S Czarske

Eckert

V Galindo
email: v.galindo@hzdr.de

R Nauber

S Franke

D Räbiger

J Czarske

S Eckert

RMF-driven

Keywords: Liquid metals, rotating magnetic field, spin-up Experimental setup

published or not. The documents may come

The experimental setup consists of the eutectic metal alloy GaInSn, which is liquid at room temperature, enclosed in a cubic container with an edge length of 2L = 67.5 mm. The container is centered in the MULTIMAG (MULTI purpose MAGnetic field) facility, which is capable of generating different types of magnetic fields with varying strength and frequency [3]. For the given experiment a horizontal RMF in clockwise direction with a frequency of 50 Hz is applied to the fluid at the time t = 0. Due to the action of the Lorentz forces, a vortex develops in the horizontal plane (primary flow) as well as a secondary flow in the meridional plane. Both planes are instrumented simultaneously with the ultrasound array Doppler velocimeter (UADV) as shown in Fig. 1. In the given configuration, the UADV employs four linear arrays. An array consists of 25 transducers with the dimensions 2.5 × 5 mm 2 resulting in a total sensitive length of 67.5 mm. A pairwise driving of neighboring transducers results in an active surface of 5 × 5 mm 2 , which leads to a sound beam width of approximately 3 mm in GaInSn. The excitation signal is eight periods of a sine wave at f = 8 MHz resulting in an axial resolution of about 1.4 mm [1]. In order to acquire a planar velocity map with one velocity component a electronic scanning of the linear sensor arrays is performed. The frame-rate is increased over a simple sequential scan by using a parallelized time division multiplexing (TDM) scheme [1]. In the given experiment, two sensor arrays span a centric horizontal plane that captures the primary flow, as depicted in Fig. 1. To avoid crosstalk between the sensor arrays in this configuration, all four arrays are driven consecutively.

RMF coils

Ultrasound array sensors

B RM F container
The velocity information is extracted from the amplified and digitized US echo signals via the Kasai autocorrelation method. A typical mean bandwidth after digitalization is 1.2 GB/s, which is beyond the limit that can be acquired and stored continuously with common PC-hardware. Therefore a real-time data compression is performed by offloading parts of the signal processing to a field-programmable gate array (FPGA, NI PXIe-7965R). The preprocessing reduces amount of data by 10:1 and enables continuous streaming for a practically unlimited duration. This allows investigating flows with fast transients with nondeterministic onset, as predicted by the numerical simulation in the next section.

Governing equations

Let us consider the flow of an electrically conducting fluid with kinematic viscosity ν, density ρ and electrical conductivity σ in a cubic container with an edge length 2L driven by a uniform magnetic field of induction B 0 rotating in a horizontal plane (around the z -axis) with a constant angular frequency ω.

In the scope of the low-frequency and low-induction approximation (very small shielding parameter R ω = µ 0 σωL 2 << 1 and very small magnetic Reynolds number R m = µ 0 σuL << 1, respectively) a quasi-analytical expression for the time-averaged electromagnetic force density acting on the liquid metal in a square cavity can be given. Here is µ 0 the magnetic vacuum permeability) and u is a characteristic velocity.

The clockwise rotating magnetic field can be expressed as:

B = B 0 {cos(ωt)e x -sin(ωt)e y } (1)
The corresponding magnetic potential A = B 0 (y cos(ωt) + x sin(ωt)) e z has only an axial component. Within the considered approximation the induced electric current can be calculated using the Ohm's law and the Maxwell equations:

j = σE = σ{-∇Φ - ∂A ∂t } = σ {-∇Φ -B 0 ω (-y sin(ωt) + x cos(ωt)) e z } (2)
We compute the instantaneous electromagnetic force density j × B taking into account the expressions 1 and 2. Until this point equations show the same form as in [4]. Now we can obtain a semi-analytical expression for the electromagnetic force density using a proper ansatz for the electric potential Φ: Φ(x, y, z, t) = ωB 0 (a(x, y, z)cos(ωt) + b(x, y, z)sin(ωt))

The functions a(x, y, z) and b(x, y, z) have the dimensions m 2 . The Kirchhoff's law for the electric charge conservation ∇ • j = 0 leads to the equation for the electric potential ∇ 2 Φ = 0. Taking into account the ansatz 3, we obtain following system of equations:

∇ 2 a = 0 and ∇ 2 b = 0 (4)
which should be solve under the boundary condition that no electrical current flows through isolating walls (j n = 0). It means for example for the functions a:

∂a ∂z | z=±H = -x and ∂a ∂x | x=±L = ∂a ∂y | y=±L = 0 (5)
We perform now the time average of the force density over a period T = 2π/ω and take into account that 1/T T 0 dt sin 2 (ωt) = 1/T T 0 dt cos 2 (ωt) = 1/2 and obtain following final expression for the time averaged electromagnetic force density:

f = σωB 2 0 2 (- ∂b ∂z + y) e x + (- ∂a ∂z -x) e y (6)
Electromagnetic driven flow in a square container

The flow in a closed cube with L = 33.75 mm containing the liquid metal GaInSn at 20 o C driven by a rotating magnetic field was simulated numerically and results were compared with measurements in the experiments. We use following material properties for the numerical simulation: density ρ = 6403 kg/m 3 , viscosity ν = 0.34 × 10 -6 m 2 /s and electrical conductivity σ = 3.29 × 10 6 S/m. The numerical simulation of the liquid metal flow is performed using the code library OpenFOAM c . The flow was computed solving the incompressible Navier Stokes equation which in a dimensionless form, with L, L 2 /ν and ρ(ν/L) 2 being the distance, time and pressure scale, respectively, is given by

∂u ∂t + (u • ∇)u = -∇p + ∇ 2 u + T a f EM (7)
together with incompressibility condition ∇ • u = 0.

The last term at the right side is the non-dimensional electromagnetic force density

f EM = {(- ∂b ∂z + y) e x + (- ∂a ∂z -x) e y } ( 8
)
where T a = σωB 2 0 L 4 2ρν 2 denotes the magnetic Taylor number. The functions a and b are solutions of the equations 4 under the boundary conditions 5.

Fig. 2 shows the spatial distribution of time-averaged non-dimensional electromagnetic force density f EM , which was added as an additional body force to the Navier Stokes equation. A computational grid with at least one million volume elements was used depending on the boundary layer thickness. A second-order discretization scheme was used for the convective term in Eq. 7.

In order to estimate the critical Taylor number T a c for the transition from steady to oscillatory or unsteady flow regimes, we observe the time evolution of the velocity field. Fig. 3 shows the time evolution of the velocity component v x at the monitoring point x = 0.9, y = 0, z = 0.1 for different values of the magnetic Taylor number T a. The transition to a time-dependent flow regime appears to occur for a value of the magnetic Taylor number in the range 1.3 × 10 5 < T a c < 1.4 × 10 5 . The corresponding critical value for the case of a finite circular cylinder of aspect ratio 1 is T a c = 1.232 × 10 5 (Grants et al. [7]). Fig. 4 shows the time evolution of the kinetic energy of the primary flow

< u 2 x + u 2 y > | z=0 = 1 N x N y Nx,Ny i,j u x 2 ij + u y 2 ij (9)
in the middle horizontal plane z = 0. The red colored solid line shows values computed from the numerical simulation at the same positions as measured in the experiments for T a = 2 × 10 5 .

The secondary flow reaches the maximal values near the top and the bottom of the container. Due to saturation effects, the measurement system yields no valid velocity measurements near the walls and therefore the measurement volume does not include these regions. This leads to a systematically lower value of the mean kinetic energy obtained from measurement compared to the simulation. Fig. 5 shows a comparison of the flow pattern in the meridional plane x = 0 for T a = 2 × 10 5 . In the left part, the values coming from the numerical simulation are shown. We can observe, that the maximal velocity magnitudes are found outside of the measurement volume of the ultrasound instrumentation.

Conclusions

We present a semi-analytical expression for the induced electromagnetic force density in a electrically conducting medium contained in a square cavity in the presence of a rotating magnetic field. This allows a direct numerical simulation of the flow. We found the transition from the steady state to time dependent flow structures to occurs for T a > 1.3 × 10 5 . Velocity distributions in two perpendicular planes were measured using a two-component ultrasound array Doppler velocimeter for a large time interval for the first time. The mean flow structures and the evolution of the flow in time in experiments and numerical simulation are in very good agreement. More details concerning the instability structures will be presented in a subsequent paper.

