MAGNETOHYDRODYNAMICS Vol. 53 (2017), No. 4, pp

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1 MAGNETOHYDRODYNAMICS Vol. 53 (2017), No. 4, pp NUMERICAL STUDY OF THE INTERACTION BETWEEN A BUBBLE RISING IN A COLUMN OF CONDUCTING LIQUID AND A PERMANENT MAGNET N.Tran 1, T.Boeck 2, U.Lüdtke 1, Z.Lyu 2, C.Karcher 2 1 Institut für Elektrische Energie- und Steuerungstechnik, Technische Universität Ilmenau, Ilmenau, Germany 2 Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Ilmenau, Germany ninh.tran-thi-hang@tu-ilmenau.de Electromagnetic induction in a conducting liquid that moves in an external magnetic field can be used for contactless flow measurement. In Lorentz Force Velocimetry (LFV), the induced force on the magnet is determined to obtain velocity information. This measurement principle may also be applied to conducting flows with gas bubbles encountered in metallurgical processes. This provides the motivation for our work, in which we study a single bubble rising in a liquid metal column as a model problem for LFV in two-phase flows. By using a small permanent magnet, one can not only detect the presence of a bubble but also obtain information on its position and velocity. Our numerical investigation aims at reproducing experiments with Argon bubbles in GaInSn alloy and at studying the electromagnetic induction in the flow in more detail. For three-dimensional and phase-resolving simulations we use the Volume of Fluid method provided by ANSYS FLUENT. The induction equation in the quasistatic limit is an elliptic problem for the electric potential. It is implemented in FLUENT with a user-defined scalar. The electric conductivity varies between the phases, and the magnetic field is given by an analytical expression for a uniformly magnetized cube. The comparison with the experiments also helps to validate the numerical simulations. Introduction. Multiphase flows occur in a range of processing technologies, such as cavitating pumps and turbines, electro-photographic processes, and transport of granular materials, such as grain or ore [1]. Multiphase flows are also common in the metallurgical industry. For example, bubbly flows are injected into the molten metal in order to stir, refine the melt and to homogenize the physical and chemical properties of the alloy[2]. The quality of alloys and the efficiency of those processes depend, among other things, on the quantity of individual molten metal components acting in the production process. For process control it may be advantageous to measure the flow rate of the liquid metal at several locations. Since liquid metals are often opaque, hot and aggressive, the traditional flow measurement techniques cannot be applied, especially when there is a mechanical contact between the flow and the measurement device [3]. To overcome these challenges, the measurement techniques involving electromagnetic induction have been greatly improved. Among a variety of electromagnetic flow measurement techniques, the force flowmeter helps to overcome the mechanical contact between the measurement system and the fluid flow also in high-temperature conditions [4]. There are different force flowmeter developments, such as the single-magnet rotary flow meter [5, 6]. Another type of contactless flow measurement technique is Lorentz force velocimetry (LFV) [7]. LFV measures the reaction force in an externally arranged magnet system due to the induction between the electrically conducting liquid and the magnetic field. LFV with large and complex magnet systems whose fields penetrate deeply into the flow and produce strong forces are able to measure the volume flux [8]. However, many metallurgical processes require the detailed 619

2 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher flow behavior at certain locations to optimize the quality of materials and other crucial phenomena, such as heat transfer, passive scalar transport and thermal treatment [9]. Therefore, we examine local Lorentz force velocimetry (LLFV), in which a small magnet system with highly localized magnetic fields is applied. In the present work, we focus on a rising bubble as a simple model problem for LLFV in gas liquid flows. It has already been studied experimentally with a small permanent magnet placed next to a vertical tube filled with the GaInSn alloy in which an Argon bubble rises. In this work we briefly describe how the ANSYS FLUENT solver can be adapted to solve this problem by using the phaseresolving Volume of Fluid method in combination with our own separate module for the induction equation. We then validate the code with literature results for rising bubbles without magnetic field. Finally, we examine the full problem including the magnetic field and obtain a favorable comparison with the bubble LLFV experiments. 1. Mathematical model and numerical method Governing equations. Magnetohydrodynamics(MHD) deals with electrically conducting flows which interact with magnetic fields. The governing equations describing such phenomena are the combination of Maxwells equations of electromagnetism and the Navier Stokes equations of mass and momentum conservation. All the equations are explained in textbooks on MHD, e.g., [10, 11]. For an electrically conducting fluid moving in a magnetic field B with a velocity field u, the eddy current density j that is generated in the fluid is governed by Ohm s law j = σ(e+u B), (1) where σ is the electrical conductivity of the fluid and E is the electric field. In most industrial applications, the magnetic field induced by the flow is negligible if compared with the imposed field, since the magnetic Reynolds number [10] Rm = µ 0 σu 0 L (2) is small. In this definition, u 0 is a characteristic velocity and L is a characteristic length of the flow. According to Faraday s law, the electric field is then curl-free. Therefore, the electric field can be represented through an electric potential φ as E = φ. (3) This simplification is also called the quasistatic approximation. In addition, the bulk of the conducting liquid remains charge-free due to the high conductivity, i.e. j = 0. (4) The final version of Ohm s law for MHD in the quasistatic approximation becomes j = σ( φ+u B). (5) From the charge conservation (Eq. (4)), the governing Poisson equation for the electric potential can then be derived by taking the divergence of Eq. (5): (σ φ) = (σu B). (6) The Lorentz force density is computed by taking the cross product of the current density j and the external magnetic field B: 620 f L = j B. (7)

