G Pottlacher
email: pottlacher@tugraz.at

K Aziz

A Schmon

Containerless experimental techniques to obtain thermophysical properties of liquid metals and alloys at high temperatures and density data for liquid aluminium

Keywords: Liquid metal/alloy, thermophysical properties, containerless techniques, density liquid aluminium

High temperature experimentation with solid and especially liquid specimens faces numerous difficulties. Traditional static steady state techniques for the measurement of thermophysical properties are generally limited to temperatures of about 2300 K. This limitation is a result of chemical interaction of the specimens with the containers, the loss of mechanical strength, problems with heat transfer, evaporation and electrical insulation while the sample and its environment are kept for times up to hours at high temperatures. Containerless investigation methods have been developed to avoid these difficulties and to permit the extension of the measurements to higher temperatures. The methods used by our workgroup are on the one hand ohmic pulse heating and on the other hand an electromagnetic levitation technique both can be called containerless techniques. Pulse heating delivers thermophysical properties of electrically conducting materials far into the liquid phase. The measurements allow the calculation of specific heat capacity and the temperature dependencies of electrical resistivity, enthalpy and density of the samples at the end of the solid phase and in the liquid phase. Measurements of normal spectral emissivity at 684.5 nm increase the accuracy of the pyrometric temperature measurements. Thermal conductivity and thermal diffusivity as a function of temperature are estimated from resistivity data using the Wiedemann-Franz-law. Electromagnetic levitation, as the second experimental approach used, delivers data for surface tension (which is not available by means of pulse heating technique) and for density of liquid metals as a function of temperature. Properties of matter at high temperatures are useful for high-temperature technologies such as aerospace, nuclear energy and the establishment of temperature reference points, including applications which are subjected to high temperaturehigh pressure conditions, as well as input data for modeling, which got very popular recently in steel working industry to simulate casting and welding processes and in jewelry industry to reduce reject due to defects.

Introduction

For about 70 % of all industrially formed metal parts the production starts with the liquid metal/alloy. To reduce rejection parts due to casting defects computer-based simulations allow modeling of casting, heat transport, solidification shrinkage, residual stress, heat treatment, welding, forging, rolling and cutting or even predictions of microstructures. But the quality of the simulation might be restricted when wrong thermophysical parameters are used. Thus experimental obtained thermophysical properties of pure metals and also of binary and ternary alloy systems as are of great importance as input parameter for various simulation tools. The most relevant properties for casting simulations are heat of fusion and heat capacity, and for the liquid phase electrical resistivity, density, thermal conductivity and thermal diffusivity, thermal expansion, hemispherical emittance, viscosity and surface tension.

Pulse heating

A dynamic calorimetric pulse experiment utilizes resistive self-heating of an electric conductor -typically wire shaped samples (with diameters ranging from some hundred micrometres up to a few millimetres), rectangular shaped samples (if the material cannot be drawn into wires and test samples have to be cut), foils, or tubes -by passing a large current pulse over the sample. As a result of the materials' resistivity, the test specimen can be heated from room temperature up to melting, further through the liquid phase and finally up to the boiling point in a split second. The following parts are common for all pulse heating experiments: energy storage (typically battery banks or capacitor banks) with charging unit, main switching unit (i.e. high-voltage mercury vapour ignition tubes), experimental chamber with windows for optical diagnostics and the ability to maintain a controlled ambient atmosphere. Pulse heating experiments are commonly performed under inert ambient atmosphere, e.g., nitrogen or argon at ambient pressure or in vacuum. Quite often data recording equipment is placed in a shielded room (see Figure 1). The quantities typically recorded and/or accessible (as a function of experimental duration, t) during such an experiment in the solid and in the liquid phase are the current trough the sample, the voltage drop across the sample, the temperature using surface radiation for optical thermometry and further optical quantities such as thermal expansion of sample, sound velocity measurements, polarization information for optical properties measurements. Based on the initially measured quantities, thermophysical properties such as the change of enthalpy, resistivity, density, thermal conductivity, and thermal diffusivity as a function of temperature can be deduced assuming that the mass, m, or the combination of density and volume of the sample at room temperature are known. Only viscosity and surface tension cannot be obtained by pulse heating.

Our working group has rendered outstanding services in the field of dynamic pulse calorimetry within the last twenty years [1 -11]. A summary of thermophysical data for 22 Elements has been published recently [START_REF] Pottlacher | High temperature thermophysical properties of 22 pure metals[END_REF].

Levitation techniques

Levitation is the process where a sample is suspended against gravity, in a stable position, by an external force without physical contact [START_REF] Egry | [END_REF]. All that is required on earth for levitation is a force vertically upwards equal to the object's weight. This can be achieved by many different means, for example, electrical, magnetical [14], acoustic [15], optical, electrostatic [START_REF] Paradis | Microgravity sci. technol. XVI[END_REF] and aerodynamic forces [START_REF] Millot | [END_REF]18]. A different approach is levitation in microgravity such as parabolic flights or in space [19,[START_REF] Aune | Microgravity sci. technol. XVI[END_REF]. A summary and description of different levitation techniques is given in [START_REF] Egry | Measurement of the Thermodynamic Properties of Single Phases[END_REF] and reviews can be found in [START_REF] Baykara | [END_REF][23][24]. For metallic melts electromagnetic levitation provides an elegant method of noncontact containerless measurement, eliminating most interactions between the sample and its environment [25]. This technique has been used in the past mainly for the study of highly reactive melts at high-temperatures. The sample, assuming a simple spherical shape, is contained in a clean environment and can be studied over a large temperature range [25]. Temperatures again have to be measured with a pyrometer. An electromagnetic levitation apparatus employs inhomogeneous radio-frequency electromagnetic fields to heat and position the samples. Such a field has two effects on a conducting, diamagnetic body. First, it induces eddy currents within the material, which, due to ohmic losses, eventually heat up the sample by inductive heating, and second, it exerts a Lorentz force on the body, pushing it towards regions of lower field strength. The latter effect can be used to compensate the gravitational force [26]. The significant difference between dynamic pulse heating (for this technique it is easy to measure enthalpy and electrical resistivity as a function of temperature) and levitation is the fact, that the actual energy input during a levitation experiment cannot be determined. But on the other hand, levitation is not limited to a subsecond timescale and is required to measure high temperature thermophysical properties such as viscosity, surface tension and density of liquid samples, which cannot be obtained by means of pulse heating. Due to the influence of gravitational forces reliable viscosity measurements only can be performed by levitation in microgravity. These measurements are possible by digital image processing and frequency analysis of induced surface waves of a levitated liquid sample. A schematic display of the experimental technique and some main items of instrumentation for an electromagnetic levitation system as established at TU Graz are given in Figure 2. The details of the components vary for different measurements and for different techniques. The sample is levitated in a special designed high frequency coil, which is supplied by a RFgenerator (100 kHz to 1 MHz). The levitation coil is usually water-cooled. The levitated sample is placed in a vacuum chamber with quartz windows for various types of optical diagnostics such as a fast CCD -camera to monitor the sample shape and thus obtain density or surface tension results, a pyrometer for temperature measurements, and a probe laser, which sometimes also can be used for heating. The ambient cooling-gas also has to be supplied to the experimental chamber.

Uncertainty of liquid aluminium density

The uncertainty of our density results obtained with the levitation technique (k = 2) is 1 % and indicated in Figure 3 with an uncertainty bar, the uncertainty for the data given by Assael (k = 2) is stated in [30] with 0.65 %.

Figure 1 :

 1 Figure 1: Schematic plot of components in a typical pulse heating experiment at TU-Graz.

Figure 2 :

 2 Figure 2: Schematic plot of components of an electromagnetic levitation experiment.The underlying concept of the oscillating drop technique is that the surface oscillations of the liquid drop around its equilibrium shape can be related to surface tension as the restoring force. Translation frequencies and oscillation frequencies allow to calculate surface tension. With the electromagnetic levitation device at TU Graz surface tension and density of liquid copper and nickel[27 -29] have been determined recently. Recent results for density versus temperature for liquid aluminium obtained by the levitation technique are presented in Figure3. The linear fit (full line) to our data (full triangles) is D liquid (T) = 2569 -0.255•T in the temperature range 1000 K < T < 1370 K.

Figure 3 :

 3 Figure 3: Density versus temperature for liquid aluminium obtained by the levitation technique. Full line with triangles this work, dashed line recommended values from Assael [30].

Acknowledgment

Work partially funded by the Austrian Science Fund (FWF) Project P 23838-N20


B Willers

S Eckert
email: s.eckert@hzdr.de

M Barna

J Reiter

Flow measurements in a liquid metal model of a bloom caster under the effect of rotary electromagnetic stirring

Keywords: Continuous casting, electromagnetic stirring, rotating magnetic field, flow measurements

This paper presents an experimental study of electromagnetically stirred flow in the mold using a 1:3 scale acrylic glass model of the round bloom caster from voestalpine Stahl Donawitz GmbH. An electromagnetic stirrer was installed at the mold producing a rotating magnetic field (RMF). Flow measurements were performed by means of the ultrasound Doppler velocimetry (UDV) at room temperature in the eutectic alloy GaInSn. Up to 10 ultrasonic transducers were employed simultaneously in order to obtain a two-dimensional reconstruction of the flow structure. The experiment provides an extensive and valuable data base for validation of numerical methods.