Fig. 1 :

 1 Fig. 1: Measurement configuration: cube instrumented with four ultrasound array sensors

Fig. 2 :

 2 Fig. 2: Vector plot (left) and amplitude (right) distributions of the non-dimensional electromagnetic force density f EMBoundary conditions for the calculation of the flow field have been the no-slip condition u = 0 at the solid container walls. For the upper surface either u = 0 or the conditions for a stress-free, non-deformable surface u n = 0 and ∂u t /∂n = 0 were used depending on whether the melt flow was evaluated in an open or in an enclosed container.A computational grid with at least one million volume elements was used depending on the boundary layer thickness. A second-order discretization scheme was used for the convective term in Eq. 7.In order to estimate the critical Taylor number T a c for the transition from steady to oscillatory or unsteady flow regimes, we observe the time evolution of the velocity field. Fig.3shows the time evolution of the velocity component v x at the monitoring point x = 0.9, y = 0, z = 0.1 for different values of the magnetic Taylor number T a.

Fig. 3 :

 3 Fig. 3: Time evolution of the velocity component u x at the monitoring point x = 0.9, y = 0, z = 0.1.

Fig. 4 :

 4 Fig. 4: Spin-up of the kinetic energy of the primary flow at the middle horizontal plane z = 0 for T a = 2 × 10 5

Fig. 5 :

 5 Fig. 5: Flow pattern in the meridional plane x = 0 for T a = 2 × 10 5 . Both pictures show the mean values of the velocities |u y e y + u z e z | (left: numerical simulation, right: measurement).

Acknowledgment

Financial support for this research from the German Helmholtz Association in frame of the Alliance "Liquid Metal Technologies (LIMTECH)" is gratefully acknowledged.


E Takeuchi
email: takeuchi@mat.eng.osaka-u.ac.jp

H Konishi

H Kawabata

H Ono

Electrocapillary Phenomena in Material Processing

Keywords: Interfacial tension, Electrocapillarity, Emulsification, Refining, Continuous casting ■

Formation mechanism of emulsification is being studied by using "electrocapillarity" which had been once developed to study the structure of "electrical-double layer". First the characteristics of electrocapillary curve of KCl aqueous solution/Hg system used for the cold model have been investigated to understand the ionic structure near the interface in comparison with the case of real (liquid CaO-SiO 2 -Al 2 O 3 /liquid steel) system. Then the emulsification phenomena at the meniscus(the interface) with and without imposing the electric potential have been observed by using a high speed camera. Finally the mechanisms of newly found phenomena have been discussed by taking account of the theory of electrocaoillarity.

Introduction

Interfacial phenomena between the ionic liquid(e.g. molten slag and molten flux) and the liquid metal(e.g. molten steel) are closely related both to the refining efficiency and to the cast qualities through the emulsification between two phases at the meniscus. Because the direct observation of the interfacial phenomena of slag/steel system is difficult, various simulation experiments using (1)the water, oil or the other transparent liquid for slag and (2)the mercury or other liquid metal for steel [1,2] have been conventionally performed to propose hypothesis of the emulsification mechanism. However the different properties of the materials for the cold model experiment often gives rise to the erroneous understanding for these phenomena. Applying different materials for cold model also does not allow the quantitative explanation for the effect of the specific property, such as interfacial tension or viscosity, on the meniscus phenomena. This study is first undertaken for more profound comprehension on the meniscus disturbance inducing emulsification by using the electrocapillary method mainly from the view point of interfacial tension. As the first part, the fundamental data, a relationship between the interfacial tension and the imposing electric potential, has been investigated. Then the change of interfacial phenomena by imposing the electric potential has been examined by using a high speed camera to discuss the mechanism of emulsification.

Electrocapillarity of Electrolyte Solution/Liquid Metal System

Review of Conventional Studies Fig. 1 shows the effect of electric potential(φ) on the interfacial tension(δ) between slag and liquid iron reported by Esin et al [3]. It shows that δ decreases monotonically with increasing φ. This tendency is explained by the simple models, such as Helmholtz model and Gouy-Chapman model [START_REF] Sotojima | Electrochemistry[END_REF] shown in Fig. 2. According to these models the cation layer on the interface electrically attracted by imposing negative potential to metal decreases the interfacial tension due to the repulsion of the same polarity of ions(cations) in the layer. On the other hand Patrov reported the different results [START_REF] Patrov | [END_REF] as shown in Fig. 3, in which the interfacial tension has a peak near 0v and decreases with increasing the electric potential both by negative and positive imposition. Also it is reported that the chemical composition of slag affects the interfacial tension.