3 Numerical study of the interaction between a bubble rising in a column of... B (a) z z B x u u B x B, [T] y x (b) (c) Fig. 1. Schematic of the problem setup: (a) side view, (b) top view. The mean velocity u of the container points in the positive z-direction, the magnetization direction of the permanent magnet is along the positive x-axis. (c) The magnetic field distribution in the domain. This force is added as an additional body force to the Navier Stokes equation ( ) u ρ t +(u )u = p+µ 2 u+ρg+j B, (8) where ρ is the fluid density and µ is the dynamic viscosity. The vector g denotes the acceleration due to gravity with g 9.81m/s 2. The continuity equation which determines the pressure field is u = 0. (9) Eqs. (6) and (8) are supplemented with the following boundary conditions: 1. no-slip boundary conditions on the walls: u = 0, 2. electrically insulating and non-magnetic walls: j n = Geometry and material properties. The flow is confined to a rectangular geometry shown in Fig. 1a. The origin of the Cartesian coordinate system is at the lower left corner in Fig. 1b that represents the bottom wall of the container. The dimensions of the container are H x H y H z = mm 3 ). The z-direction is antiparallel to the direction of gravity. The magnet is a cube with equal sides of length D = 12mm. Its edges are aligned with the coordinate axes and its center is located at the position C x, C y, C z = 10mm, 13.8mm, 90mm. It is homogeneously magnetized along the x-direction. The magnetic field is given by an analytical expression that can be found in section of the book by Furlani [12]. Fig. 1c shows an inhomogeneously distributed magnetization that induces a maximum magnetic field induction 621

4 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher Table 1. Physical properties of GaInSn and Argon at a temperature of 20 o C and atmospheric pressure. Material properties GaInSn Argon Density ρ, [kg/m 3 ] Dynamic viscosity µ, [kg/(m s)] Electrical conductivity σ, [S/m] Surface tension γ, [N/m] in the vincinity of the magnet s surface. The magnetic induction at the center of the magnet surface in the (y,z)-plane is B x = 0.58T. In our simulations with single rising bubbles, both the liquid GaInSn phase and the gaseous Argon phase are assumed incompressible. In the initial state, both phases are quiescent and the bubble has a spherical shape with the given diameter. The relevant material properties of the liquid GaInSn phase and gaseous Argon phase are listed in Table Numerical method. The time-dependent liquid gas flow is computed with the pressure-based solver and VOF method provided by the commercial software ANSYS FLUENT 17.0 [13]. FLUENT is based on a finite-volume discretization. The VOF method is used because it is well suited for a precise representation of the free surface. The volumetric data is used to store the interface location and the conservation of volume is guaranteed to ensure that simulations are physically accurate. With the VOF method, the two phases are spatially resolved and treated as a single incompressible fluid with variable properties. The volume fraction of the liquid phase C in a grid cell characterizes the position of the interface and the spatial distribution of the two phases. The phase continuity is solved through the explicit time discretization and the geometric reconstruction scheme is applied to track the interface. The local properties in the grid cell are represented as a C- weighted arithmetic mean of the properties of the individual phases. The surface tension is recast as a body force term located on the grid cells with the interface. FLUENT does not provide a ready-to-use implementation of the induction problem given by Eq. (5), Eq. (6) and Eq. (7). Therefore, to compute the electric potential, eddy currents and Lorentz force density, we implement user-defined functions. The elliptic Eq. (6) for the electric potential is implemented as a transport equation α ρφ t + ( F i φ Γ φ ) = S φ (10) x i x i of a User-Defined-Scalar (UDS) with the coefficients α and F i equal to zero. One can define a UDS transport equation by setting the parameters for these four terms: unsteady, convection, diffusion and sources, where Γ and S φ are, respectively, the diffusion coefficient and the source term for the scalar φ. Γ is defined as a tensor in the case of anisotropic diffusivity. Eq. (10) is solved after each time step for the velocity field. In the simulations, we use numerical grids composed of cubical cells. This is advantageous for the VOF method. A bubble is resolved by at least 20 cells per diameter. The grid sizes exceed 10 7 cells. For numerical convenience, the conductivity of the Argon gas is artificially increased to a value that is 10 3 times lower than the conductivity of the liquid metal. Nonetheless, we have observed that for grid cells closest to the conductivity jump, the current density computation becomes incorrect when it is based on the built-in gradient approximation and local conductivity values. Due to our choice of the structured uniform grid, these 622