Introduction

Electromagnetic stirring (EMS) by AC magnetic fields is a valid method for contactless flow control in billet/bloom casters since about 40 years [1][START_REF] Asai | Proc. of the 6 th Int. Iron and Steel Congress, ISIJ[END_REF][START_REF] Spitzer | Proc. of the 3 rd Int. Symp. Ele. Proc. of Materials, ISIJ[END_REF][START_REF] Kunstreich | [END_REF], where the rotary stirring of the flow in the mold by means of a rotating magnetic field (RMF) can be considered as a standard application. The stirring is supposed to improve the homogeneity of the molten steel and the quality of the solidified steel strand by significantly decreasing the casting defects. In particular, it is assumed that non-metallic particles are prevented from entrapping in the solidified shell by the so-called washing effect [5]. Particles caught between columnar dendrites are washed out by the forced flow and transported into the bulk liquid. Then, the stirring shall stimulate agglomeration and the formation of larger particles which move up more easily towards the free surface. Moreover, it is expected that forcing of the melt flow in the solidification region promotes a transition from columnar to equiaxed solidification. An intense flow along the solidification front causes fragmentation of dendrites and hence a multiplication of nuclei. Su et al. [6] suggest a dendrite fragmentation criterion for low carbon and high carbon steels casted under the influence of EMS. Industrial experience has shown that the particular stirrer design, position and operating conditions have a strong influence on the metallurgical quality. First of all, a high stirring intensity should be guaranteed by a suitable coil design and appropriate magnetic field parameters. A stirrer operation at low magnetic field frequencies (typically 2 to 8 Hz) is required because the thick copper walls of the mold impede a deep penetration by an AC magnetic field at higher frequencies. The use of low frequencies limits the resulting electromagnetic Lorentz force for driving the rotary flow. On the contrary, an intensive swirling flow at the free surface poses the risk of slag or mold powder entrapment due to the depression of the melt surface at the nozzle. The resulting absorption of impurities is known to lower the quality of steel products significantly. A specific electromagnetic stirring system of two stirrers was considered to influence the meniscus stability [7]. Two independent rotating magnetic fields superimposed upon each other are applied to achieve a flexible control of the stirring motion in the meniscus zone regardless whether an intensive stirring is generated deeper in the mold. Thus, even the opposite direction of stirring of the near meniscus field can be chosen with respect to the main stirrer. A great deal of previous research work was dedicated to the prediction of the mold flow and related heat and mass transfer under the action of EMS, mainly by numerical simulations (see for example [8][9][10][11][12][13]). Numerical calculations can provide a better understanding of the complex flow behaviour, but, experimental data are indispensable for the validation of CFD models. The motivation of our study is to provide a rather generic experiment equipped with advanced ultrasonic flow measuring technique for detailed investigations of the liquid metal flow in a cylindrical column driven by an RMF. Especially, the effect of variations of the field strength and frequency on the stirring intensity is considered. Furthermore, we compare the case of a pure RMF-driven flow with the realistic configuration of a vertically aligned submerged jet in the swirling flow. The flow measurements were conducted in a 1:3 scale Perspex model of the round bloom strand caster from voestalpine Stahl Donawitz GmbH.

Experimental set-up and procedure

The experiments were conducted at the mini-LIMMCAST facility at HZDR. A detailed description of this experimental equipment can be found in previous publications [14,15]. About 12 litres of the ternary eutectic alloy Ga 68 In 20 Sn 12 were used as a model fluid.

A schematic view of the experimental facility can be seen in Fig. 1. The tundish is a cylindrical vessel made of stainless steel connected with the submerged entry nozzle (SEN). A stopper rod controls the melt flow through the SEN. The SEN is an ordinary pipe with an inner diameter of 10 mm and a length of 300 mm. A SEN immersion depth of 35 mm below the surface level of the liquid metal was chosen. The circular mold has an inner diameter of 80 mm and a length of 800 mm. The mold and SEN consist of acrylic glass. The melt discharges from the SEN into the mold as a submerged jet. The experiments were performed in a continuous mode, i.e. after filling the tundish with the melt the stopper rod was lifted to drain the fluid into the SEN. A magnetic pump conveys the melt from the vessel back into the tundish. The liquid levels in both the tundish and the storage vessel were monitored using a laser sensor. The flow measurements have been started after achieving fully developed flow conditions. The velocity inside the SEN could be estimated to be in the range 1.2-1.5 m/s. The electromagnetic system has a bore diameter of 200 mm and a height of 310 mm. This system of 12 coils generates a rotating magnetic field (RMF) with a maximum magnetic flux density of B 0 =20 mT. Flow measurements were conducted for different values of the magnetic field strength varying in the range between 5.8 mT and 18.3 mT. A magnetic field frequency of 2.5 Hz was chosen for the flow measurements. The flow velocity in the mold was measured by means of the ultrasound Doppler velocimetry (UDV). This method is based on the pulse-echo technique and delivers instantaneous profiles of the velocity component projected onto the propagation direction of the ultrasonic beam [16,[START_REF] Eckert | Velocity measurements techniques for liquid metal flows[END_REF]. The measurements in the present study were performed using the DOP2000 velocimeter (Signal Processing SA, Lausanne). A vertical line array of ten 4 MHz transducers (TR0405AS, diameter 8 mm) was assembled with a distance of 10 mm between two adjacent transducers. This vertical arrangement was installed at positions of R=15 mm and R=30 mm parallel to the mold axis to determine the tangential velocity component (see Fig. 2). The outer cylindrical wall of the mold was partly shaped to a plane surface in order to allow an exact positioning of the US sensors which were attached at the wall using a technical vaseline.

Results

A first set of velocity measurements was carried out for the situation of a rotating flow driven by the magnetic field without submerged jet. Tangential velocities are measured at a radius of R=15 mm and R=30 mm. On both measuring lines at a distance of a=43 mm from the sensor the measured velocity equals the tangential velocity. At this position there is no disturbance of the measurement due to projections of the radial velocity. Vertical profiles of the mean tangential velocity obtained at R=30 mm for four different magnetic field intensities and a stirring frequency of f=2.5 Hz are shown in Fig. 3(a). The time-averaged values are calculated from 100 to 300 flow profiles recorded over a period of ca. 30 s to 60 s. The magnetic stirrer was installed in the domain highlighted by the two red lines in the diagram. The largest values of the mean tangential velocity were detected at a height which approximately coincides with the geometric centre of the stirrer. The flow is directly driven by the Lorentz force within the stirrer region whereas the magnetic field intensity decreases rapidly outside the stirrer. Internal friction and secondary flows are responsible for the rotating motion of the fluid above and below the magnetic stirrer. The volume of liquid in the model is distributed asymmetrically with respect to the stirring system. The free surface of the liquid metal is 60 mm above the stirrer. Measurements in the lower area of the mold reveal that the tangential velocities is about 50 % of the maximum detected within the stirrer zone. Fig. 3(b) shows corresponding measurements of the mean tangential velocities for the radius R=15 mm. As it was expected for the lower radial position the velocities are smaller compared to the data obtained at R=30 mm. However, the respective rotation frequencies are almost the same indicating a solid body rotation in a wide domain of the liquid metal column. A fluid rotation frequency of about 1 s -1 as found here corresponds to 40 % of the magnetic field frequency. A remarkable difference is the fact that the vertical velocity profiles do not show the highest velocities in the region of the magnetic stirrer. The maximum of the mean tangential velocity of V t15 =120 mm/s was detected at a position of approx. 200 mm below the stirrer. This velocity distribution can be explained by the conservation of angular momentum. Besides the primary rotating motion a secondary flow exists in the meridional plane of the liquid which is directed from the sidewalls towards the cylinder axis in the regions outside the stirrer. That means that rotating fluid is transported inwards and increases its angular velocity along this way. This mechanism results in an almost uniform vertical profile of the mean tangential velocity at R=15 mm. It is interesting to note that at a position far below the stirrer (z =-440 mm) higher velocities can be observed close to the axis than at radius R=30 mm. flow being approximately two orders of magnitude stronger. Therefore, the acceleration of the rotation near the SEN and the deflection of the free surface are much more pronounced with the jet flow as for the case of the pure RMFdriven flow discussed in the previous section. It is interesting to note that the fluid above the stirrer rotates much faster than inside the magnetically forced region, the angular velocity even exceeds the magnetic field frequency f=2.5 Hz.

On the other hand, the submerged causes lower tangential velocities at R=15 mm in the upper stirring zone (between z=120 mm and z=-40 mm) in comparison to the situation without jet. The conservation of angular momentum is the reason for this phenomenon. The jet widens after leaving the SEN and pushes the rotating fluid from the core towards the side wall. This effect is not significant for positions farther down the strand (z -40 mm).

Conclusions

This paper describes laboratory experiments in a 1:3 scale model for continuous casting of round blooms at voestalpine Stahl Donawitz GmbH. Fluid velocities inside the cylindrical mold were measured non-invasively by means of the ultrasound Doppler velocimetry (UDV). The measuring setup enables the detection of the horizontal flow being a combination of tangential and radial velocity component. Flow measurements of time-averaged velocity profiles can be summarized by the following conclusions:

1. The tangential velocity resulting from the application of a rotating magnetic field (RMF) without jet flow shows different dependencies on the radius with respect to the vertical positions of the measurements. 2. The mold flow is composed of a primary, swirling flow and a meridional secondary flow which consists of two toroidal vortices lying on top of each other. This secondary flow is responsible for a redistribution of angular momentum within the mold. 3. Measurements of the tangential velocity for the situation of a pure RMF-driven flow at R=30 mm show a maximum rotation speed in the stirrer region and a continuous decrease of the tangential velocity with growing distance to the stirrer. On contrary, the tangential velocity does not vary significantly over the height at R=15 mm. In fact, the tangential speed outside the stirrer is even slightly higher as in the magnetic field region which can be attributed to the angular momentum transfer by the secondary flow. 4. A discharging jet through the SEN intensifies the meridional flow near the free surface. The superposition with the RMF-driven flow leads to a drastic acceleration of the rotating flow and a vortex formation near the SEN. This is accompanied by a strong deflection of the shape of the free surface of the liquid metal.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Schematic view of the experimental setup showing the test section and the liquid metal loop

Fig. 3 :Fig. 4 :

 34 Fig. 3: Vertical profiles of the time-averaged, tangential velocity at radial positions of (a) 30 mm and (b) 15 mm

Acknowledgment

The research program of the Competence Center for Excellent Technologies in "Advanced Metallurgical and Environmental Process Development" (K1-MET) has been financially supported within the Austrian competence centre programme COMET (Competence Center for Excellent Technologies) by the Federal Ministry of Economy, Family and Youth; by the Federal Ministry for Transport, Innovation and Technology; by the provinces of Upper Austria, Styria and Tyrol, by the Styrian Business Promotion Agency and by the Tiroler Zukunftsstiftung. The authors from HZDR acknowledge additional financial support from the German Helmholtz Association in the framework of the Helmholtz Alliance LIMTECH.


