).
Fig. 62: Reduced velocity in the streamwise direction as a function of the spanwise coordinate along line B (left) and line C (right) for two different sizes of magnets.
Eddy current density
We now would like to discuss results of our numerical simulations concerning
the distribution of the induced eddy current density represented by the coeffi-
67
cient cjy = jy/(Bmaxu0). In our case the y-component, i.e. jy is the most important
one as it is the dominant contribution to the braking Lorentz force. Fig. 63
shows the variation of cjy along line A for various values of the Reynolds number and the Hartmann number.
Fig. 63: Distribution of the coefficient of the eddy current density along line A. Other parameters are d = 1.25 and a = 0.25. Vertical lines indicate the position of the permanent
magnets.
We observe that in the regions under the magnets, strong eddy current are
flowing into the negative y-direction. This gives rise to strong braking Lorentz
forces that contribute to the magnetic obstacle effect. Due to the solenoidal
constraint, i.e. j = 0, these eddy currents must loop back in regions right in the
front and right behind the magnets. By that they turn their direction as well do
the induced Lorentz forces. Hence, in these regions the flow is accelerated. We
expect that the magnitude of the eddy current density strongly increases upon
increasing both Reynolds number and Hartmann number. However, due to the
used scaling the non-dimensional coefficient cjy remains almost constant. An
example of the eddy current distribution in the free-surface plane is given in Fig.
64. Here, parameters are fixed at Re = 21.000, Ha = 460, a = 0.25, and d =
1.25. We observe the formation of two eddy current loops. Underneath the permanent magnets, the currents run in the positive y-direction and loop back in
field-free regions. As it can be seen, the current density induced under magnet
1 is much higher than that under magnet 2. This finding reflects the fact that the
streamwise free-surface flow velocity is much smaller under magnet 2 as the
flow has already been broken by magnet 1.
68
LKA 1
LKA 2
Fig. 64: Distribution of eddy current density at the free-surface.
Fig. 65: Coefficient of the Lorentz force density in the streamwise direction for various
Reynolds number and Hartmann number. Other parameters are fixed at a = 0.25 and
d = 1.25.
69
Drag force coefficient Cfx
According to the relation fL  j B the Lorentz force density is generated as the
result of the interaction of the eddy current and the applied magnetic field. Fig.
65 shows the Lorentz force coefficient cfx = fx L0/(½u02) at the free-surface
along line A for various Reynolds numbers and the Hartmann numbers. We observe that underneath the magnets strong braking Lorentz forces are generated. According to the equation above, this is the result of the eddy currents running in the positive y-direction and the applied magnetic field basically pointing
in the negative z-direction. However, in the regions in the front and behind the
magnets, the drag force coefficient becomes positive indicating accelerating
Lorentz forces. As mentioned before, this finding is due to the fact that in these
regions the induced eddy currents loop back, i.e. running in the negative ydirection. Moreover, we find that cfx increases upon increasing the Hartmann
number. This finding reflects the fact that Ha is a non-dimensional parameter of
the electromagnetically induced drag force. However, from Fig. 65 we conclude
that cfx decreases with increasing Reynolds number. Mathematically, this finding
is due to the scaling of the Lorentz force using the dynamic pressure being proportional to u2. Physically speaking, this finding demonstrates that in the low-Re
case, flow profiles are more sensitive to deformation due to the presence of
electromagnetic forces. Finally, we observe that the magnitude of cfx is much
higher underneath magnet 1 than underneath magnet 2. Again, this result indicates that magnet 1 acts as an obstacle forming a wake. This reduces the velocity and although the induced Lorentz forces underneath magnet 2.
Lift force coefficient and surface deformation Cfz
Finally, we would like to have a look at the electromagnetically generated lift
forces. This force basically results from the interaction of the induced eddy current density jy in the y-direction and the applied magnetic field in the z-direction.
Although, we have assumed that the free-surface in non-deformable flow, the
evaluation of this force may give some practical information about the surface
deformation and the generation of surface waves that have been observed in
the laboratory experiments. To this end, neglecting curvature effects, we make
a simple balance of the hydrostatic and the electromagnetic pressure across the
free-surface that is elevated in the vertical direction by some height h. This
balance reads as
gh  fzL0.
(51)
Upon rearranging Eq. (51) we obtain for the reduced magnitude of deformation
h/L0 = fz/(g).
(52)
Fig. 66 shows some results of this evaluation. Here, the reduced surface deformation along line A is given for various Reynolds numbers and Hartmann numbers. Other parameters are fixed at a = 0.25 and d =1.25. We observe a nonsymmetric distribution of lift force and the possible resulting surface deformation. In front of the magnets there are negative lift forces resulting in the formation of depressed area. Behind the magnets, respective positive lift forces
70
result in the formation of amplified area on the free surface. Similar deformations were observed in laboratory experiments [31]. Moreover, a strong positive lift force underneath the left-hand side and strong negative lift force underneath the right-hand side of the magnets while the force underneath the vertical
symmetry line of magnets is zero. This is expected since the streamwise component of the magnetic field is likewise changing its direction being zero in the
vertical symmetry line. Another expected result is that the surface deformations
become more pronounced when both Reynolds number and Hartmann number
are increased. Larger values of Reynolds number lead to higher eddy current
densities and larger values of Hartmann number lead to stronger magnetic
fields.
Fig. 66: Reduced surface deformation along line A for various Reynolds number and the
Hartmann number. Vertical lines indicate the location of the magnets.
Both effects contribute to higher electromagnetic forces. Eventually, underneath
magnet 1 the magnitudes of the lift forces are much higher than underneath
magnet 2. Again, this reflects the fact that due to the obstacle effect behind
71
magnet 1 a wake is formed that reduces the streamwise velocities under magnet 2.
Summary
We have numerically investigated free-surface liquid metal flow influenced by
two localized magnet fields that are produced by small cube-type permanent
magnets that are arranged in some distance above the surface and that are
arranged one behind the other separated by a certain distance in the streamwise direction. Such an arrangement serves as a model to study ToF LFV for
the non-contact measurement of local surface velocities in metallurgic applications. We have analyzed the effects of the Reynolds number and the Hartmann
number as well as geometric parameters on the distributions of streamwise surface velocity, induced eddy current density, and generated Lorentz force density. Our main findings main be summarized as follows.
i.
Magnet 1 acts like a magnetic obstacle. Due to strong braking Lorentz forces acting underneath the magnet, the streamwise velocity is
reduced and fluid is pushed aside, by-passing the region in which the
magnetic field is present. A wake is formed. These effects are more
pronounced upon increasing Ha and decreasing Re.
ii.
Magnet 2 is located in the wake produced by magnet 1. Hence,
streamwise velocities, eddy current densities, and Lorentz forces are
reduced underneath magnet 2.
72
5 Application of ToF LFV to free surface velocity
measurement in metallic melt
5.1 Introduction
Global and local control of liquid metal flow is crucial for success of many metallurgic processes. 90% of the worldwide steel is produced by continuous casting.
During this process, non-steady melt flow in the mould may lead to highly unwelcome slab surface defects. Both DC and AC electromagnetic fields are utilized to improve the ability of controlling the melt flow by induced Lorentz forces.
Plant measurements of melt flow are important to record the electromagnetic
effects. Of special interest is knowledge of both the direction and the magnitude
of the flow velocity at the surface in the mold are critical for the final product
quality [1].
To provide the continuous casting machine with liquid steel, scrap is melted in
electric furnace. The primary melt is delivered to the tundish where it is heated
by plasma torches. During the casting process, liquid steel is continuously fed
from the tundish to the mould through the submerged entry nozzle (SEN). The
flow rate of steel in the tundish and in the mould is controlled either using a sliding gate mechanism or a stopper rod device, which is mounted beneath the
tundish. The surface of steel in both ladle and tundish is usually covered with
casting powder to avoid oxidation upon contact with the atmosphere. The mould
oscillates to prevent sticking of the solidifying shell at the mould. Mould walls
consist of water-chilled copper plates. Hence, upon contact with the wall, liquid
steel immediately solidifies due to intense heat transfer provided by the mould
cooling system. When the solidified shell is thick enough to bear up against the
ferrostatic pressure of the liquid steel inside the strand, it leaves the mould and
is further cooled by a secondary cooling system using water sprays. Rollers
keep the strand in shape until it is completely solidified. The strand is cut into