5 Numerical study of the interaction between a bubble rising in a column of... Fig. 2. Numerical algorithm to calculate the Lorentz force. particular cells could be treated separately using a local reconstruction of the discontinuous potential gradient from the potential and conductivity values in the neighboring cells. A flow chart of the augmented computational procedure is provided in Fig. 2. The numerical scheme is explained briefly here: 1. The first step is to initialize the solution set up at t =0. The magnetic field was calculated and the user-define-memory values were defined and allocated in the graphical user-interface. The fraction vectors were initialized to track the interfaces between the Argon and GaInSn phases. 2. We set t n+1 = t n + t and the Computational Fluid Dynamic (CFD) calculation loop was solved. 3. The electric potential method is executed at the end of the time step in a transient run and improves the accuracy of the eddy current density at the interface. The force acting on the bubble is determined using the information from all loops. Initial tests of the code are reported in [14]. 2. Results. We now present the results for two different configurations. First, we compare our simulation results with experiments and other simulations for a single Argon bubble rising in GaInSn without a magnetic field. After that, we analyze the velocity field, eddy current and Lorentz force distribution of the bubble interacting with a permanent magnet Single Argon bubble in GaInSn without magnetic field. For validation, we used the results obtained in [15 17] by Schwarz, Zhang et al. and Ni et al. We performed simulations in a smaller domain with H x H y H z = mm in order to reduce the computational cost. The domain was still sufficiently tall for the bubble to reach terminal velocity. At the initial time t = 0, the bubble has a spherical shape with the diameter d 0 and is located on the central vertical 623

6 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher Re Experiment, Zhang (2005) Simulation IBM, Schwarz(2013) present t Fig. 3. The bubble Reynolds number as a function of the non-dimensional time. Comparison with numerical and experimental results [15, 16] without a magnetic field. axis a short distance away from the bottom. The initial coordinates of the bubble center correspond to x,y = 13.8,13.8mm with z a few mm above the bottom wall. The velocity of the bubble u was determined for the centroid of the bubble. At t = 0, it is zero. The time step was set to 10 4 s or less. For a first comparison, we used a bubble diameter d 0 = 4.6mm. Our uniform grid had 24 cells across the bubble diameter, i.e. the grid step was δx = d 0 /24 and the total number of cells was For this case, there were time series for the evolution of the instantaneous bubble velocity from a simulation by Schwarz [15] using an immersed boundary method (IBM) and from an experiment by Zhang et al. [16]. For comparison, we defined the bubble Reynolds number Re = ud 0ρ µ, (11) with its instantaneous velocity u. Time is made non-dimensional in units of t ref = d0 /g. Fig. 3 showsthe results. The black line showsthat the bubble rise velocity starts to oscillate after the initial acceleration period before it saturates. This is due to the shape oscillations of the bubble, which does not remain spherical. The final value of Re is comparable to the other studies, which exhibit much stronger oscillations. This could be attributed to different ways of obtaining the bubble velocity. The Reynolds number is fairly large due to the small kinematic viscosity. In another simulation, we determined the terminal rising velocity for a larger bubble with d 0 = 7mm. Fig. 4 shows the predicted rise velocity of 4.6mm and 7 mm bubbles as well as the measured terminal rising velocities [16], other numerical predictions [15, 17]. The line in Fig. 4 is the correlation 2γ u t = +2gd (12) ρd suggested by Mendelson [18], who noted a similarity between the velocities of inviscid surface waves and rising bubbles. Our numerical results as well as other simulations and experiments are in good agreement with the Mendelson formula(12). The deviation between the simulation and the experimental data is typically between 10 13%. The measured terminal 624