M Weidner
email: markus.weidner@tu-ilmenau.de

B Halbedel

E Rädlein

Expectable Forces of Lorentz Force Velocimetry in low conducting and slow flowing solutions by the use of high-T c superconductors

Keywords: Lorentz force velocimetry, high temperature superconductor, electrolytes, salts, glasses

    Expectable Forces of Lorentz Force Velocimetry in low conducting and slow flowing solutions by the use of high-T c superconductors M Weidner, B Halbedel, E Rädlein

Introduction

The Lorentz force velocimetry (LFV) is a novel method for flow measurement of low conducting and slow flowing fluids [1]. In this application the velocity or the conductivity of a fluid is measured by the accelerating force on a strong magnet system generated by magnetic induced eddy currents in the fluid. Because of the contactless measuring this method can stand very hot and aggressive fluids like molten salt or glass. In former phases investigations of permanent magnet systems were done. Here a Halbach array system based on NdFeB magnets was successfully used [2]. The measurement of the very small forces which are in µN range is done by an electromagnetic compensation system. This very sensitive system limits the weight of the magnet system to 1 kg so that there must be constructed an efficient system. To generate higher eddy currents in the fluid and get a higher resolution of this application magnets with higher energy densities are necessary. A promising method is the use of high-temperature superconducting bulk materials (HTS) because of their very high energy density [START_REF] Halbedel | Proceedings Workshop Elektroprozesstechnik[END_REF]. To get to know what Lorentz forces can be expected numerical simulations were done with variations of the trapped field in the HTSs. Also investigations on possible arrangements of several bulks were analyzed.

Modell

Based on the experimental setup the fluid volume which must be to consider has a quadratic cross section with sides of 50 mm and a length of 400 mm. Due to the low velocity of 5 m/s and the poor conductivity of 4 S/m the fluid profile is negligible disturbed by the magnetic field interaction and the magnetic field is not deformed by the induced secondary field and consequently for the modeling of the fluid a solid body approximation can be used [2]. The cylindrical HTS bulks are positioned symmetrical left and right of the cannel with distance of 3 mm. Fig. 1 shows a Modell with normalized current density, y component of the magnetic flux density of the HTS bulks and the induced eddy currents (red arrows) by the use of a center line between bulk pairs.

Fig. 1: Model of a five HTS bulk arrangement

Because of the almost linear depending magnetic flux density in perpendicular direction of cylindrical HTSs the bulks were simulated based on Bean's Modell with a radial depending current density J C (r) inside like Eq. ( 1).

(1) By variation of the constant c the trapped field of the HTS bulks can be simulated for changes of their operating temperature and material properties.

Results

To estimate Lorentz forces of a superconducting magnet system the trapped field of commercial available YBCO bulk pairs with diameter of 46 mm and height of 17 mm was simulated from zero to 5 T which is the maximum generated field in our laboratory. As shown on fig. 2 the Lorentz force increases by two oppositely arranged HTS bulks increasing trapped field. How the approximation of the graph indicates a quadratic behavior which agrees with the scaling law of Thess et. al. [1] at constant velocity and electric conductivity can be seen. By cooling these superconductors with liquid nitrogen at temperatures of 77 K a field of approximately 1 T can be trapped. It is reported that YBCO bulks are able to trap 5 T by cooling them down to the temperature range of 45 to 50 K [START_REF] Prigozhin | [END_REF]. With this magnetic flux density the resolution of the LFV can be increased by the factor of 25. Another way to improve the trapping behavior of HTS bulks is to irradiate them with neutrons [5]. Especially for small diameters of bulks trapped field can be increased dramatically. So a magnet system based on small improved HTS bulks could be arranged to work with liquid nitrogen at temperatures of 77 K which means resource saving and easier handling. For estimation of the potential of this idea HTS bulks with a diameter of 14 mm and a height of 6 mm were simulated with a trapped field of 1 T in different arrangements. Because of comparison and the limited mass m of the magnet system an efficiency factor e was defined as shown in Eq. ( 2).

(

) 2
Hereby it is possible to evaluate the magnet system in quantitative way. For a symmetric rectangular (fig. 3) and a rhombic (fig. 4) arrangement of four pairs of HTS bulks on each channel side with same magnetization direction the generated Lorentz force increases with increasing horizontal (y direction) and decreasing vertical (z direction) displacement. This leads to the point that for same magnetic directed HTS bulks one dimensional chains with a definite horizontal displacement would be the best magnet system. Such magnet systems are realizable with multi-seeded HTS bulks.

Fig. 3: Efficiency factor e of Lorentz force in channel direction of a symmetrical rectangular arrangement of four HTS bulk pairs with variation of horizontal and vertical distances

Fig. 4: Efficiency factor e of Lorentz force in channel direction of a symmetrical rhombic arrangement of four HTS bulk pairs with variation of horizontal and vertical distances By using a HTS bulk system with different directions of magnetization it is possible to find optimal compact arrangements which are able to generate higher Lorentz forces. Fig. 5 shows the generated Lorentz forces of the symmetric rectangular system from figure 2 with on additional HTS bulk with opposite magnetization in the center. At this arrangement also the vertical displacement should be as small as possible. For the horizontal direction there is an optimum which depends on the bulk diameter. For a symmetric rhombic arrangement with an additional HTS bulk in the center fig. 6 shows that there is an optimum for vertical and horizontal displacement. Although this arrangement is the smartest one it generates the relative to its weight the smallest Lorentz force. Concerning this results a by magnetization alternating one dimensional chain with defined horizontal distances is a promising configuration for magnet system of HTS bulk for the LFV. For comparison a single HTS bulk pair with diameter of 47 mm and height of 17 mm at 77 K has an efficiency factor of 79.4 µN/kg. This shows that already simple geometrical arrangements of improved HTS bulks have a 17.5% better efficiency factor which is 93.3 µN/kg. Fig. 7 shows the Lorentz force in channel direction of two adjoined bulk pairs (diameter 47 mm and height 17 mm) with same and opposite magnetization direction under variation of their horizontal distance at 77 K with a trapped field of 1.1 T. For same directed magnetization the generated force increases with distance until the influence of the HTS bulk pairs does disappears and the measured force will be twice of the force of on pair. For low distances the field in the middle of the two pairs superimposes so that there is a low field gradient between the HTS bulk pairs in the channel. Hence the induced eddy currents in the center of the channel are very low. For large distances there are two regions with strong eddy currents more and the Lorentz force increases. In case of one opposite magnetized HTS bulk pair at low distances the field gradient between the bulk pairs is very high so that the induced eddy currents in this region of the fluid are very high and highest Lorentz forces are generated. For increasing distance the force approaches to the double force of one HTS bulk pair. In the arrangement without a distance between the bulks the efficiency factor of this system amounts 105.1 µN/kg.

Conclusion

The analyzed cases show that it is possible to generate higher Lorentz forces with optimized dimensions and arrangements of HTS bulks. Therefor systems with different magnetized HTS bulk must be used. To increase the resolution of the LFV dramatically by the magnet system the biggest effect will be achieved by cooling these materials below 77 K. In interaction with optimized dimensions and arrangements a strong magnet system based on HTS can be built. Future works will validate these numeric results with experiments and show the feasibility of such conditions with special designed cryostats.

Fig. 2 :

 2 Fig. 2: Generated Lorentz force F Lx in flow direction of HTS bulk pair with diameter of 46 mm and a distance of 56 mm depending on trapped field B y at 5 m/s and 4 S/m of the fluid

Fig. 5 :Fig. 6 :

 56 Fig. 5: Efficiency factor e of Lorentz force in channel direction of a symmetrical rectangular arrangement of four HTS bulk pairs with a opposite magnetized bulk pair in the center with variation of horizontal and vertical distances Fig. 6: Efficiency factor e of Lorentz force in channel direction of a symmetrical rhombic arrangement of four HTS bulk pairs with a opposite magnetized bulk pair in the center with variation of horizontal and vertical distances

Fig. 7 :

 7 Fig. 7: Lorentz force of two HTS bulk pairs with diameter of 47 mm depending on the distance and magnetization direction

Acknowledgment

This work is supported by the German Research Foundation (DFG) within the Research Training Group "Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing" (RTG-1567/2).


N Dubovikova
email: nataliia.dubovikova@tu-ilmenau.de

Ch Karcher

Y Kolesnikov

Electromagnetic technique for liquid metal flow measurement by force time-of-flight principle

Keywords: flow rate, liquid metal, Lorentz force

Flow rate and velocity measurement in opaque and aggressive liquids such as metal melts is a challenging task in scopes, where flow should be controlled. Optical measurement techniques as well as submerging probes or control units cannot be applied in this highly severe environment; hence, contactless methods are of interest for practical flow control. One branch of such techniques is flow rate estimation by measurement of electromagnetic parameters. In this paper we present working principle and experimental results of a such contactless electromagnetic technique -time-offlight Lorentz force velocimetry (LFV), which allows determine the flow rate of liquid metals. The principle is based on measurement of Lorentz force that generated inside liquid metal under its motion through magnetic field, because the force linearly depends on flow rate. The method needs neither additional calibration nor information about fluid properties and magnetic field magnitude.