specified lengths at a slitting station and further treated by rolling. This continuous casting of steel process is schematically shown in Fig. 67.
73
Fig. 67: Continuous casting of steel process. The figure is taken from Ref. [55].
The steel flows from the tundish to the mould through SEN by gravity. Hence,
the SEN has an important influence on mould flow and thus on steel quality. To
prevent problems such as generation of surface waves, meniscus freezing, and
crack formation, steel should be delivered constantly into the mould. If the liquid
steel jet leaving the SEN impinges too strongly on the narrow face of the mould,
the jet may split to flow upwards along the narrow face. This may lift the level of
the molten steel, changing its profile and also generating large level fluctuations
near the meniscus. Mould flux may be pushed away from the narrow face, leading to surface quality problems [1]. Hence, non-contact measurement of both
the magnitude and the direction of the velocity in the melt flow near the surface
is advantageous for casting control.
However, the surface is covered by a non-transparent layer of mould powder to
prevent formation of slag. Thus, surface flow cannot be registered by optical
measurement techniques. Moreover, due to aggressiveness of metal melt at
high temperature, only non-contact measurement methods can be applied. The
aim of the present experimental study is to investigate the feasibility of the meniscus velocity measurement based on ToF LFV for high-temperature liquid
metal.
In this chapter we present a series of test experiments conducted at North
Eastern University (NEU) in close cooperation with Key Laboratory of Electromagnetic Processing of Materials Ministry of Education. The experiments aim to
demonstrate that the MVS is also feasible to measure free-surface velocity of
the melt in open channel under nearly industry-oriented conditions, i.e. hot liquid
metals. In more detail, we present test measurements using SnPbBi at about
74
210°C and liquid steel at 1700°C. During these experiments the melts are
transported by gravity in an open channel from an upper ladle to a lower holding
container. We record the voltage of the force sensors by varying the distance of
the sensors to the free-surface flow and the dimension of the used magnet systems. The evaluation of the data shows that meniscus velocity sensor (MVS)
works well in producing signals in the solid body and the SnPbBi experiments.
However, in the liquid steel experiments no clear cross-correlations were found.
We attribute this finding to the relatively low number of individual experimental
runs and the relatively short duration of each run.
5.2 State of the art of surface velocity sensor for
melt flow
Various methods to measure velocities in high-temperature metal melts have
been developed and reported. For instance, one similar measurement method
in ToF LFV is the Mass Flow Control (MFC) sensor [18] [80] which consist of
two probes located close to each other behind the copper mould plates. Each
probe consists of a permanent magnet and a detector, see Fig. 68. As the steel
travels through each associated magnetic field, an electrical signal is induced in
each detector caused by the electrical eddy currents induced by the fluid motion. The liquid steel velocities were measured by computing the time delay of
the signals recorded by the two probes. The time shift of the two signals is a
measure of the time taken by the flow to convect from one probe to the other.
The average velocity in the region between the probes is then the distance between the probes divided by this time shift. The drawbacks of this sensor are
twofold: First, flow past one probe often does not even reaches the other probe.
Second, this MFC sensor should be placed in regions of steady horizontal flow,
such as found near the top surface.
Fig. 68: Mold Flow Control Sensor (MFC) from german company AMEPA. The figure is
taken from Ref. [80].
On the other hand, simpler methods using basic fluid mechanics principles were
also developed. A Karman vortex probe [19] was developed by Iguchi et al. to
75
measure the liquid steel velocities near meniscus based on the linear relationship between molten steel velocity and the shedding frequency of Karman vortex streets that is formed by the immersed cylindrical probe in the mold. The
deficit of this method is as following: it is not contactless and requires a complicated signal processing process to filter the noises in the signal.
Kubota et al. [20] [76] utilized a simpler technique with a rod dipped into the
molten steel and the deflection angle of the rod and the torque acting on it was
measured. The quantities were then transformed into surface steel velocities. A
much simpler method to measure meniscus velocities using nail boards was
pioneered by Dauby et al. Rietow and Thomas [21] extended this method to
acquire velocity information by analysing the thicknesses of deposition layers on
the nails. As molten steel flows past the nail, liquid steel builds up at the impinging point on the nail lump as it solidifies, and all kinematic energy is converted
into potential energy at the stagnation point. Liquid steel level drops at the opposite side of the nail lump. After dipping the nail board into the steel liquid pool
and removing it, the steel skulls that solidified on the end of each nail. This deformation of the meniscus is recorded by the shape of the solidified lump. By
investigating the lump shape and lump height difference between the flowfacing side and its opposite side, the magnitude and direction of the surface
steel velocity can be determined.
76
Fig. 69: Continuous casting mold showing steel flow, top-surface slag layers and location of nail-board insertion (top) and Nail-board method to measure steel surface velocity
and direction (below). The figure is taken from Ref. [55].
The Swedish company MEFOS has already developed a respective VelocityMeasurement-Level-Measurement sensor. It operates also according to the
principle of the LFV. The velocity is measured by measuring the force which is
acting on the magnetic element. However, a fully functional prototype has not
successfully produced yet [22].
77
Fig. 70: Velocity-Measurement-Level-Measurement Sensor developed by the company
MEFOS. The figure is taken from Ref. [22].
5.3 A sensor for high-temperature surface velocity
measurement
Recently, we suggested an alternative and much more compact design of such
a surface velocity sensor termed Meniscus Velocity Sensor (MVS). The present
meniscus velocity sensor operates according to the non-contact measurement
of Lorentz forces acting on magnet systems. Its working principle relies on the
ToF LFV, which has been successfully tested at Ilmenau University using both
solid body movement and GaInSn as a low-melting model melt. This sensor can
also be used to record local surface velocities in high-temperature metal melts.
The principle design of MVS prototype is shown in Fig. 71. Basically, it consists
of two cubic permanent magnets of edge length A, each of which equipped with
a strain gauge based force sensor and separated by a certain distance. It can
be equipped with 4 different sizes of cubic permanent magnets with edge
lengths of 20mm, 15mm, 12mm, and 10mm. The magnets are fixed on special
holders of adjusted lengths so that the gap between the bottom face of the
magnet and the bottom of the housing is constant for all magnet sizes. Power
supply for the velocity sensor is provided by a 12V rechargeable car battery.
The total dimensions of the MVS are 105mm x 62mm x 110mm (length x width
x height). The distance between two magnets is fixed at 65mm.
78
Fig. 71: Design of Meniscus Velocity Sensor (MVS).
The MVS is positioned close to the surface of the melt flow. The strain gauge
measures the forces acting on the magnets and generated by the melt flow.
Both force sensors are fixed at a common holding plate. To protect the MVS
against heat and dust, it is embedded in a double housing of which the gap is
filled with thermally insulating material “Superwool 607 HT Paper” from company D&W. Cables and wires for power supply and data transfer are guided
through a pipe outside of the housing. The diameter of this pipe is 20 mm. This
pipe also serves to fix the MVS at an external positioning system. Temperature
of the magnet system is measured by a thermocouple. All parts of the velocity
sensors are made of non-magnetic stainless steel.
Force sensor
The most important part of MVS is the force sensor. For force measurement, we
use commercial strain gauge based force sensors produced by the company
MAUL alpha, see Fig. 72. The used sensors were selected according to the
necessary level of accuracy, resolution, and critical frequency.
To select the suitable sensor the following considerations were taken into account:
i.
The quantity of the dead load (magnet and holder) and the maximum deformations in the direction of load measurement.
79
ii.
The quantity of the desired resolution and sampling rate of measurement.
iii.
The measurement uncertainties due to the influencing effects of temperature, wind, and mechanical vibration.
Fig. 72: Digital strain gauge from the precision scale MAUL alpha fixed on cube magnets.
Amplifier
The amplifier is an important part of signal processing devices. Here, parameters like power supply, connectivity, signal filtering, sensitivity, resolution, offset,
and interface are important for proper selection. As a cost-effective solution we
choose the amplifier Soemer LDU 78.1, see Fig. 73. The performance parameters are shown in Appendix 3:
Fig. 73: Amplifier Soemer LDU 78.1.
80
Magnets
As the generate Lorentz forces are proportional to the square of the applied
magnet field, usage of strong permanents magnets is favorable. We choose
NdFeB magnet showing the following properties:




Material:
Edge length:
Remanent flux:
Coercivity:
NdFeB N45
20mm, 15mm, 12mm, 10mm
Br = 1.33 to 1.37T
Hcb => 955A/m
Fig. 74: Cubic NdFeB magnet.
Radiative heat transfer simulation
To check the transient thermal behavior of the prototype, heat transfer simulations are performed. In steel application, the magnet temperature may not exceed 80°C when the sensor is exposed to the hot melt surface. To this aim, a
simple numerical model has been built up using the commercial software ANSYS, see Fig. 75. Here, an iron cube with edge length 30mm is arranged in a
single housing made of stainless steel. The wall thickness of the housing
amounts 4mm. The distance between magnet and the bottom of housing is
26mm. The housing is filled with air to allow for thermal convection inside. As a
thermal load we apply arranged a hot plate at a fixed temperature of 1000°C
some 100mm below the bottom of the housing. Heat is delivered by thermal
radiation from the hot plate to the bottom of the housing. The selected arrangement results in a view factor of 0.451. An initial temperature of 20°C is applied
to all parts of the sensor.
81
Fig. 75: Model of heat transfer simulation.
82
Fig. 76: Temperature distributions of radiation exposure across the sensor are plotted
after 30s (above) and after 90s (below).
Fig. 76 shows some results of our transient simulations. As expected, the hot
spot is located in the centre of the bottom plate of the housing. After 90s, we
obtain a hot spot temperature of about 780°C. More interestingly, we observe
that the temperature of magnet is only 37.1°C at the bottom face and 31.4°C at
the upper face. Therefore, we conclude that our design of MVS prototype using
passive double-housing cooling with insulating material in between is well suited
for some minutes of measurements above molten steel surface.
Measurement procedure
As the entire equipment has not been intensively tested before, it has been
agreed to proceed in three experimental steps by continuously increasing difficulty and temperature. These steps are:

Solid body experiments at room temperature with a copper rod rolling in
a controlled manner underneath the meniscus flow sensor.

SnPbBi at 210°C flowing in an inclined trapezoidal open channel of dimensions L x B (top) x B (bottom) x H = 1550mm x 50mm x 40mm x
18mm underneath the meniscus flow sensor.
83

Molten steel at 1700°C flowing in an inclined ceramic channel of dimensions L x B x H = 300mm x 80mm x 20mm underneath the meniscus flow
sensor.
All experimental runs were conducted using a sampling frequency of 25Hz.
84
5.4 Preliminary test measurement under industryrelevant condition
5.4.1 Open channel flow measurement using SnPbBi at
210°C
We now turn to the experiments aiming to measure the surface velocity in melt
flow using MVS. As a first test melt we use SnPbBi at 210°C. The experimental
set-up is schematically shown in Fig. 77. Photographs of the experiments are
provided in Fig. 78. The flow experiments are carried out in a trapezoidal open
channel of dimension L x B (top) x B (bottom) x H = 1550mm x 50mm x 40mm x
18mm. The angle of inclination is 2.277°. Before each run, a volume of 22.95l of
SnPbBi (density 9.9g/cm³) was melted in a gas-fired crucible. The melt was
poured into the channel by tilting the crucible, lifted by a crane, by using a handle wheel.
Fig. 77: Set-up of SnPbBi experiment.
Fig. 78: Picture of the set-up for open channel flow experiments with SnPbBi (left). After
the run the melt was put back into the ladle (right).
The main object of these experiments is to find cross-correlations between the
force signals. To this aim we carried out experiments with and without a submerged copper cylinder of diameter 20mm. The idea of using the cylinder is to
85
trigger extra vortices to enhance cross-correlation signals. Snapshots of the
flow without and with submerged cylinder are shown in Figs. 79. We observe
that without the use of the cylinder, a smooth free-surface flow adjusts in the
channel. Using the cylinder, clear vortices are created in the wake. However,
the obstacle additionally contributes to dam oxides. By that the flow rate slows
down, see Fig. 80.
Fig. 79: Smooth flow when no cylinder was submerged in the melt; The MVS is placed
above the free surface liquid metal channel flow. The gap height is 30mm. The angle of
inclination is 2.277° (left), with cylinder was submerged in the melt (right).
Fig. 80: Wake formation behind the submerged cylinder. In the upstream direction the
cylinder causes piling up of oxides slowing down the flow rate.
A total of 18 individual runs, each lasting about 200s, were performed. For a
given parameter set, the experiments were repeated 3 times. An experimental
procedure matrix is shown in Appendix 2.
The following parameters were varied:



The gap height between channel and MVS (30mm, 20mm, 10mm).
The edge length of the magnets (20mm, 15mm).
With and without submerged cylinder.
86
Using data of a typical run, the duration of the run and the melt volume at hand,
the following rough calculations can be conducted:
.
Q 0,02295m3 / 164s
V 
 0,2 m / s.
A
0,02  0,04m2
V
D 0,065

 0,26 m / s .