7 Numerical study of the interaction between a bubble rising in a column of Terminal rising velocity, [m/s] Mendelson (2005), theoretical Zhang et al. (2005), UDV Zhang&Ni (2014), VOF present Bubble diameter, [m] Fig. 4. Terminal velocity of a bubble as a function of the bubble diameter without a magnetic field and comparison to previous work [16 18]. velocities in the UDV (Ultrasonic Doppler Velocimetry measuring technique is a method which enables the measurement of velocity components simultaneously along a line) experiments are always lower than those theoretically predicted, which may be due to the impurities in GaInSn that lead to a lower surface tension Single Argon bubble in GaInSn with a permanent magnet. For this case, experiments with an LFV setup have been performed. They are described in more detail in [19]. The position of the magnet and its parametersare as described in sec Bubbles were released from different positions indicated in Fig. 1b and the Lorentz force signal on the magnet was measured. No velocity information was available from the experiment. To compare the experimental and simulation results, we first analyze the measurements for several bubbles with d 0 = 7mm released from different positions in Fig. 5. The vertical Lorentz force component increases with time as the bubble approaches the region with a significant magnetic field and generates a Fz, [µn] Time, [s] Fig. 5. Measurements of the z-component of the Lorentz force as a function of the time for a bubble rising at different positions. 625

8 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher significant velocity there. It decays after the bubble passes through that region. Although the conditions were identical, the measured Lorentz forces varied significantly between the different realizations. This is likely due to the unsteady bubble wake at large Reynolds numbers, which is associated with chaotic or turbulent dynamics. From these results it is clear that the integral Lorentz force signals from the simulation and experiment can only agree in a statistical sense. Nonetheless, the forces agree reasonably well for the initial time period, where the force component turns negative in Fig. 5. It seems plausible that this is due to the more organized flow structure near the front of the bubble. For the simulations, the domain size was increased to 220mm in the z- direction and the bubble had d 0 = 7mm. The grid was uniform with 32 cells across the bubble diameter, i.e. the grid step was δx = d 0 /32. The total number of grid cells was The initial position of the 7mm bubble was varied according to the different positions in Fig. 1b. Since the positions pos4, pos5, pos6 were symmetrical to the positions pos7, pos8, pos9 when the magnet was in mid-position with respect to the front wall, we only examined pos1 pos6. Fig. 6 shows the simulation results for the total Lorentz force on the magnet as a function of the time for the initial bubble positions pos1 pos6. The Lorentz force magnitude varies from 80µN to +80µN, which agrees reasonably well with the measurements in Fig. 5. The influence of the bubble position is also apparent. Proximity to the front wall in pos1 and pos4 seems to lead to a faster force signal decrease, whereas a larger distance from the magnet tends to reduce the force amplitude, at least on the central line connecting pos1 and pos3. The pos5 differs from the other cases, since the force has a strong negative peak followed by a positive peak. This seems to be related to the presence of the lateral wall. In pos6, the force was much weaker, since the bubble then had the largest distance from the magnet. Although the force magnitude agrees reasonably well with the experiments and simulations, the time scales in Figs. 5 and 6 are significantly different. A plausible explanation for this discrepancy is the time resolution of the force measurement system. This resolution is limited by the sensor intertia and by the low-pass filtering of the signals required to eliminate vibration noise [20, 21] Fz, [µn] pos1 pos2 pos3 pos4 pos5 pos Time, [s] Fig. 6. Simulation results of the z-component of the Lorentz force as a function of the time for a bubble rising at different positions. 626

9 Numerical study of the interaction between a bubble rising in a column of... pos1 Bubble velocity, [m/s] pos2 t=0.28s t=0.38s z t=0.50s t=0.72s Fig. 7. Instantaneous velocity streamlines and distributions of the velocity magnitude on the plane y =0 for bubbles released from pos1 and pos5 (see Fig. 1b). x For a better understanding of the force signals, we finally considerd the spatial pattern of the flow at different times for the bubbles released from pos1 and pos5. TheresultsareillustratedinFigs.7, 8,9. Inbothcases,thebubblehadadeformed ellipsoidal shape that changed with time. This observation is in agreement with the expected wobbling regime for the present Eötvös and Morton numbers [22]. These numbers characterize the geometry and material parameters of a rising bubble without a magnetic field. Their values are Eo 6 and Mo in our case. 627