Introduction

According to the principles of magnetohydrodynamics, videlicet that as a result of the relative movement of an electrically conducting liquid in an externally applied magnetic field, Lorentz force 𝐹 𝐿 is induced. The induced forces act on both: the conductor and the system producing the magnetic field, they tend to slowdown the relative motion and are proportional to mean velocity 𝑣 and magnetic field 𝐵 [START_REF] Davidson | An introduction to Magnetohydrodynamics[END_REF]:

F L = 𝑞v × B
(1) where 𝐵 = 𝐵 0 + 𝑏 -applied and induced values of magnetic field; 𝑞 -the charge, which is carried with liquid. And 𝑞v = 𝑗 represents eddy currents within the flow, initiated by magnetic field. According to Ohm's law for moving conductive liquid:

j = 𝜎(E + v × B),
(2) where 𝜎 -electrical conductivity of liquid; E -electric field. Eddy currents are followed by creation of Lorentz force, which opposes the flow. By virtue of the fact that permanent magnets are caused appearance of the force and in accordance with the third law of Newton, a reaction force, equal in magnitude and opposite in direction to Lorentz force, acts on magnets. The reaction force is measured by commercial force sensors, which are rigidly coupled with permanent magnets. Hence, according to (1), [START_REF] Davidson | An introduction to Magnetohydrodynamics[END_REF], by applying Faraday's law, measured Lorentz force can be also estimated by [START_REF] Jain | [END_REF], where only mean velocity 𝑣 or volumetric flow rate 𝑄 𝑣 are unknown:

𝐹 𝐿 ∝ 𝜎𝑣𝐵 0 2 𝐿 3 , (3)
where 𝐿 -characteristic length of system magnets-liquid.

Working principle

Time-of-flight Lorentz force velocimetry (Fig. 1) allows to measure the flow rate in liquid metal and the technique is unaffected by physical properties of fluid or by outer conditions. The time-of-flight principle [START_REF] Jain | [END_REF] is based on correlation of two Lorentz force measurements. Two flowmeters consist from a pair of permanent magnets and force sensor each. The sensors detect reaction of Lorentz force on magnets. Sensors are mounted on a duct at a certain distance 𝐷 between each other. A vortex moves sequentially through magnetic field of both flowmeters and disturbs flow and, hence, measured force. Correlation of the signals gives time delay value 𝜏 that vortex needs to pass distance between flowmeters. The distance between sensors per time difference give mean velocity of liquid metal within a duct with cross-section area 𝐴, which, in turn, allows get liquid metal flow rate 𝑄:

𝑄 = 𝑣𝐴 = 𝐴 𝐷 𝜏 (4)
3D strain gauge sensors are used for force measurements, the measurement range of each direction is 2 N, so magnet holding system should be as light as possible to not overload strain gauge component in gravity direction. The goal was reached by applying standard aluminum profiles for construction. Besides this, new magnet holding system allows mount permanent magnets with 1 mm distance to the duct, which increased sensitivity of force measurement.

The model experiments described here demonstrate that time-of-flight LFV is feasible non-contact electromagnetic technique for measuring and control of volumetric flow rates in turbulent liquid metal flow.

Results

To validate time-of-flight LFV method before liquid application, dry test with a copper plate was provided. The plate was moving through the gap between permanent magnets of flowmeters (fig. 2a Measurement system and signal processing enabled to estimate mean velocity of liquid metal in the duct (Fig. 3). Electromagnetic pump creates flow within the duct, so its rotation frequency is used as horizontal coordinate to refer to velocity. The measurement results show high error value, as one can see by error bars on both diagrams. This is observed due to low signal-to-noise ratio. More specifically, signal is getting lost against the background noise, what is easy to see by comparison between cross-correlation function of two signals and their autocorrelation functions: if peaks of cross-and auto-correlations coincide we are able detect only noise. Another observation, that can be made due to presented diagrams -the time-of-flight velocity value mostly lays lower than measurement results of Vives-probe [START_REF] Ricou | Local velocity and mass transfer measurements in molten metals using an incorporated probe[END_REF]. This can be explained by the velocity difference between mean flow in the duct center and moving vortices or other disturbances that are slowed down according to their size and direction either by boundary layers or by internal friction in liquid mostly. The time delay values, calculated by cross-correlation of x-and y-directions, don't represent clear change of velocity with increase of pump rotation frequency and, thus, cannot be used in velocity estimation. In x-direction constant bulk flow force is in evidence and it is a complicated task to distinguish useful signal from strong turbulent disturbances. In y-direction case flow is parallel to magnetic field lines and if this occurs there is no Lorentz force generation possible. The only component, that shows visible change of time delay during experiment, is z-component of force, so the mean velocity was estimated by this component.

Conclusions

Experimental research of time-of-flight Lorentz force velocimetry is very challenging and interesting task. And the method itself is a prospective technique for mean velocity and volumetric flow rate measurements especially in relation to its industrial applications. This method is successfully used in laboratory conditions for velocimetry of turbulent liquid metal and gained results show sufficient correlation between time-of-flight and reference velocity estimation without applying any additional calibration. The time-of-flight Lorentz force velocimetry does not directly depend on physical properties of liquid metal and gives wide opportunities to investigate behavior of vortices of different nature in turbulent duct flow under moderate and high Reynolds number.

Fig. 1 :

 1 Fig. 1: Working principle of time-of-flight LFV. Two identical flowmeters, consist of permanent magnets pair and force sensor each, measure Lorentz force to calculate flow rate.

2 :

 2 ,b). The test's results demonstrate the highest value of normalized correlation coefficient C at time equal 0.32 sec, what with the distance 0.15 m between flowmeters gives us rough estimation of mean velocity of plate 0.47 m=sec and proved that method is working and can be applied for more physically complicated liquid flow. Dry test's time-of-flight LFV results. Force signals with clear peaks (a), caused by copper plate passing through magnetic field of measurement systems and normalized cross-correlation coefficient C of the signals (b), which shows time-of-flight value

Fig. 3 .

 3 Fig. 3. Time-of-flight measurements results in comparison with reference method -Vives-probe. Velocity on (a) and (b) are presented for normal position of sensors with and without artificial vortex generation correspondingly

Acknowledgment

We would like to express appreciation to DFG for funding of our Research Training Group Lorentz force velocimetry and Lorentz force eddy current testing and this research particularly (GRK 1567).


T Wondrak
email: t.wondrak@hzdr.de

J Pal

F Stefani

V Galindo

S Eckert

Contactless inductive flow

Keywords: flow measurement, contactless inductive flow tomography, temperature driven convection

come

Introduction

The Czochralski (Cz) crystal growth method delivers about 95% of the mono-crystalline silicon production worldwide. Due to the melt temperature of more than 1420 °C and the highly corrosive impact of liquid silicon on most materials, any contactless method for flow determination would be highly attractive and welcome. Especially the poloidal flow structure below the meniscus in the melt plays a key role for the quality of the crystal since it controls the temperature gradient and mass transport. The flow structure, in turn, is governed by the temperature gradient and the differential rotation of crucible and crystal. For stabilizing the flow, external magnetic fields are frequently used [1]. One promising candidate for flow determination is the contactless inductive flow tomography (CIFT) [2,3,4] which is able to reconstruct the mean three-dimensional flow structure in liquid metals. By applying primary magnetic fields to the melt and by measuring the flow induced magnetic field perturbation outside the fluid volume the flow is inferred from solving a linear inverse problem. Appropriate regularization methods are necessary in order to mitigate the nonuniqueness problem. Challenges for applying CIFT to Cz crystal growth arise from the weak poloidal flow velocity which is only in the order of a few cm/s, and from the relatively large distance of the sensors to the melt of about 20 cm, because the sensors have to be placed outside of the puller due to the high temperature of about 1500 °C inside. These conditions cause a high ratio between the applied magnetic field and the measurable induced field and demand a magnetic field measurement technique with a dynamic range of at least 5 orders of magnitude. Motivated by some preliminary test measurements at real industrial pullers, we would like to investigate in more detail the applicability of CIFT for determining small velocities as those occurring in thermal convection. In the first instance, we omitted to generate any rotation of the setup and focused on a modified Rayleigh-Bénard (RB) configuration which had already been used to study the effect of local temperature gradients [5]. The goal of this first experiment was to check whether the small induced magnetic fields are detectable at all. Beside the low velocities longer measurement times of more than 4 hours have to be considered as difficult because the environmental field might also change during that time. In our test case, we used one magnetic field applied in vertical direction which allows for identifying the position of the wind within the fluid container, a flow pattern that is typical for this kind of RB flow. The deployment of more applied fields for a full three-dimensional reconstruction of the flow is left for further investigations.

Contactless inductive flow tomography

CIFT relies on the following integral equation system which provides the flow induced magnetic field 𝑏 ⃗ outside the fluid volume 𝑉 for an applied magnetic field 𝐵 ⃗ 0 under the assumption of insulating boundaries [2,3]:

𝑏 ⃗ (𝑟 ) = 𝜎𝜇 0 4𝜋 ∭ [𝑣 (𝑟 ′ ) × 𝐵 ⃗ 0 (𝑟 ′ )] × (𝑟 -𝑟 ′ ) |𝑟 -𝑟 ′| 3 𝑉 𝑑𝑉 ′ - 𝜎𝜇 0 4𝜋 ∯ 𝜑(𝑠 ′ )𝑛 ⃗ (𝑠 ′ ) × (𝑟 -𝑠 ′ ) |𝑟 -𝑠 ′| 3 𝑆 𝑑𝑆 ′ (1)
𝜑(𝑠 ) = 1 2𝜋 ∭ [𝑣 (𝑟 ′ ) × 𝐵 ⃗ 0 (𝑟 ′ )] • (𝑠 -𝑟 ′ ) |𝑠 -𝑟 ′| 3 𝑉 𝑑𝑉 ′ - 1 2𝜋 ∯ 𝜑(𝑠 ′ )𝑛 ⃗ (𝑠 ′ ) • (𝑠 -𝑠 ′ ) |𝑠 -𝑠 ′| 3 𝑆 𝑑𝑆 ′ (2)
In this equation system 𝜑 denotes the potential and 𝑣 denotes the velocity of the liquid metal which is assumed to have uniform conductivity 𝜎 in the volume 𝑉 with the surface 𝑆. In the volume integrals, 𝑑𝑉 ′ denotes the volume element and 𝑟 ′ the position vector in the volume. In the surface integrals, 𝑑𝑆 ′ denotes a surface element and 𝑛 ⃗ (𝑠 ′) represents the normal vector of the surface at position 𝑠 ′.