0,25
These calculations serve to check the plausibility of the data obtained later on.
Results and discussion
A total of 18 individual runs were carried out. Each run lasts about 115-341s.
Only two runs were successfully completed, i.e. cross-correlations between the
force signals were detected. In both of these runs the gap height was fixed at
30mm. One successful run was with using the submerged cylinder, the other
one was without.
4630
4625
4620
0
x 10
50
100
150
200
4500
4480
4460
4440
250
4
-1.532
LKA2
-1.538
50
100
Time, s
150
200
100
150
200
250
LKA2
-1.534
-1.536
-1.538
-1.54
250
0
50
100
150
200
250
Time, s
LKA1
LKA1
4490
Voltage , m V
4628
4626
4624
4488
4486
4484
4482
4622
0
5
10
15
20
25
4
x 10
-1.5355
-1.537
-1.5375
-1.538
0
0
x 10
5
10
15
20
25
4
LKA2
LKA2
-1.5365
V oltage , m V
Voltage, mV
50
x 10
4492
4630
-1.536
0
4
-1.537
-1.539
0
Voltage, mV
LKA1
Voltage, mV
Voltage, mV
-1.536
4520
LKA1
Voltage, mV
Voltage, mV
4635
5
10
Time, s
15
20
25
-1.536
-1.5365
-1.537
-1.5375
0
5
10
15
20
25
Time, s
Fig. 81: SnPbBi_gap height 30mm without cylinder, Data-04_41_28 (left); with cylinder,
Data-07_05_00 (right). Raw signals (above), magnified signals (below).
Fig. 81 (above) presents the raw signals as the voltages delivered by the two
force sensors are plotted against the total running time. Fig. 81 (below) shows a
magnification of the raw signals taken during a consecutive time interval of 25s.
Overall, the signal conditioning is poor. Signal to noise ratios are not sufficient.
The signals of LFV 2 indicate that propagating waves are present, probably triggered by uncontrollable parasitic vibrations. On the other hand, the crosscorrelation function reveals that there is a delay time of 0.25s. This particular
value corresponds surprisingly very well with the correlation time of 0.26s calculated beforehand. Using an estimate of the mean velocity based on total volume
87
and total running time. However, it should be noted that only in 2 of the 18 runs
cross-correlating have been detected.
The following observations were made:
(i) Clear cross-correlations were found for 20mm magnets at gap height
30mm.
(ii) There is no obvious difference between with or without the submerged
cylinder.
(iii) The surface velocity, obtained by evaluating the transit time out of crosscorrelation function, corresponds well with a simple calculation using the
total time of the run, the melt volume, and an estimate of the flow crosssection.
(iv) The submerged cylinder generates a vortex street; however, it also
causes a piling up of oxides diminishing the flow rate.
(v) During the experiments, the MVS heats only slightly up.
We conclude that the MVS is feasible to measure surface velocities in liquid
metal free-surface flow. Having a closer look on the data, we decided to choose
the 15mm magnets for the final experiments in molten steel. No submerged cylinder will be used.
5.4.2 Open channel flow measurement using steel at 1700°C
Finally, we would like to present results of open channel free-surface flow
measurement using steel at 1700°C. A total of six runs were performed. An experimental procedure matrix is shown in Appendix 2. Before each run, a solid
steel rod of 8.72kg was melted in a 45kW induction furnace. The steel was
poured into the ceramic channel with dimension L x B (top) x H = 300mm x
80mm x 50mm, see Fig. 82, by lifting and tilting the furnace via handles by men.
The angle of inclination of the channel is 2.06°.
88
80
50
70
Fig. 1: Dimension of ceramic channel.
Fig. 82: Dimension of the ceramic channel.
The experimental set-up for open channel steel flow experiments is shown in
Figs. 83.
Fig. 83: Set-up for open channel steel flow experiments.
89
During the experiments we vary the gap height (10mm, 20mm). For each gap
height 3 individual runs were performed. A typical run lasts about 20s. The
preparation time for one individual run is about 60min. Snapshots of a typical
experimental run are shown in Fig. 84.
Fig. 84: Snapshots of a typical run at gap height 20mm (a) and 10mm (b).
Results and discussion
Fig. 85 presents the raw signals of three selected individual runs. The peaks in
the raw signals are due ambient parasitic tremors that do not contribute to the
evaluation of the delay time. More measurements are necessary to support the
conjecture that the MVS is capable to detect surface velocities in liquid steel
free-surface flow. However, also in these experiments no clear cross-correlating
signals were detected.
Voltage, mV
-9050
LKA1
-9100
-9150
0
2
4
6
8
10
12
14
16
18
Voltage, mV
-5200
20
LKA2
-5250
-5300
-5350
0
2
4
6
8
10
12
14
16
18
20
Time, s
Fig. 85: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of MVS
with gap height 20mm, Data-05_40_42.
90
Voltage, mV
-9000
LKA1
-9050
-9100
-9150
-9200
0
2
4
6
8
10
12
14
16
Voltage, mV
-5500
18
LKA2
-5600
-5700
-5800
0
2
4
6
8
10
12
14
16
18
Time, s
Fig. 86: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of MVS
with gap height 20mm, Data-07_07_35.
Voltage, mV
-9350
LKA1
-9400
-9450
-9500
0
2
4
6
8
10
12
14
16
18
20
Voltage, mV
-5300
22
LKA2
-5400
-5500
-5600
0
2
4
6
8
10
12
14
16
18
20
22
Time, s
Fig. 87: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of MVS
with gap height 10mm, Data-07_55_00.
The following observations were made:
(i) The MVS heated up to a maximum temperature of 55°C by steel temperature 1700°C with measuring time 26s. We conclude that the passive
cooling system was sufficient to protect the magnets from overheating.
(ii) Due to short running times, limited test conditions, and limited force sensor quality, no clear cross-correlating signals were detected.
91
5.5 Sub-summary
A first prototype of a Meniscus Velocity Sensor, which operates according to the
Time-of-Flight Lorentz Force Velocimetry method, has been tested in both solidbody experiments and free-surface model experiments. In the solid-body experiments we use an aluminium sheet and an aluminium disk both of which rotating
at controlled speed underneath the two Lorentz force flow meters. Best results
were obtained for the case of the rotating sheet. Here, the quality of the force
signals is good and the measured transit times correspond well with the applied
rotation speed. In the case of the rotating disk we observe that usable measurement signals are obtained due to the natural unevenness of the disk. However, the quality of the signals depends strongly on the applied rotation speed
and the gap height between sensor and disk.
In the case of free-surface melt flow we observe that the magnetic obstacle effect restricts the determination of the local surface velocity as the applied magnetic field strongly influences both the magnitude of the velocity and the shape
of the free surface. At the present time the reproducibility of these measurements is quiet moderate. Although some first results look promising, further
model experiments using improved equipment and advanced signal processing
method are required to demonstrate the feasibility of Time-of-Flight Lorentz
force velocimetry in practical applications. All these efforts aim to improve the
quality of the cross-correlation signals.
To improve the existing prototype in term of suitability for plant application and
extension of capability, the following tasks are to do aiming to improve the quality of the cross-correlation signals:
I. Tailored magnet system (temperature-resistant, lightweight and high-field
magnet systems, low Integration of magnetic field lines by small distance,
strong magnetized, shielding and guiding magnetic flux in defined direction)
II. Highly sophisticated force sensor (fast response time, dynamic instead of
static sensor)
III. Electronic signal processing (damp noise signals from mechanical vibration)
IV. High-resolution electronic signal processing packages/and data acquisition
systems
V. Damping system for preventing mechanical vibration
VI. Insulating measurement device against overheating (active: water or air
cooling; passive: ceramic housing)
VII. Signal strength vs. distance (explained below)
92
As a consequence, the most important factor that could be addressed during
the present study is the effect of the distance between the sensors and melt
flow. Upon decreasing the distance between the sensor and the liquid steel surface, clear cross-correlations are more probable. However, in this case active
cooling of the sensor may be necessary.
93
6 Conclusion and outlook
We experimentally and numerically study turbulent liquid metal flow in the test
facility EFCO using the eutectic alloy GaInSn as a test fluid. Our model experiments demonstrate that Time-of-Flight Lorentz Force Velocimetry is a feasible
non-contact tool for measuring flow rates in such flows. The present technique
measures the transit time of a tagging vortex that is transported by the flow and
registered by two Lorentz force flow meters that are arranged in a certain distance one behind the other. The measurement is based on just crosscorrelating the force data registered by the two flow meters. By that it becomes
independent of any fluid properties and the magnetic field strength. However,
the flow rate experiment show that intense calibration of the measuring device
is necessary as the ratio between transit time and characteristic flow time depends strongly on the separation distance of the flow meters and, presumably,
on the geometry of the obstacle which is submerged into the flow for producing
detectable vortex structures. Instead of permanent magnet in LFF1, an electromagnet may be employed in the further investigations with the intention to produce controllable excitation by pulse generation.
Furthermore, the meniscus velocity sensor (MVS) has been evaluated during
industrial-relevant conditions. The test experiments show that ToF LFV is a potential non-invasive method to measure surface velocities in high-temperature
metallic melts. The challenge in the development of MVS is to obtain a weakly
noisy sampling rate due to the small and weak nature of the force signals as
well as to find applicable pulse generators, which provide more clearly crosscorrelation peaks even in laminar flow. Further experimental and numerical investigations have to be performed to transfer the non-contact electromagnetic
flow rate and free-surface velocity measurement technique relying on ToF LFV
into industrial applications, for instance, continuous casting of steel and the production of secondary aluminium.
Listoffigures
Fig. 1: Principle of Lorentz Force Velocimetry. It is based on measuring the
Kelvin force that pulls at an externally arranged magnet system. The Kelvin
force is the counteracting force to the Lorentz force that is generated in the melt
due to the movement of the electrically conducting melt through the magnetic
field. .................................................................................................................... 2 Fig. 2: Principle of Time-of-Flight Lorentz Force Velocimetry. Two Lorentz force