Electrocapillarity of KCl Aqueous Solution/Hg system Direct observation of the interfacial phenomena at KCl aqueous solution/Hg by using electrocapillarity has been conducted as described in the next chapter. In order to obtain the quantitative data for the electrocapillary characteristics of this system, dependence of interfacial tension on the electric potential has been investigated by using the apparatus as in Fig. 5. A droplet of Hg(≈1g) was placed on the horizontal glass plate in the pool of KCl aqueous solution filling the transparent container. Both phases were electrically connected to the potentiostat by fine Pt wires insulated by alumina coating. The imposed electric potentials were in the range of -3V to +3V, and the concentrations of KCl aqueous solution(C KCl ) were 0, 0.5, 1.0, 1.5mol/ℓ. The effect of imposed electric potential in different concentrations of KCl aqueous solution on the interfacial tension is summarized in Fig. 8. In the case of pure water(without KCl) the electric potential has no effect on the interfacial tension, while the electric capillary curves appear in case of KCl electrolyte aqueous solution. The curve has a peak named "ECM; Electrocapillary Maximum" and the electric potential which gives ECM is named "Electric potential for ECM; E ecm " The value of ECM seems to be the same when C KCl is below 1mol/ℓ, while it decreases if C KCl is more than 1mol/ℓ. These results also could be explained by the Grahame's model [6] shown in Fig. 4; (i)even in case of no imposition of the electric potential, the specific adsorption layer of anion(Cl -) decreases the interfacial tension, (ii) increasing the electric potential to E ecm increases the interfacial tension by attracting cation(K + ) outside the specific adsorption layer which neutralizes the effect of cation to increases the interfacial tension up to ECM. Then, (iii)when the electric potential exceeds the E ECM , the accumulated population of cation(K + ) near the interface increases the repulsion force to decrease the interfacial tension again. These tendencies are similar to those of the iron/slag system reported by Patrov. Results and Discussions Fig. 10 shows the Hg particles formed on the meniscus of Hg pool by injecting Ar gas at the flow rate of 0.6ℓ /min. They are generated by Ar bubbles coming out to the meniscus. The size of particles distributes from 0.5mm to several mm in diameter. The particles are scarcely combined with the Hg pool in case of no imposition of the electric potential.

Effect of Interfacial Tension on

On the other hand they disappear(combined with Hg pool) at instance just after imposing the electric potential of 2.0v as shown in Fig. 11.

It is also noted that the small local convections have been observed at the imposition of electric potential.

Conclusions

The characteristics of electrocapillary curve of KCl solution/Hg system has been investigated to understand the ionic structure near interface in comparison with the case of liquid CaO-SiO 2 -Al 2 O 3 /liquid steel system. The mechanism of behavior of Hg particles at the meniscus has been then investigated with "electrocapillary method" by using a potentiostat and a high seed camera. It is proposed that the specific adsorption layer prevents the involvement of particle formed by Ar injection into the meniscus, while the electric potential decreases the interfacial tension in a non-uniform manner to cause a micro-local marangoni convection to take these particles into the lower phase(Hg) through the meniscus.

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :

 1234 Fig. 1: Electrocapillary curve of Fe-C alloy in molten slag (1; 2.5%C, 2: 3%C, 3:4%C-Fe) Fig. 2: Two models for the electrical double layer structure

Fig. 5 :Fig. 6 :

 56 Fig. 5: Experimental apparatus for measurement of Hg droplet shape(interfacial tension) in KCl aqueous solution. Fig. 6: Effect of KCl concentration on the interfacial tension between KCl aqueous solution and Hg.

Fig. 6

 6 Fig.6 represents that the interfacial tension decreases linearly with increasing KCl concentration. Fig.7 shows the shape of Hg droplet; (a) φ=0.8v, (b) φ=1.8v. The droplet shape becomes flatter by increasing the imposing electric potential.

Fig. 8 :

 8 Fig. 8: Effect of imposed electric potential and KCl concentration on the interfacial tension between KCl aqueous solution and Hg droplet.

Fig. 9 :

 9 Fig. 9: Experimental apparatus for the observation of interface between KCl aqueous solution and Hg.

Fig. 10 :

 10 Fig.10: Accumulation of Hg particles generated by gas injection(no imposition of electric potential) .

Fig. 11 :

 11 Fig.11: Extinction of Hg particles just after the imposition of electric potential.

Fig. 12 :

 12 Fig. 12: Proposed mechanism of behaviour of Hg particles on the meniscus without and with imposing the electric potential.