In order to reconstruct the velocity field 𝑣 in the volume 𝑉 this integral equation system has to be inverted. The intrinsic non-uniqueness problem of the emerging linear inverse problem, which mainly concerns the detailed depth-dependence of the velocity, is circumvented by utilizing the so-called Tikhonov regularization [3,6].

The main challenge of CIFT is the reliable measurement of the induced magnetic field which is typically 2 to 5 orders smaller than the applied magnetic field. Therefore, a very stable current source for generating the excitation magnetic field and magnetic field sensors with a high dynamic range and a very linear response are needed.

Experimental setup

The setup consists of a cylindrical column of aspect ratio 1 filled with GaInSn which is homogeneously heated from below through a copper plate (Fig. 1). In order to emulate the growing crystal of a Czochralski system, only partial cooling of the top is realized by means of a circular heat exchanger mounted concentrically within the upper lid. This partial cooling covers approximately the same relative area as the growing crystal in a real puller. Both heat exchangers (heater and cooler) are made of copper with branched channels to minimize remaining temperature gradients. PIDcontrolled thermostats with a large reservoir were used to supply both the top heat exchanger and the bottom plate with a coolant/heating fluid at high flow rate.

Preliminary results

Before the experiment, we had simulated the typical flow structure in this setup for a temperature difference of 40 K using OpenFoam, and calculated the induced magnetic field outside the vessel assuming a homogeneously applied vertical magnetic field of 3 mT, as seen in Fig. 2. A mean velocity of about 2 cm/s was predicted. The flow induced magnetic field 𝑏 𝑟 was calculated at r = 75 mm, which corresponds to the chosen positions of the magnetic field sensors. The induced magnetic field there is expected to be in the range of 100 nT which is about 5 orders of magnitude smaller than the applied magnetic field. Therefore, we concentrated on reliable magnetic field measurement with vertical excitation magnetic field. For this configuration it is supposed that the angle of the wind can be easily identified. The maximum of the induced magnetic field over the azimuth corresponds to the position of the wind (see Fig. 2). Once the safe measurement of the induced magnetic field is guaranteed, we can later add a horizontal magnetic field for a full three dimensional flow reconstruction. We conducted our experiments during the night in order to avoid magnetic field changes generated by any moving ferromagnetic parts or switching currents in the experimental hall. With this precaution we were able to measure the magnetic field over 8 hours with a drift of about 20 nT. In a first test setup we had observed magnetic field changes in the order of 800 nT resulting simply from ambient temperature changes in the experimental hall occurring between evening and midnight. These changes result from the temperature expansion of the experiment which modifies the relative position and alignment of the excitation coils and the magnetic fields sensors. Therefore, we used copper for the mounting frame of the excitation coils, and cooled the excitation coils in order to maintain a constant temperature during the experiment. For mounting of the magnetic field sensors we tried a variety of different constructions made of PVC, copper, and Macor ceramics. Due to the very small temperature expansion coefficient of Macor we fixed the sensors on a Macor plate using brass screws, and mounted the plate on the coils. Additionally, we replaced all ferromagnetic parts in the experiment by parts made of stainless steel, brass or copper. Ultimately, all these arrangements allowed us to measure the flow induced magnetic field over about 12 hours. Fig. 3 illustrates a typical measurement using a temperature gradient of 30 K. From the 20 magnetic field sensors we selected 4 sensors at positions which are shifted by 90°. In Fig. 3(a) it can be clearly seen that the magnetic field rises after generating the temperature gradient (marked by the black arrow), and is maintained until the temperature gradient approaches zero again (marked by the grey arrow). At the end of the experiments, all 4 sensors return approximately to zero, with an offset between 20 and 40 nT. In the long period when the temperature gradient is switched on, several transitions between different flow regimes can be detected. Comparing the magnetic field data with the temperature measurement in the melt near the top gives a good correlation, as shown in Fig. 3(b). For instance, in all periods, when sensors 3 and 13 show a strong signal, we also see strong fluctuations in the temperature measurement. This can be attributed to different positions of the large convection roll (wind) within the cylindrical melt volume (see also Fig. 2). Another important feature of such kind of RB configuration is the existence of long-wave oscillations of both the temperature and the flow pattern. Analysing the temperature measurements in the flow, as depicted in Figure 3(b), we get a typical frequency of 0.052 Hz. The same frequency is present also in the magnetic field data which confirms that the recorded signals are generated by the flow.

Conclusions

In this paper, we have demonstrated the measurement of the flow induced magnetic field in a modified Rayleigh-Bénard (RB) configuration with maximum velocities in the order of 2 cm/s. Taking into account the thermal expansion of the setup during the measurements, experiments over 12 hours during night time are possible with a magnetic field drift of about 20 nT. Using 20 magnetic field sensors over the azimuth at one height, the position of the wind can be detected robustly over time. The comparison of the magnetic field measurements with temperature measurements in the liquid gives a good correlation. Moreover, recurring long-wave changes of the flow pattern can be detected by the magnetic field measurements. The next step will be the comparison of the magnetic field measurements with Ultrasound Doppler Velocimety measurements, in order to validate the position and the velocity of the wind. Additionally, we will reduce the temperature gradient for estimating the lowest velocity which can be detected with CIFT. After that, we will develop a new setup for deploying a second applied magnetic field to enable a full three-dimensional reconstruction of the flow.

During the preparation for the presented experiments, it turned out that even measurements over the weekend (72 hours) are possible. This enables long time measurements to investigate long term oscillations of the liquid metal. Therefore, CIFT complements nicely Ultrasound Doppler Velocimetry measurements which are usually restricted to approximately 2 hours measurements time due to the sedimentation of scattering particles. Hence, a remixing of the fluid is required in order to allow further measurements. However, such a remixing disturbs the actual flow structure.

Fig. 1 :

 1 Fig. 1: Schematic sketch and photos of the modified Rayleigh-Bénard configuration.

Fig. 2 :

 2 Fig. 2: Flow induced magnetic field b r at r = 75 mm as computed from a CFD simulation of the velocity field for a temperature difference between the top and the bottom of 40 K.

Fig. 3 :

 3 Fig. 3: A typical experiment with a temperature gradient of 32 K: magnetic fields at 4 selected positions (a), and temperature in the fluid close to the top of the container (b).

Acknowledgment

Financial support of this research by the German Helmholtz Association in the frame of the Helmholtz-Alliance LIMTECH is gratefully acknowledged.


O Andreew
email: o.andreev@hzdr.de

T Gundrum

T Wondrak

S Eckert

G Gerbeth

Sensitivity of electromagnetic method for gas bubble detection in liquid metal flows

Keywords: electromagnetic induction technique, two-phase flow, bubble detection

Electromagnetic induction (EMI) is known as a technique which is very sensitive to variation of electrical conductivity of materials. In the present study we simulate numerically the work of electromagnetic sensor which consists of two differential coils. Such a detector rejects the basic signal and monitors the perturbations of electromagnetic field caused by the heterogeneity of electrical conductivity. We investigate the capability of this method for detecting non-metallic inclusions, such as gas bubbles in liquid metal (GaInSn) and sensitivity of differential detector to variation of its configuration.

Introduction

Liquids metals are widely used in industrial processes and scientific investigations. For example, the homogeneity of liquid sodium flows in the cooling systems of fast reactors is very important for the safety of nuclear power plants. The presence of any gaseous admixture is an indicator of very undesirable processes in the sodium flows: leakage, cavitation and so on. It can make the work of hydraulic system unstable or lead to ignition. Thus, it become mandatory to develop sensors capable of detecting any gaseous voids in the sodium liquid which can be used as emergency indicators of boiling or gas inflow into the loop. In the present work we develop a simple sensing technique for electronic detection of gas bubbles in liquid metal flow using the electromagnetic inductance (EMI). This method has been proved [1,2] to be an inexpensive, non-intrusive systems with low but sufficient resolution of the internal distribution of material parameters. The technique can deliver information on the distribution of electrically conductive and magnetically permeable material within the object space. EMI is the most appropriate method for monitoring metallic objects and media with inhomogeneous distribution of electrical conductivity. In particular, liquid metal flows with gas bubbles inside can be easily monitored by this method. Measurements can be acquired by passing through the liquid metal an alternating primary magnetic field. This magnetic field interacts with conductive objects in the medium to produce eddy currents, which, in turn, produce a secondary magnetic field, which is measured by sensing coils. In our numerical experiments we used a simplified system, which consists of one excitation coil and two detector coils. The excitation is a rectangular coil while the detector should be a gradiometer type of coil. In the case of no bubble, the receiver voltage should be zero. When the bubble enters the sensing area in the duct, the receiver will produce a voltage. The main task of present investigation is to define a proper configuration of receiving coils in order to ensure the maximal sensitivity of detector to the presence of gas bubbles (non-metallic enclosures) for the whole interior of a liquid metal target.

Numerical model

A formal description of the instrument system can be derived from Maxwell's equations in complex amplitude of vector potential A  notation as described in [2, 3] for a practical case:

                          x A x A i x x A i x A             2 2 2 2 2    
Here,  ,  averaged and perturbed electrical conductivities,

2 0 2 2 L      the dimensionless "skin depth"
defined by the magnetic permeability of vacuum 0, frequency of applied magnetic field  and length scale L of the problem. The frequency =100 Hz has been selected as a trade-off to provide sufficient primary field penetration across the pipe (skin depth -27.45 mm) and to have a good induction multiplication factor for the measured signal on the receiver coil. The cross-sectional scheme of calculation domain is shown in Figure 1(a). It consists of four principle zones. The interior zone 3 (-1x1) contains the rectangular duct, which is filled with the working fluid. The half-width of the duct is used as length scale L=25 mm. The electrical conductivity varies in this zone from zero (outside of the duct) to the maximal value, which corresponds to conductivity of GaInSn taken for the dimensionless statement of the problem as a scaling value. Other three zones 1, 2 and 4 (Fig. 1(a)) are considered to be a non-conducting. The excitation coil (inductor in Fig. 1(a)) is installed on the half-length of the column filled with eutectic alloy GaInSn. In dimensionless notation it has size of 2×2. The detector consists of two rectangular coils (Fig. 1(a)). It was posed on the opposite side of the column. The top (magenta) and bottom (blue) coils are wound in series opposition. Actually the resulting output signal from such a detector, often known as a planar gradiometer, is proportional to the difference of magnetic fluxes passing through the coils. In the numerical experiments we vary the vertical ℓ z , and horizontal ℓ y size of the coils in the range of (ℓ z , ℓ y )=0.6…3 in order to define the maximal value of output signal for the current position of a single gas bubble inside the liquid metal. In this study, instead of the real gas bubbles we use a single non-conducting spherical target (0.2) placed inside the fluid as it is shown in Figure 1(b). Position the sphere was varied in three directions.