flow meters are arranged one behind the other. This flow measurement
technique is based on just cross-correlating the force signals recorded by the
two flow meters. The method is independent of melt properties and magnetic
field parameters. ................................................................................................. 2 Fig. 3: Secondary aluminum production process. The aluminium melt flux is
indicated by blue arrows. Scrap is melted in furnaces. The primary melt is
delivered to converters within which the final melt is prepared. The figure is
taken from Ref. [71] (in German)......................................................................... 4 Fig. 4: Meniscus flow in Continuous Casting of Steel (left). Double and single
roll flow pattern (right). The figure is taken from Ref. [55]. .................................. 5 Fig. 5: Magnetic fly wheel (left). Single-magnet rotary flow meter (right). The
figure is taken from Ref. [11] and [12]. ................................................................ 6 Fig. 6: Contactless electromagnetic phase-shift flow meter for liquid metals. The
figure is taken from Ref. [13]. .............................................................................. 7 Fig. 7: Design of Vives Probe (left) and Potential Probe (right). The figure (right)
is taken from Ref. [54]. ........................................................................................ 8 Fig. 8: The US transducer is submerged at an angle of 60° into the liquid metal
duct EFCO (left). Multiple reflections of US wave may result in imaginary
velocity values outside the region of the liquid flow (right). The right figure is
taken from Ref. [17]. ............................................................................................ 9 Fig. 9: Principle of Lorentz Force Velocimetry for the measurement of flow rate
(left) and local surface velocities (right). ............................................................ 10 VII
Fig. 10: Principle of Time-of-Flight Lorentz Force Velocimetry for the
measurement of flow rate (left) and local surface velocities (right). The transit
time  of a tagging signal flowing through the two flow meters is determined by
cross-correlating the voltage data provided by the two force sensors. .............. 12 Fig. 11: Sketches of the experimental facility EFCO (left) for flow rate
measurements, LiMeSCo for surface velocity measurement (middle), and MVS
for measurement of surface velocity in high-temperature metallic melt (right). . 14 Fig. 12: Set-up of test facility EFCO for non-contact flow rate measurement in
turbulent liquid metal flow using single LFV. ..................................................... 25 Fig. 13: Lorentz force FL as a function of volumetric flow rate Q. ...................... 26 Fig. 14: Velocity profiles at the mid plane of the channel measured by UDV.
Counter-clockwise mode: turbulent hydrodynamic profile. Clockwise mode: Mshaped MFD profile. .......................................................................................... 27 Fig. 15: Numerical simulation for Re = 1.54 x 106 and Ha = 140. Top graph:
geometry of the channel wall. Bottom graph: velocity profiles in entrance, inside,
and at the exit of the magnet field. .................................................................... 28 Fig. 16: Induced eddy currents in the region of the localized magnetic magnetic
field. .................................................................................................................. 29 Fig. 17: Eddy currents in the region of the localized magnetic field at half-height
of the channel.................................................................................................... 30 Fig. 18: Lorentz force density in the region of the localized magnetic field at halfheight of the channel. ........................................................................................ 30 Fig. 19: Lorentz force density in the region of the localized magnetic field at the
mid plane of the channel. .................................................................................. 31 Fig. 20: Set-up of the test facility EFCO for non-contact flow rate measurement
in turbulent liquid metal flow using Time-of-Flight Lorentz Force Velocimetry. .. 32 Fig. 21: Velocity profiles at the mid plane of the channel measured by UDV. The
green curve gives the profile when both flow meters are present. The blue curve
gives the profile when LFV 2 was removed. ...................................................... 33 Fig. 22: Flow velocity (in m/s) in the horizontal plane at y = 2.5mm around the
cylinder submerged at an angle of  = 60° into the flow. Parameters are
VIII
Red = 1.6 x 104, ReC = 7.42 x 103, and the blockage ratio of the cylinder of
 = 0.8. .............................................................................................................. 34 Fig. 23: Raw voltage signals U1(t) and U2(t) delivered by the strain gauge of flow
meters LFF 1 and LFF 2 over a period of 10s. .................................................. 35 Fig. 24: Auto- and cross-correlation functions calculated according to Eq. (7)
from the voltage signals U1(t) and U2(t) shown in Fig. 23 above. Experimental
parameters are fixed at f = 23Hz and D = 170mm. ........................................... 35 Fig. 25: Averaged relation between the vortex velocity Vvortex measured by ToF
LFF and the velocity of the flow V measured by UDV and Vives probe. The
separation distance is fixed at D = 170mm. The slope of the linear fitting curve
is k2 = 1.90......................................................................................................... 36 Fig. 26: Averaged relation between the vortex velocity Vvortex measured by ToF
LFF and the velocity of the flow V measured by UDV and Vives probe. The
separation distance is fixed at D = 220mm. The slope of the linear fitting curve
is k2 = 0.95......................................................................................................... 37 Fig. 27: Computational domains and meshes used in the numerical simulations.
Within the volumes Fluid 2 and Fluid 4 a constant spanwise magnetic field is
applied. ............................................................................................................. 38 Fig. 28: Flow velocities (in m/s) in the mid plane of the channel at Red = 2.0×104
(V = 0.4m/s) and Ha = 140. ............................................................................... 38 Fig. 29: Flow velocities (in m/s) in the mid plane of the channel at Red = 3.0×104
(V = 0.6m/s) and Ha = 140. ............................................................................... 39 Fig. 30: Flow velocities (in m/s) in the mid plane of the channel at Red = 3.0×104
(V = 0.6m/s) and Ha = 205. ............................................................................... 40 Fig. 31: Reduced velocity profiles at various positions in the streamwise
direction: Line 1: inlet, Line 2: right behind LFF1, Line 3: between LFF1 and
LFF2, Line 4: right behind LFF2, Line 5: outlet. The separation distance of the
flow meters is D = 150mm................................................................................. 41 Fig. 32: Induced eddy current density (in A/m2) in the mid plane of the channel.
The applied localized magnetic fields are pointing out of the plane. ................. 41 IX
Fig. 33: Lorentz force density (in N/m3) in the mid plane of the channel. The
applied localized magnetic fields are pointing out of the plane.......................... 42 Fig. 34: Velocity profiles in streamwise direction (a) and in spanwise direction
(b)...................................................................................................................... 43 Fig. 35: Experimental set-up of test facility LiMeSCo. ....................................... 45 Fig. 36: Single LFV with two rotating aluminum sheets, one of which with drilling
hole (left); The iron wire (circled in red) is used to trigger controlled signals for
counting the experimental runs (right). .............................................................. 45 Fig. 37: Voltage signals of the iron wire (1); the Aluminum sheets with hole (2)
and without hole (3) and iron wire (4) as function of time. ................................. 46 Fig. 38: Lorentz force FL vs. driving velocity V and height H. ............................ 47 Fig. 39: Coefficient of Lorentz force CL as a function of magnetic Reynolds
number Rem and height H. ................................................................................ 48 Fig. 40: Set-up of solid-body rotation experiments using Time-of-Flight LFV. ... 48 Fig. 41: Raw voltage data U1(t) (blue curve) and U2(t) (green curve) recorded by
the force sensors LFV 1 and LFV 2. The velocity is fixed at V = 30cm/s. .......... 49 Fig. 42: Auto- and cross-correlation functions calculated according to Eq. (7)
and using the raw voltage signals U1(t) and U2(t) shown in Fig. 41. .................. 50 Fig. 43: Set-up of the solid-body rotation experiments using an aluminum sheet
(left) and an aluminum ring-type plate (right)..................................................... 50 Fig. 44: Raw signals (top) and cross-correlation function (below) for the case of
a rotating aluminum sheet at V = 41.27cm/s and H = 28mm. ........................... 51 Fig. 45: Raw signals (top) and cross-correlation function (below) for the case of
a rotating aluminum sheet at V = 27.35cm/s and H = 33mm. ........................... 52 Fig. 46: Comparison of measured velocity using MVS with measured velocity
using laser; channel rotating counter clockwise. ............................................... 53 Fig. 47: Comparison of measured velocity using MVS with measured velocity
using laser; channel rotating in clockwise. ........................................................ 54 Fig. 48: Experimental set-up for measuring the rotating speed using an
aluminium sheet without housing. ..................................................................... 54 X
Fig. 49: Comparison between the measured velocity using MVS with the one
measured using laser by varying height above the rotating channel. ................ 55 Fig. 50: Graphs of the y-component By of the magnetic field (red curve), the zcomponent jz of the induced eddy current density (blue curve), and the xcomponent Fx of the generated Lorentz force (purple curve). Parameters are
fixed at V = 30cm/s, H = 2cm and D = 10mm.................................................... 56 Fig. 51: Graphs of the y-component By of the magnetic field (red curve), the zcomponent jz of the induced eddy current density (blue curve), and the xcomponent Fx of the generated Lorentz force (purple curve). Parameters are