10 pos1 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher Current density, [A/m 2 ] pos5 t=0.28s t=0.38s z t=0.50s t=0.72s Fig. 8. Instantaneous distributions of the eddy current density magnitude on the plane y =0 for bubbles released from pos1 and pos5 (see Fig. 1b). x The flow fields visualized by streamline projections and velocity magnitude in Fig. 7 do not reveal profound differences in velocity magnitude between pos1 and pos5 although the wake region seems to be more pronounced for pos5. The different times correspond to bubble locations well below the magnet (t = 0.28 s), slightly below the magnet (t = 0.38s), at the same height (t = 0.5s) and well above the magnet (t = 0.72s). The distribution of the eddy current magnitude in Fig. 8 does not suggest significant differences between pos1 and pos5, at least in the vicinity of the bubble. Only for t = 0.72s the eddy currents are generally stronger for pos5. 628

11 Numerical study of the interaction between a bubble rising in a column of... pos1 Lorentz force density, [N/m 3 ] pos5 t=0.38s z t=0.50s t=0.72s x Fig. 9. Instantaneous distributions of the Lorentz force density magnitude on the plane y =0 for bubbles released from pos1 and pos5 (see Fig. 1b). The bubble released from pos1 remained close to the front wall for times up to t = 0.5s. When compared with pos5, the velocity near the wall is, therefore, reduced and the extended downflow zone is absent. This zone may well be responsible for the somewhat reduced Lorentz force for pos5 at t = 0.5s. The Lorentz force density magnitudes are shown in the vicinity of the magnet in Fig. 9. Here it is apparent that a significant contribution at the late time t =,0.72s occurs only for pos5, which is responsible for the second peak in the time signal in Fig. 6. It may be related to the structure of the distant bubble wake that is affected by the walls, which is not apparent from Fig Conclusions. We have studied the behavior of rising bubbles in the GaInSn alloy in the presence of a localized magnetic field. This is a model problem for when the LFV flow measurement technique is applied to conducting liquid-gas flows. The goal of the work was to compute the flow induced by an Argon bubble and to correlate it with the measured Lorentz force. To this end, the VOF method in the ANSYS FLUENT commercial flow solver was augmented with a module to solve the induction equation in the quasistatic limit. Simulations without a magnetic field were performed for validation. Several simulations with bubbles released from different positions in a tall cylinder with the square cross-section 629

12 N.Tran, T.Boeck, U.Lüdtke, Z.Lyu, C.Karcher indicated blockage effects from the walls that also affected the eddy currents. The magnitude of the computed Lorentz force signals agree reasonably well with experiments. Acknowledgements. The authors are grateful to Henning Schwanbeck for discussions and technical support. They also acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Research Training Group Lorentz Force Velocimetry and Lorentz Force Eddy Current Testing(GRK 1567) at Technische Universität Ilmenau, Germany. Computer resources were provided by the TU Ilmenau Computing Center. References [1] C.E. Brennen. Fundamentals of multiphase flow (Cambridge university press, 2005). [2] J. Fröhlich, et al. Influence of magnetic fields on the behavior of bubbles in liquid metals. The European Physical Journal Special Topics, vol. 220 (2013), no. 1, pp [3] A. Thess, E.V. Votyakov, and Y. Kolesnikov. Lorentz force velocimetry. Physical Review Letters, vol. 96 (2006), no. 16, p [4] J.A. Shercliff. The theory of electromagnetic flow-measurement (CUP Archive, 1962). [5] J. Priede, D. Buchenau, and G. Gerbeth. Force-free and contactless sensor for electromagnetic flowrate measurements. Magnetohydrodynamics, vol. 45 (2009), no. 3, pp [6] J. Priede, D. Buchenau, and G. Gerbeth. Single-magnet rotary flowmeter for liquid metals. Journal of Applied Physics, vol. 110 (2011), no. 3, p [7] A. Thess, E. Votyakov, B. Knaepen, and O. Zikanov. Theory of the lorentz force flowmeter. New Journal of Physics, vol. 9 (2007), no. 8, p [8] Y. Kolesnikov, C. Karcher, and A. Thess. Lorentz force flowmeter for liquid aluminum: laboratory experiments and plant tests. Metallurgical and Materials Transactions B, vol. 42 (2011), no. 3, pp [9] S. Eckert, G. Gerbeth, and V. Melnikov. Velocity measurements at high temperatures by ultrasound doppler velocimetry using an acoustic wave guide. Experiments in fluids, vol. 35 (2003), no. 5, pp [10] P.A. Davidson. An introduction to magnetohydrodynamics, vol. 25 (Cambridge university press, 2001). [11] R.J. Moreau. Magnetohydrodynamics, vol. 3 (Springer Science and Business Media, 2013). [12] E.P. Furlani. Permanent magnet and electromechanical devices: materials, analysis, and applications (Academic press, 2001). [13] A. Fluent theory guide. Ansys Inc, vol. 5 (2009). 630

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