Result and discussion

In Figure 2 (a) we show three-dimensional distribution of sensitivity of the differential detector to the presence of a gas bubble. The numerical experiment was performed in the following way. On the first step, the single spherical target was settled in a certain place and the corresponding distribution of magnetic field was calculated in the plane of detector. On the second step the voltage U generated in the coils was evaluated according to the following equation:

          s n s n ds B ds t B U 
Here B n is the amplitude value of the normal to the coil component of magnetic field, s=ℓ z ℓ y is area of the rectangular coil. The size of coil was varied in the range of (ℓ z , ℓ y ) = 0.6…3 as it is shown by the green lines in Figure 2 (a). The maximal value of the signal in the range of variation was accepted as an "optimal" one for the certain position of the spherical target. In Figures 2(b,c) the yellow isosurfaces confine the regions of liquid metal, where this "optimal" detection of bubbles requires a coil with the sizes which do not exceed the values of ℓ z =1 and ℓ y =1. It is a rather narrow region situating very closely to the rear wall of the channel near the detector. The red shells in Figure 2(d,e) indicate the regions, where the gas bubbles can be effectively detected by the coils with ℓ z =2 and ℓ y =2. In Figure 3 illustrated by 2D distributions of "optimal" coils sizes ℓ z and ℓ y in two vertical cross-sections at y=-0.2 and x=-0.5.

According to Figures 3 (a,b) the region, which is situated closely to the driving inductor, requires for the effective monitoring of gas bubbles installation of a measuring coil with the vertical and horizontal sizes equal at least to (ℓ z , ℓ y )=3 (or even more). This region (colored in blue in Figures 3 (a,b)) occupies almost 55% of the plot areas.

For any real gas bubble measurements one should choose an appropriate constant shape of the detector coils. In Figure 4 we show distributions of sensitivity for three fixed configurations of detector with ℓ z =ℓ y =2 (Fig.

Conclusions

The differential electromagnetic detector considered in this work is not sensitive to the basic symmetrical magnetic field driven by the external excitation coil. The electrical signal delivered by this probe is generated by possible perturbations of magnetic field caused by a local impurity of electrical conductivity (gas bubble). Magnitude of such perturbations depends on the value of electrical current in its origin (position of gas bubble). Thus, the detector is very sensitive to the bubbles which are situated near the excitation coil and in the vicinity of the lateral walls, where the electrical current concentrates due to the "skin" effect. In particular case, when a bubble occurs exactly on the horizontal central plane (z=0), the detector does not "recognize" the bubble, because the distribution of perturbed magnetic field becomes symmetrical with respect of the central plane and both positive and negative coils generate the same signal. In order to obtain a recognizable signal caused by the gas bubbles occurred in the whole volume of the liquid metal domain, it is reasonable to use the relatively large detector coils. Their size should be commensurable with the cross-sectional area of the domain, namely, 2-3 its length scales.

[1] A.J. Peyton, M.S. Beck, A.R. Borges, et al.,(1999), Development of Electromagnetic Tomography (EMT) for Industrial Applications. Part 1: Sensor Design and Instrumentation, 1st World Congress on Industrial Process Tomography, Buxton, Greater Manchester, April 14-17 [2] A.J. Peyton, Z.Z. Yu, G. Lyon, S. Al-Zeibak, A.J. Fereira, J. Velez, et al.,(1996), An overview of electromagnetic inductance tomography: description of three different systems. Meas. Sci. Technol., 7 261-

[3]

A.R. Borges, J. E. de Oliveira, J. Velez, C. Tavares, F. Linhares, A. J. Peyton, (1999), Development of Electromagnetic Tomography (EMT) for Industrial Applications. Part 2: Image Reconstruction and Software Framework. 1st World Congress on Industrial Process Tomography, Buxton, Greater Manchester, April 14-17.

  Fig. 1: (a) Sketch of numerical domain, (b) scheme of a spherical target (gas bubble) positioning.

  Fig. 2: (a) 3D distribution of sensitivity U of differential detector obtained for "adaptive" configuration of the coils. U=0.310 -5 -yellow, 110 -5 -red, 210 -5 -green, 310 -5magenta. Isosurfases define the regions where the bubbles can be detected with the maximal sensitivity by the coils with (b) ℓ z =1, (c) ℓ y =1, (d) ℓ z =2, (e) ℓ y =2.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Cross-sectional distribution of the "optimal" sizes of detector coils. (a) ℓ z , (b) ℓ y for the vertical plane y=-0.2, (d) ℓ z , (e) ℓ y for the vertical plane x=-0.5. Yellow -(ℓ z , ℓ y )=0.6 -1, red -(ℓ z , ℓ y )=1 -2, green -(ℓ z , ℓ y )=2 -3, blue >3

  4(b)), ℓ z =1, ℓ y =2 (Fig.4(c)), ℓ z =ℓ y =1 (Fig.4(d)) in comparison to the experiment with the "adaptive" detector coils (Fig.4(a)). The small detectors under evaluate the central region of the liquid metal column, especially on the vertical mid-plane y=0, where the eddy currents decay. The maximum of sensitivity displace toward the detectors, otherwise in the case of relatively large measuring coils, the most sensitive region is situated in the region of strong eddy currents, namely, near the inductor driving the magnetic field.


T Gundrum
email: th.gundrum@hzdr.de

P Büttner

A Peyton

B Dekdouk

T Wondrak

S Eckert

Contactless inductive bubble detection in a liquid metal column

Keywords: bubble detection, two phase flow, liquid metal, inductive contactless measurement, void fraction

The detection of bubbles in liquid metals is important for many technical applications. The opaqueness and the high temperature of liquid metals set high demands on the measurement system. The electrical conductivity of the liquid metal is relatively high, which can be exploited with contactless methods based on electromagnetic induction. We will present a measurement system which consists of one transmitting coil and a planar gradiometer on opposite sides of the pipe. With this sensor we were able to detect bubbles in a sodium flow inside a stainless steel pipe.

Introduction

Bubble detection and void fraction measurement is of huge interest in two-phase flow research and in industrial applications like continuous casting of steel, liquid metal cooled reactors or cracking of Methane and Ethane into Hydrogen inside a liquid metal reactor [1]. For example, in continuous casting argon gas is injected into the submerged entry nozzle (SEN) in order to reduce nozzle clogging and to withdraw impurities from the melt [2]. Important parameters are the filling degree of the SEN and the bubble distribution in the melt. Both issues have a strong influence on the flow structure in the mold and therefore on the quality of the produced steel. Due to the opaqueness and the high temperature of the liquid steel only a limited number of measurement systems are available. One of them is the Mutual Inductance Tomography (MIT) which is able to reconstruct the electrical conductivity distribution in one cross section of a pipe using Faraday's induction. It was already used to visualize the position of the liquid metal strand in the submerged entry nozzle in a real caster [3] and in a cold model [4,5].In case of the cold model experiments the gas distribution in a liquid metal two-phase flow in a pipe is reconstructed by solving a non-linear inverse problem for magnetic field data recorded by 8 sensors. In order to reduce the complexity of this system for detecting single bubbles in a pipe filled with liquid metal, we developed a new system consisting only of one sendingcoil and oneplanar gradiometer coil on the opposite sides of the pipe (see Fig. 1).In this paper we will briefly describe the working principle of the sensor and present preliminary measurements in the eutectic alloy GaInSn and liquid sodium.

Working principle

In contrast to the well investigated electromagnetic identificationof conducting objects in a non-conducting volume [6] the task to reconstructan argon bubble inside the liquid metal issimilar to the case of the detection of cracks using nondestructive eddy current testing [7].Instead of moving the sensor along the object containing the cracks, in our case the bubble detector is fixedon the pipe and the bubble moves through detector. The basic idea of this sensor is that the bubble in the liquid metal acts as an obstacle to the eddy currents generated by the magnetic field from the sending coil. The slight change of the current distribution can be detected outside the liquid metal by a magnetic field sensor. The challenge of this method is the reliable detection of these small changes which demand a sophisticated measurement system. Figure 1 shows a schematic sketch of the sensor. On the left hand side of the pipe the sending coil (dashed line) is located which generates the magnetic field for the measurement indicated by the field lines. On the opposite side, a planar gradiometer (dottedand solid lines) is mounted symmetricallywith respect to the transmitter coil. The planar gradiometer is composed of two coilswith opposite turning direction, markedwith solid and dotted lines. The advantage of this arrangement is that the planar gradiometer reject the magnetic field generated by the sending coils as well asby the eddy currents in the liquid metal. If there is no bubble in the measurement volume, the output signal of the planar gradiometer is zero, because the induced voltages in bothpartsare equal and compensate each other. Only if a bubble is entering the measurement volume, as depicted in Fig. 2(a), the eddy currents in the liquid melt are deformed, resulting in a non-symmetrical magnetic field distribution in the gradiometer which generates a voltage signal at the gradiometer. If the bubble moves from the bottom to the top, then the typical time-dependent voltage signal shows the typical shape as depicted in Fig. 2(b). From this signal the vertical position of the bubble can be deduced. One important feature of this setup is the zero crossing of the induced voltage exactly at that time, if the bubble passes the middle of the gradiometer, as marked by an arrow in Fig. 2(b). This allows for a robust automatic detection of bubbles without any calibration. Even the rising velocity of the bubbles can be inferred by taking the residence time of the bubble in the sensor into account.