fixed at V = 30cm/s, H = 2cm and D = 80mm.................................................... 57 Fig. 52: Set-up for measuring surface velocities in free-surface flow using ToF
LFV. .................................................................................................................. 58 Fig. 53: Visualization of liquid metal free-surface flow influenced by a localized
magnet field that is produced by a small permanent magnet located above the
surface. Surface deformation and surface waves are created due to the
magnetic obstacle effect. .................................................................................. 58 Fig. 54: Comparison between the measured velocity using MVS and measured
velocity using laser; magnet in LFV 1 with edge length 10mm and magnet in
LFV 2 with edge length 15mm........................................................................... 59 Fig. 55: Submerged styrofoam as a disturbing body in liquid metal free-surface
flow measurement. ............................................................................................ 60 Fig. 56: Comparison of measured velocity using MVS with measured velocity
using laser; magnet of LFV 1 with edge length 10mm and magnet of LFV 2 with
edge length 15mm. ........................................................................................... 60 Fig. 57: Comparison of measured velocity using MVS with measured velocity
using laser; both magnets in LFV 1 and LFV 2 are with edge length 15mm. .... 61 Fig. 58: Geometry of the problem. Two cubic-type permanent magnets of edge
length A are arranged in a row separated by a certain distance D and some
height H above the free surface. Numerical solutions will be given along the
lines A (symmetry line in the streamwise direction), lines B (spanwise direction
in front of the first magnet), and lines C (symmetry line in the spanwise
direction). .......................................................................................................... 62 XI
Fig. 59: Magnetic field strength B0 [T] in streamwise direction. ......................... 63 Fig. 60: Reduced streamwise velocity along line A for various Reynolds number
and the Hartmann number. Vertical lines indicate the locations of the permanent
magnets. ........................................................................................................... 65 Fig. 61: Streamwise velocity distribution (in m/s) at the free surface. Flow
direction is from left to right. Squares illustrate locations of the permanent
magnets. Parameters are Re = 21.000 (top graph) and Re = 63.000 (below
graph). ............................................................................................................... 66 Fig. 62: Reduced velocity in the streamwise direction as a function of the
spanwise coordinate along line B (left) and line C (right) for two different sizes of
magnets. ........................................................................................................... 66 Fig. 63: Distribution of the coefficient of the eddy current density along line A.
Other parameters are d = 1.25 and a = 0.25. Vertical lines indicate the position
of the permanent magnets. ............................................................................... 67 Fig. 64: Distribution of eddy current density at the free-surface. ....................... 68 Fig. 65: Coefficient of the Lorentz force density in the streamwise direction for
various Reynolds number and Hartmann number. Other parameters are fixed at
a = 0.25 and d = 1.25. ....................................................................................... 68 Fig. 66: Reduced surface deformation along line A for various Reynolds number
and the Hartmann number. Vertical lines indicate the location of the magnets. 70 Fig. 67: Continuous casting of steel process. The figure is taken from Ref. [55].73 Fig. 68: Mold Flow Control Sensor (MFC) from german company AMEPA. The
figure is taken from Ref. [80]. ............................................................................ 74 Fig. 69: Continuous casting mold showing steel flow, top-surface slag layers
and location of nail-board insertion (top) and Nail-board method to measure
steel surface velocity and direction (below). The figure is taken from Ref. [55]. 76 Fig. 70: Velocity-Measurement-Level-Measurement Sensor developed by the
company MEFOS. The figure is taken from Ref. [22]. ....................................... 77 Fig. 71: Design of Meniscus Velocity Sensor (MVS). ........................................ 78 Fig. 72: Digital strain gauge from the precision scale MAUL alpha fixed on cube
magnets. ........................................................................................................... 79 XII
Fig. 73: Amplifier Soemer LDU 78.1.................................................................. 79 Fig. 74: Cubic NdFeB magnet. .......................................................................... 80 Fig. 75: Model of heat transfer simulation. ........................................................ 81 Fig. 76: Temperature distributions of radiation exposure across the sensor are
plotted after 30s (above) and after 90s (below). ................................................ 82 Fig. 77: Set-up of SnPbBi experiment. .............................................................. 84 Fig. 78: Picture of the set-up for open channel flow experiments with SnPbBi
(left). After the run the melt was put back into the ladle (right). ......................... 84 Fig. 79: Smooth flow when no cylinder was submerged in the melt; The MVS is
placed above the free surface liquid metal channel flow. The gap height is
30mm. The angle of inclination is 2.277° (left), with cylinder was submerged in
the melt (right). .................................................................................................. 85 Fig. 80: Wake formation behind the submerged cylinder. In the upstream
direction the cylinder causes piling up of oxides slowing down the flow rate. ... 85 Fig. 81: SnPbBi_gap height 30mm without cylinder, Data-04_41_28 (left); with
cylinder, Data-07_05_00 (right). Raw signals (above), magnified signals
(below). ............................................................................................................. 86 Fig. 82: Dimension of the ceramic channel. ...................................................... 88 Fig. 83: Set-up for open channel steel flow experiments. .................................. 88 Fig. 84: Snapshots of a typical run at gap height 20mm (a) and 10mm (b). ...... 89 Fig. 85: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of
MVS with gap height 20mm, Data-05_40_42. ................................................... 89 Fig. 86: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of
MVS with gap height 20mm, Data-07_07_35. ................................................... 90 Fig. 87: Raw voltage signals u1(t) and u2(t) delivered by the two force sensors of
MVS with gap height 10mm, Data-07_55_00. ................................................... 90 XIII
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[80] K. U. Köhler, P. Andrezejewski, E. Julius, H. Haubrich: „Measurement of Steel Flow in the Mould." International Symposium on Elec-
XXI
tromagnetic Processing of Materials, 1994, Nagoya, Japan, pp. 344 –
349.
[81] Discussions with P.A. Davison, 10.01.2012.
[82] M.H. Butterfield, G.F. Bryant and J. Dowsing: A new method of strip
speed measurement using random waveform correlation. Trans. Soc.
Instrum. Tech. 13. 111-123 (1961).
[83] J.C Hill, C.A Sleicher: Directional sensitivity of hot-fil sensors in liquid metals. Rev Sci Instrum 42, pp. 1461-1468. (1971).
[84] I. Platnieks, G. Uhlmann: Hot-wire sensor for liquid sodium. J Phys
E: Sci Instrum 17, pp. 862-863. (1984).
[85] T. Von Weissenfluh: Probes for local velocity and temperature measurements in liquid metal flow. Int J Heat Mass Transfer 28. pp. 15631574. (1985).
XXII
Appendix
A1Physicalpropertiesofliquidmetals
Steel
Liquidus temperature T [°C]
1510
Solidus temperature T [°C]
1480
7.89
Density
 [103 kg/m3]
-6
2
0.85
Kinematic viscosity  [10 m /s]
6
0.77 (liquid)
Electrical conductivity σel [10 /Ωm]
0.8 (solid)
Thermal conductivity λ [W/Km]
25
Surface tension
σ [N/m]
1.6
Table A1: Physical properties of liquid metals.
Sn60Bi40
170
138
8.25
0.19
1.05
Ga68In20Sn12
10.5
35
0.46
39
0.53
6.36
0.34
3.46
Klaus Timmel, Sven Eckert, Gunter Gerbeth, Frank Stefani and Thomas
Wondrak: ISIJ International, Vol. 50 (2010), No.8, pp. 1134-1141.
XXIII
A2Experimentalmatrixforopenchannelflowmeasure‐
ments
The Tables below (Table A2-1 and A2-2) summarize the experimental matrix for
open channel flow measurement using SnPbBi at 210°C.
Table A2-1: ToF LFV; edge length of magnet 20mm
Run
1
2
3
4
5
6
7
8
9
10
11
12
Gap height Cylinder Data #
H=30mm without 312
313
314
with
315
316
320
H=10mm with
317
318
319
without 321
322
323
04_41_28
05_18_57
05_54_45
06_36_46
07_05_00
09_03_46
07_33_09
08_01_25
08_32_32
03_52_44
04_11_37
04_37_17
TSnPbBi [°C]
215-120
205-157
204-147
212-155
205-149
213-147
220-157
213-144
217-154
209-148
208-169
209-154
Table A2-2: ToF LFV; edge length of magnet 15mm
13 H=20mm without 324
05_17_52 216
14
05_48_00 213
325
15
06_26_46 210
326
16
06_56_00 212
with
327
17
07_42_12 208
328
18
08_12_19 207
329
trun [s]
260
285
222
190
263
269
233
252
276
166
179
212
Thousing
[°C]
30.5-33.5
34.6-34.8
35.5-35.2
35.3-35.5
35.7-35.9
36.6-36.5
36-36.3
35.9-36.1
36.3-36.4
27.7-28.2
29.0-29.3
30.6-30.8
164
155
172
302
341
270
31.1-31.1
31.6-31.6
31.7-31.7
31.7-31.7
31.8-31.5
31.5-31.5
The Table below (Table A2-3) summarizes the experimental matrix for open
channel flow measurement using steel at 1700°C.
Run Gap height[mm]
1
H=20mm
2
3
4
H=10mm
5
6
Data #
330 05_10_56
331 05_40_42
332 07_07_35
333 07_29_07
334 07_55_00
335 08_20_38
Tsteel [°C]
1667
1700
1680
1680
1670
1680
trun [s]
26,7
19,5
16,35
18,2
21
18,65
Thousing [°C]
28,9-29,1
38,1-40
35,9-42
42-50
48-54
52,3-55
XXIV
A3Straingaugeforcesensor
The employed force measurement systems are based on measuring the deflection of a parallel aluminum spring. The aluminum parallel spring acts as the deflection element when a force is superimposed on it [73]. The sensitive direction
of the parallel spring is in streamwise direction. The deflection is measured with
a strain gauge. This strain gauge is basically a thin electrical wire that changes
its resistance upon changing its length. A picture of the force measurement system is shown in Fig. A3. By means of the basic relations we explain the functional principle of strain gauge sensor. The maximum load and the resolution of
which the aluminum deflection element bears are shown in Table A3-1. Accordingly, the performance parameters of the used D/A converter and amplifier are
shown in Table A3-2.
l
l0
relative change of length
Hooke’s law