Results for GaInSn

We set up a cylindrical vessel with an inner diameter of 40 mm filled with GaInSn which is liquid at room temperature. Figure 3 shows a photograph of the setup with the bubble sensor.Argon gas was injected intoGaInSn through a thin stainless steel tube with the inner diameter of about 1 mm which produced bubbles with a diameter of approximately 5 mm. In the first experiments we generated single bubbles and recorded the induced voltages in the gradiometer. The results of this measurement are seen in Figure 4. The typical shape of the signal for each bubble agrees very well with the simulation shown in Fig. 2(b). For this setup, we implemented a simple algorithm for the detection of bubbles. The automatically detected bubble is marked by a rectangle which gives the residence time of the bubbles in the sensor.

From that quantity the rising velocity could be detected.

S G

Fig. 4: Measured signal from argon bubbles rising in a GaInSn column with automatic bubble detection marked by the rectangles.

One interesting aspect of this measuring device is the maximum frequency of the bubbleswhich allows the detection of single bubbles. For this purpose we injected argon at different flow rates up to 750 sccm (standard cm^3 per minute). It turned out that this sensor is able to detect bubbles up to a flow rate of 47 sccmcorresponding about 1 bubble per second. For higher flow rates it is possible to uses the standard deviation as an indicator of the void fraction in the pipe.

Figure 5 shows the dependency of the flow rate and the standard deviation.

Results for sodium

In order to evaluate this new measurement system at higher temperatures we mounted the sensor on a vertical pipe at the NATAN sodium loop at the Helmholtz-Zentrum Dresden Rossendorf. Figure 6 shows a photograph of the sensor in our sodium laboratory.

Fig. 6: Setup of the sensor at the sodium loop Similar to the experiments with GaInSn we injected argon gas into the sodium below the sensor. In order to complicate the measurement we switched on the electromagnetic pump generating a mean velocity of 19 cm/s of the sodium. Figure 7 shows the results of the experiment. Even in such a noisy environment the characteristic signal of a bubble could be measured. However, the simple automatic bubble detection algorithm has to be improved to handle noisy data. Fig. 7: Measured signal from sodium pipe flow. Single bubbles could be automatically detected, which areindicated with the rectangles. Measurements were carried out at sodium velocity of 19 cm/s at 150°C.

Conclusions

In this paper, we have demonstrated the applicability of a simple contactless inductive sensor consisting of a sending coil and a planar gradiometer for detecting single bubbles in a pipe filled with liquid metal. In GaInSn as well as in sodium the single bubbles can clearly identified by the voltage measurements. Using a planar gradiometer allows for a robust automatic detection of bubbles without any calibration, because there is a zero crossing of the induced voltage exact at that time, when the bubble is in the middle of the gradiometer. We have implemented a simple algorithm for detecting the bubbles automatically. From this data even the rising velocity of the bubble can be estimated. For a more detailed validation of the new inductive method, accompanying ultrasound measurements are needed. Additionally, we will investigate if the horizontal position of the bubble could be inferredfrom measurements using this simple setup.

Fig. 1 :Fig. 2 :

 12 Fig.1: Principle of the measurement system of the system. The magnetic field lines show structure at the central plane.

Fig. 3 :

 3 Fig. 3: Photograph of the cylindrical vessel filled with GaInSn, the sending coil (S) and the gradiometer (G)

Fig. 5 :

 5 Fig. 5: Void fraction estimationin a pipe filled with GaInSn.

Acknowledgment

Financial support of this research by the German Helmholtz Association in the frame of the Helmholtz-Alliance LIMTECH is gratefully acknowledged.


INTRODUCTION

In order to predict with a high accuracy the springback in case of reversal bending, it is impossible to neglect the kinematic hardening. To identify this kinematic hardening from classical uniaxial tests, the specimen must be subjected to compression. Such a compressive test is impossible on thin sheets (thickness under 1mm) because of buckling. One solution is to perform a bending test and to identify the behaviour in compression by an inverse analysis. The simulations used for the inverse analysis are all the more accurate as the specimen is subjected to pure bending i.e. without tension and shear.

Pure bending is generally obtained by four-point bending. The maximum curvature that can be reached with this solution is so limited that maximum strain may be very limited for thin sheets. To overcome this problem, different original bending machines have been recently proposed in the literature [START_REF] Yoshida | Identification of material parameters in constitutive model for sheet metals from cyclic bending tests[END_REF][START_REF] Brunet | Nonlinear kinematic hardening identification for anisotropic sheet metals with bending-unbending tests -ASME[END_REF], but all these architectures induce some sort of friction in linkages and hence spurious forces normal and shear loads on the specimen. An original machine that enables one to perform pure bending tests on thin sheets has been developed within Brite-EURAM project FORMAS. The maximum value of the angle α (see Fig. 1) that can be reached with this machine is 90 o . For instance, this value corresponds to a value of 13% for the maximum strain in the case of a 1 mm thick and ABSTRACT: It is now commonly accepted that the so-called Baushinger effect has to be taken into account for accurate springback predictions during sheet metal forming simulations. Baushinger effect can be modelled through the introduction of a kinematic hardening law in the material model. The identification of such a behaviour requires to perform tests such that the material is subjected to reversal loadings. In the case of thin sheets, tension-compression tests are impossible because of buckling. Two alternative solutions can be found in the literature: shear tests and bending tests. Since bending tests are more representative of the behaviour of the material during many industrial metal forming processes, several authors (see for example [START_REF] Yoshida | Identification of material parameters in constitutive model for sheet metals from cyclic bending tests[END_REF][START_REF] Brunet | Nonlinear kinematic hardening identification for anisotropic sheet metals with bending-unbending tests -ASME[END_REF]) have recently proposed bending machines for such applications. This paper is devoted to the presentation of an original pure bending machine developed at LMT-Cachan (patent pending 2002 [START_REF] Calloch | European Patent pending[END_REF]). The machine architecture and kinematics is presented. It enables to bend specimens up to 90° (i.e. 13% strain at the surface of a 1 mm thick and 6 mm long specimen). The bending moment is simultaneously applied to two identical specimens so that no spurious "normal" and "shear" loads are applied to the specimens as experimentally verified through the measurement of moments in different points of the machine. Strain fields can be obtained directly from strain gauges glued on the specimens or from the motors angle and the compliance of the machine. The capabilities of this original pure bending machine are illustrated herein by tests performed on different materials including a 0.25 mm thick TRIP steel sheet. Key words: bending, testing machine, identification, metal sheet, kinematic hardening A pure bending machine to identify the mechanical behaviour of thin sheets G. Arnold 1,2 , S. Calloch 1 , D. Dureisseix3 , R. Billardon The inverse analysis that can be used to identify the constitutive laws from bending and tension tests has been applied to a 1 mm thick non-alloyed eutectoïd steel sheet [START_REF] Arnold | Identification of kinematic and isotropic hardenings using a pure bending machine[END_REF].

PRINCIPLE AND DESCRIPTION

The machine uses 2 identical specimens. The heads of each specimen are fixed with grips on arms that are attached to the shaft or the stator of a step motor.

Each aluminium arm consists in 2 torque transducers that are equipped with strain gauges. Combinations of the bending moments measured with the 8 transducers can be used to verify the pure bending state (see section 3.1).

The machine is angle-controlled by the two step motors. A program developed in Labview environment controls the step-motors and records the measurements. Figure 2 illustrates the machine just after the settingup of the specimens (α = 0°) and after symmetric bending of the 2 specimens up to α = 90°.

MEASUREMENTS AND VALIDATION

Loads evaluation

The bending moment applied to the specimens as well as the spurious loads can be derived from the strain gauges measurements made on the 8 torque transducers.

3.1.a Torque transducers

The bending moment at the centre of each torque transducer can be derived from the tension U measured as the output of an extensometry bridge connected to the 4 gauges so that (see Fig. 3):

U = K ε u1 -ε l1 + ε u2 -ε l2 [ ] (1)

3.1.b Measurement of spurious forces

It can be noticed from that expression that the transducers are not sensitive to tension and compression. However these measurements made at different points of the machine enable to evaluate the moment in the specimens as well as the spurious forces. Considering that the beam theory can be applied to the complete machine (motors and attached specimens) considered as a closed arc, the internal forces (generalised stresses) at current point P along the arc can be expressed as :

F int { } = O X r x + Y r y M o r z ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ with M P = M O + Xy P -Yx P (2)
r z being parallel to the axes of the motors. In practice, moment M can be measured at 8 different locations P i along the arc i.e. at the centre of each torque transducers (2 per arm, see Fig. 4). The values of X, Y and M 0 can be derived from the minimisation of an error function :

Min X,Y,M O ( ) G [ ] with G = M Pi (X, Y, M O ) -M Pi exp ( ) 2 i=1:8 ∑ ⎤ ⎦ ⎥ ⎥ (3)
Figure 5 illustrates the evolution of the spurious shear X and tension/compression Y forces during a cycle of amplitude +/-90°. The measured shear force X cannot be measured with accuracy for small bending angles where a singularity is observed. For higher angles, the value of shear force X, as well as tension-compression force Y, is less that 0.01 N. The corresponding stresses are negligible (less than 0.1 MPa) when compared to the tensioncompression stresses (a few 100 MPa) due to the applied bending moment (within the range +/-6 Nm for this test). These results validate that the specimens are in pure bending state.

Strain measurements

3.2.a From gauges

Strain gauges can be glued on each side (labelled + or -) of each specimen A and B. The mean value of the surface strain in each specimen can be calculated as:

ε surfA = ε A + -ε A - 2 ε surfB = ε B + -ε B - 2 (4)
In practice, the symmetry of the system allows to consider that the value of the surface strain is equal to the mean value:

ε surf = ε surfA + ε surfB 2 (5)

3.2.b From motor angle

Besides, if the specimens can be assumed in a pure bending state, their deformed state is a circle so that the surface strains can also be deduced from the measurement of the bending angle α by using the following expression :

ε surf = α specimen t 2L (6)
The value of angle α can be deduced from the angle applied by the motor according to a correction that takes into account the machine compliance S :

α specimen = α motor -S M (7)
Both procedures have been used during several experiments on different specimens. The responses of the gauges as well as the motor angle measurements are plotted on figure 6 for the first elastoplastic cycle of a bending test. The correlation between the values of the angle as deduced from the strain gauges measurements or from the motor angle is good until the rupture of the gauges. Some gauges gave good response for strains with a value up to about 10% but for a limited number of cycles. These good results first allowed for the calibration of the machine compliance. They also encourage the use of the motor angle measurements to derive the strains: indeed this approach is the only one that can be used for very short specimens on which it is impossible to glue strain gauges.

TEST RESULTS

Examples of test results are given for two different metal sheets in figures 7 and 8 where the moment is plotted vs the angle. The first material is a 1 mm thick eutectoïd steel sheet. The bending specimens are 6 mm long and 40 mm wide. The identification of its behaviour has been presented in a previous paper [START_REF] Arnold | Identification of kinematic and isotropic hardenings using a pure bending machine[END_REF]. The second material is a 0.25 mm thick TRIP steel sheet [START_REF] Scholz | [END_REF]. The bending specimens are only 2 mm long in order to reach 8% strain and 98 mm wide in order to measure torques that are in the range of the transducers.

CONCLUSION

An original pure bending machine has been built to perform pure bending tests on specimens cut in thin sheets that can be used to identify elasto-plastic models taking into account kinematic hardening. The architecture of the machine has enabled to measure the spurious loads in order to validate that the specimens are in a pure bending state.

Fig. 1 .

 1 Fig. 1. Pure bending.

Fig. 2a .

 2a Fig. 2a. Machine in initial configuration (0°).

Fig. 2b .

 2b Fig. 2b. Machine after bending to 90°.

Fig. 3 .

 3 Fig. 3. Torque transducers.

Fig. 4 .

 4 Fig. 4. Moments measured by torque transducers and moments in the specimens.

Fig. 5 .

 5 Fig. 5. Measurements of the spurious forces.

Fig. 6 .

 6 Fig. 6. Strains measurements during a cycle.

Fig. 7 .

 7 Fig. 7. Experimental bending responses for 3 cyclic tests of different amplitudes on a 1 mm thick eutectoïd steel sheet.

Fig. 8 .

 8 Fig. 8. Experimental bending response on a 0.25 mm thick unstable 301 stainless steel sheet for a cyclic test.

  1

LMGC (Université Montpellier 2/CNRS UMR 5508) CC 048, Place Eugene Bataillon, 34095 MONTPELLIER France URL: www.lmgc.univ-montp2.fr e-mail: dureisse@lmgc.univ-montp2.fr


J Barglik
email: jerzy.barglik@polsl.pl

A Smalcerz

A Smagór

Induction hardening of small gear wheels made of steel 50CrMo4

Keywords: surface hardening, induction heating, electromagnetic field, coupled problem

Analysis of coupled physical fields in the process of induction hardening of gear wheels is described. Modeling of the process is not easy because of complicated geometry of the system and also due to non-linear dependences of material properties on temperature. The computations are provided for gear wheels with diameter of 41 mm made from steel 50 CrMo4. The Flux 3D software is applied for computations of coupled electromagnetic and temperature fields in a direct coupled formulation. The single owned procedure is used for hardness calculation. Some examples concerning different temperature evolution and process parameters are calculated and compared with measurements. Quite reasonable accordance between computations and measurements is achieved.

Introduction

For induction hardening of gear wheels, a complete hardening is carried out in thin surface layer only. The transition zone and internal zones of gear wheels are characterized by incomplete hardening, tempering, remaining or the material soft respectively [1].In general, the process of induction hardening consists of two consecutive stages: induction heating and quenching. A break between heating and cooling should be as short as possible. As usual, the domain of a tooth consists of four zones A, B, C and D with different hardness and microstructure (Fig. 1). The exemplary time evolution of the temperature in particular zones is presented in Fig. 2. For the surface induction hardening of gear wheels the stage of heating is often divided into two steps differing by frequencies used. Of course, the process could be as well realized at a single frequency or simultaneous dual frequency using a single coil [2]. There are possible other patterns of heating including, for instance, three-step induction heating. For a classical dual frequency heat treatment, first a longer heating is performed at the medium frequency within a time period t = t MF . After a short technological break (t = t b1 ), high frequency induction heating is applied. Its time t HF is distinctly shorter than t MF . Then, after another short break (t = t b2 ), the gear wheel is cooled by spraying or by merging into a container with a suitable quenchant. The time of cooling t c is chosen based upon the Continuous Cooling Transformation (CCT) diagram of the investigated steel. The value of the hardness and obtained microstructure depend on the starting temperature of cooling and its velocity. For zone A, for example, cooling lines begin at a temperature bigger than A c3 and terminate at the temperature smaller than the martensitic temperature M s . If the speed of cooling is high enough, the zone A is characterized by a complete hardening with the martensitic microstructure. For the zone B the cooling lines begin from temperatures Tlocated between the lower critical temperature a A c1 and the austenization temperature A c3 (A c3 TA c1 ). It results in an incomplete hardening with lower hardness and microstructure containing not only martensite but also bainite, ferrite and pearlite. For the zone C the cooling lines begin from temperatures slightly lower than the A c1 temperature. The material in this zone is composed from the tempered martensite and its hardness is lower than in the zone B. And finally, for the zone D the cooling lines begin from the temperature distinctly smaller than the lower critical temperature A c1 . The material remains soft and the initial microstructure and hardness do not change. However, induction hardening is characterized by a high value of the heating rate  h , which could achieve 1000 -1500C/s. In order to secure the uniform austenite microstructure before quenching for such fast heating it is necessary to heat the gear wheel till a modified austenization temperature A c3m being bigger than the conventional austenization temperature A c3 . The real value of A c3m should be determined by measurements [3].

o c 3 m c 3 h ( ), 5 0 2 0 0 C A A T v T       . (1)
In practice, in order to secure the uniform austenite microstructure within the whole zone A, the hardening temperature T h should be at least 50Chigherthanthe modifiedaustenization temperature A c3m [3].

Description of the problem

The analyzed problem is the induction spin hardening of a small gear wheel made of steel 50 CrMo4. Its chemical composition is as follows: 0.46-0.54 % C, max 0.4% Si, 0.5-0.8 % Mn, 0.9-1.2 % Cr, 0.15-0. Two different arrangements of the inductor-sprayer system are analyzed:

-Dual frequency heating by MF inductor (Fig. 3) and HF inductor (Fig. 4) and water spraying.

-Single frequency heating by a MF inductor (Fig. 3) and water spraying, Due to the symmetry of the arrangement, it is sufficient to analyze just one half of the tooth (Fig. 5). A view of the gear wheel is shown in Fig. 6. The 3D computation of the task is realized by the finite element method, using the code FLUX3D [3 -6] supplemented with several own procedures [7 -9]. The mathematical model and a way of meshing are depicted in Figs. 78. The radial boundaries are characterized by the conditions of periodicity. The number of the nodes for the electromagnetic computations was over 90000, for thermal computations about 7000. Radiation is neglected [10]. The computation time of one example takes approximately 8 hours. The hardness distribution is measured after the full process of the heat treatment consisting of induction hardening and low temperature tempering (t t = 2h, T t = 160  C). There is a reason that for all points 1 -4 the calculated hardness is bigger than the measured one. More detailed description of computations, measurements and discussion of their results will be provided during the conference.

Conclusion

The paper presents the numerical analysis of single and dual frequently spin induction hardening of gear wheels made of steel 50 CrMo4. Electromagnetic and temperature field were analyzed as directly coupled. Dependences of material properties on temperature are taken into consideration. The modified austenization temperature depending on the speed of heating is determined by measurements. The Flux 3D software is applied for computations of coupled electromagnetic and temperature fields and the single owned procedures for the hardness determination. Quite reasonable accordance between computations and measurements is noticed.

Fig. 1 :

 1 Fig. 1: Zones of the tooth with different hardness andmicrostructure requirements: Acontour zone, Btransition zone, Croot of the tooth, Dbase zone.

Fig. 2 :

 2 Fig. 2: Time evolution of induction hardening for surface (A), transition (B) and internal parts (C, D): A c3austenization temperature, A c1lower critical temperature, M smartensite temperature, T hhardening temperature.

  3 % Mo, max. 0.025 % P, max. 0.035 % S. Other parameters and dimensions of the gear wheel are: diameter d = 41mm, width b = 13.8 mm, number of teeth n = 41, and the pressure angle  = 14.5 °.

Fig. 3 :Fig. 4 :

 34 Fig. 3: MF inductor for single frequency heating: 1-coil, 2-sprayer, 3-bus-bars. Fig. 4: HF inductor: 1-coil, 2-bus-bars, 3-flux concentrator.

Fig. 5 : 6 :

 56 Fig. 5: Geometry of the gear wheel made of steel 50 CrMo4. Fig. 6: View of wheel gear.

Fig. 7 :Fig. 8 :Fig. 9 :Fig. 10 :

 78910 Fig. 7: Mathematical model.

Fig. 11 :

 11 Fig. 11: Case II. Temperature distribution before HF heating: I MF = 1750 A, f = 10 kHz, t MF = 10 s, t b1 = 0.5 s.

Fig. 12 :Fig. 15 :

 1215 Fig. 12: Case II. Temperature distribution before cooling: I HF = 1200 A, f = 200 kHz, t HF = 1 s, t b2 = 0.1 s.

Acknowledgment

Financial support from National Centre for Research and Development (project PBS2/A5/41/2014) is gratefully acknowledged.