E

EE
mod ulus of elasticity
tensile stress
Poisson’s ratio   
d l
d
l
d
relative change of diameter
d
l
relative change of length
l
l
A
density of conductor
Resistance law R  
Fig. A3: Design of a strain gauge force sensor.

A cross  sec tion
l length of conductor
 



R


k
 1  2 


R





k-factor: sensitivity of strain
gauge
XXV
Nominal load
Measurement
priciple
Nominal variable
Class of accuracy
Error of linearity
Operate voltage
Fmax
Soemer
Platform load cell
Model 1004
Velomat
PBB-8A-2N-2.00
Maul Alpha scale
16405
3N
2N
4.9N
DMS full bridge
DMS full bridge
0.86mV/V±0.1%
C3
0.5mV/V±0.1%
0.1
≤0.1SN
Max. 10V
3V/10V
8N
5N
10V
3N+250%
Temperatur
compensated
Resolution
0.1g
Fmin
0.002N
Total Error
±0.02% of load
Table A3-1: Performance parameters of the used force sensor.
Soemer
LDU 78.1
InstruNational
Instru- National
ments
ments
NI USB 9237
NI USB-9219
Resolution
19bit
24 bit
24bit
Bridge voltage UB
5V
0,125--60V
2,5-10V
Channel
1
4
4
Sampling rate
25Hz(real)
100Hz
50kHz
pro channel
Linearity
<0,002%
Input signal
2,2mV/V
Max. voltage range
-60V/60V
-10V/10V
Min. voltage rage
-125mV/125mV
-2,5V/2,5V
Accuracy by UB=
+/-0,4%
4V
Connection
RS-422 USB
USB
USB
Table A3-2: Performance parameters of the used D/A converter and amplifier.
XXVI
A4Errorestimationofexperiments
During any experiments, various random and systematic errors appear. To eliminate all errors in an experiment is impossible. Hence, the experimentalist has to
keep the influences of the errors as less as possible. Therefore, it is necessary
to know the possible sources of errors and to calculate quantitatively the values
of errors, if possible. Systematic errors include errors that occur because of incorrectly calibrated measuring instruments or the influence of the measured
object on the measurement device. Such errors occur repetitively in the experimental runs. It can be captured by estimation of average and standard deviation
of a large number of repeated measurements. The arithmetic average x and
the standard deviation s of a measured quantity x are defined by the following
relations.
Arithmetic average:
x
1 n
 xi .
n i 1
(A1)
Standard deviation:
s
n
1
 ( xi  x)2 .
n (n  1) i 1
(A2)
Random errors are in principle not controllable. Due to fluctuations of voltage in
the electrical power supply, the opening and the closing of doors and windows
in the experimental environment, stochastic vibrations of the building, or acoustic noise triggered by the air conditioning unit in the laboratory. Since there are
many potential interferences and influence factors that cannot be exactly determined -such as humidity, temperature inside the laboratory, discontinuity in
the probe, bucking the driving engine, and externally generated mechanical vibrations by walking- the random errors can only be roughly estimated.
Error estimation of the measured Lorentz force FL using single LFV
We start with the error estimation in the model experiment LiMeSCo where the
measured force acting on a magnet due to the rotation of solid aluminum sheet.
The relation between the measured Lorentz force FL and the influencing factors,
i.e. the rotating speed u of the aluminum sheet, the height H between magnet
and solid body, the magnitude of magnetization m and the electrical conductivity
 of the metallic body is given by the relation
02um2
FL 
.
H3
(A3)
See chapter 1.4 for the detailed derivation of this formula. Here we take the values for both m and  from respective data sheets and suppose that the uncer-
XXVII
tainty in these values is negligible compared to the uncertainty in the measurement of H. Due to the positioning system used in the experiment, H can be
measured with an uncertainty of  0.01 mm. Table A4 summarizes error estimation for a given constant rotation speed of the metallic body u = 32.76cm/s.
Here, we list the measured quantities H and FL along with the average and the
standard deviation of as well as the error in FL by varying the height H.
H [mm]
arithmetic average
of FL [mN]
standard deviation standard
of FL [mN]
in %
deviation
1.4
398.5497
41.3585
10%
1.6
374.6205
41.5839
11%
2
335.3014
35.8837
11%
2.4
300.8547
44.4736
15%
2.8
271.4194
25.7318
9%
3
255.1280
50.2966
20%
3.4
227.0289
22.5932
10%
3.8
205.6078
18.5867
9%
4.2
188.0563
17.2553
9%
4.6
167.9492
18.1923
11%
5
142.4282
18.8820
13%
8
76.7906
13.0395
17%
10
51.1089
13.5545
27%
12
34.7239
11.7275
34%
14
24.8306
3.7032
15%
16
16.4672
8.1893
50%
19
10.2717
6.1984
60%
23
5.5654
2.2559
41%
27
3.2810
2.1739
66%
29
2.5572
1.5512
61%
31
1.8834
1.1239
60%
33
1.5669
0.9578
61%
37
1.0019
1.8128
181%
Table A4-1: Error estimation of the measured FL using single LFV.
From these data we can clearly conclude that an increase in H results in an increase of the standard deviation of measured FL.
Error estimation of the measured velocity using ToF LFV
For the Time-of-Flight method, the two important influencing quantities are the
separation distance D of the two Lorentz force flow meters and the measured
time shift  . With these two values at hand we calculate the vortex velocity Vvortex according to the relation
XXVIII
Vmeasured 
D
.

(A4)
See chapter 1.4 for details. The uncertainty V of the calculated velocity can be
determined by propagation of uncertainty [74]:
∆D D
 d

d

 2 .
∆V  
V(D,) ∆D   V(D,) ∆ 


 dD

 dτ

(A5)
Here we have omitted the subscript in representing the vortex velocity. The uncertainty of the two input parameters and their standard deviation in the measured velocity for the electromagnetic free surface velocity measurement in annulus flow is shown in the following table.
Input parameters
Distance of
magnetsystem D
Transit time 
Typical quantities
Respective
uncertainty
Standard
deviation
7.5cm
0.2cm
2.67%
0.24s
0.01s
4.17%
+
Measured
V =33.48cm/s
7.14%
31.25cm/s
V =29.20cm/s
Velocity V
6.56%
Table A4-2: Error estimation of the free surface velocity measurement.
First we estimate the uncertainty of the measurement of D overall to be
 0.2cm. The uncertainty of the reference photoelectric sensor from the manufacturer is 0.12mm. The uncertainty of the driving unit of the channel (electrical
motor) is ±0.2%. The standard deviation is less than 5% when the rotating
speed of the channel smaller than 15cm/s. The deviation of the measured velocity comparing MVS sensor to reference sensor is approximately about 3%.
Interestingly, the smaller the disturbing body the higher is the discrepancy from
the measured velocity. We find an uncertainty of about 10% for a styrofoam diameter of 40mm and an uncertainty about 20% for a diameter of 10mm.
Input parameters
Typical quantities
Respective
uncertainty
Distance of
22cm
0.5cm
magnetsystem D
Transit time 
0.73s
0.04s
Measured
V+=32.60cm/s
30.14cm/s
V-=27.29cm/s
Velocity V
Table A4-3: Error estimation of flow rate measurement.
Standard
deviation
2.27%
5.48%
8.16%
9.45%
Erklärung
(gemäß Anlage 1 der Siebten Änderung der Promotionsordnung der TU
Ilmenau – Allgemeine Bestimmungen)
Ich versichere, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter
und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe.
Die aus anderen Quellen direkt oder indirekt übernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet.
Bei der Auswahl und Auswertung folgenden Materials haben mir die nachstehend aufgeführten Personen in der jeweils beschriebenen Weise entgeltlich/unentgeltlich 1) geholfen:
1. .........Numerische Untersuchungen im Rahmen von Projektseminar:
Stefan Buhl, Mohsen Habqa
2. .........Numerische Untersuchungen im Rahmen von Masterarbeit:
Abdallah Mansour
3. .........Numerische Untersuchungen im Rahmen von Studienarbeit:
Yang Li
Weitere Personen waren an der inhaltlich-materiellen Erstellung der vorliegenden Arbeit nicht beteiligt. Insbesondere habe ich hierfür nicht die entgeltliche
Hilfe von Vermittlungs- bzw. Beratungsdiensten (Promotionsberater oder anderer Personen) in Anspruch genommen. Niemand hat von mir unmittelbar oder
mittelbar geldwerte Leistungen für Arbeiten erhalten, die im Zusammenhang mit
dem Inhalte der vorgelegten Dissertation stehen.
Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher
Form einer Prüfungsbehörde vorgelegt.
Ich bin darauf hingewiesen worden, dass die Unrichtigkeit der vorstehenden
Erklärung als Täuschungsversuch bewertet wird und gemäß § 7 Abs. 10 der
Promotionsordnung den Abbruch des Promotionsverfahrens zur Folge hat.
(Ort, Datum) (Unterschrift)
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