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Numerical calibration of a Lorentz force flowmeter
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IOP PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 23 (2012) 045005 (9pp)
doi:10.1088/0957-0233/23/4/045005
Numerical calibration of a Lorentz force flowmeter Xiaodong Wang 1,2 , Yurii Kolesnikov 1 and Andr´e Thess 1 1
Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, PO Box 100565, 98684, Ilmenau, Germany 2 College of Materials Science and Opto-Electronic Technology, Graduate University of Chinese Academy of Sciences, 100049, Beijing, People’s Republic of China E-mail:
[email protected],
[email protected] and
[email protected]
Received 9 September 2011, in final form 26 January 2012 Published 14 March 2012 Online at stacks.iop.org/MST/23/045005 Abstract Numerical simulation of complex-shaped devices for contactless electromagnetic flow measurement in metallurgy is a challenge for computational magnetohydrodynamics. We report a series of numerical simulations which demonstrate for the first time that it is possible to predict the calibration constant of a generic Lorentz force flowmeter (LFF) with an uncertainty close to the requirements of real-life industrial applications. Our simulations involve both magnetostatic computations of a complex-shaped magnet system and magnetohydrodynamic computations of the flow of a liquid metal in a nozzle under the influence of a predominantly transverse magnetic field. In order to assess the role of turbulence, the simulations have been performed both for laminar and for turbulent flows using Reynolds-averaged Navier–Stokes equations in the latter case. In addition to the numerical simulations we have measured the calibration constant of the considered LFF using room-temperature liquid metal instead of liquid aluminum. A comparison between the numerically predicted and the measured values of the calibration constant shows that they differ by only 3.4%. This result suggests that numerical calibration of a LFF may become an economic alternative to expensive full-scale experimental calibration. Keywords: Lorentz force flowmeter, Halbach-array magnet, liquid metal, turbulent flow,
calibration factor
1. Introduction
channels or oil channels which have the same size as the flow geometry in the desired application. However, such fullscale calibration becomes increasingly difficult at elevated temperatures. This work is concerned with flow metering in metallurgy at temperatures between 500 and 1500 ◦ C where full-scale experimental calibration is either prohibitively expensive or altogether impossible. The goal of this paper is to demonstrate that for the specific example of a Lorentz force flowmeter (LFF) [9, 10], a numerical simulation including the liquid metal flow and its interaction with a complexshaped non-contact electromagnetic flow measuring device offers an attractive opportunity to overcome the difficulty of experimental calibration. The measurement of the flow rate of opaque, hot and aggressive molten metals or molten glasses is a challenging task in metallurgy, semiconductor crystal growth and nuclear industries. In view of the harsh working conditions it is
The accurate measurement of the mass flow rate of liquids such as water, beverages, chemicals, gasoline, liquid steel and molten glass is important for two reasons, namely (i) to improve the reliability of accounting and (ii) to facilitate process control. Nowadays, there exist a variety of commercial flowmeters such as Faraday flowmeters [1–4], Venturi flowmeters [5], Coriolis flowmeters [6], vortex shedding flowmeters (VSF) [7] and ultrasonic Doppler flowmeters [8]. A key to the successful application of all these types of flowmeters is calibration—the accurate determination of the relation between the measured quantity, for instance a vortex shedding frequency in a VSF, and the desired mass flow rate. As long as the temperature of the fluids is lower than about 100 ◦ C, the calibration of a given flowmeter can be performed in a straightforward manner using water 0957-0233/12/045005+09$33.00
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desirable that the flowmeters developed for these applications be contactless, robust and reliable. Flowmeters based on electromagnetic induction are capable of meeting these requirements. The development and study of electromagnetic flowmeters can be traced back to Faraday [1–4] and has continued to be a subject of intensive investigation (Shercliff [1], Cervantes et al [11], Bucenieks [12], Thess et al [9, 10], Stefani et al [13] and Priede et al [14]). The present work is concerned with a contactless electromagnetic flow measurement method termed Lorentz force velocimetry and its embodiment which is called LFF [9, 10, 15]. The LFF can be successfully applied for flow measurement in secondary aluminum production at 700 ◦ C and is currently under development for applications at even higher temperatures [16–18]. Recently, direct industrial calibration of a LFF and net mass flow rate measurements at an operating temperature of 780 ◦ C in a recycling aluminum production process at a reasonable accuracy of the flow rate definition with an error of 2.34% were performed (Kolesnikov et al [19]). Full-scale calibration is difficult to control and demands much more costly work, while numerical calibration might be a better choice. In what follows we first explain the principle of a LFF, the experimental set-up and an outline of the mathematical structure of the calibration problem. We second describe our numerical model and present its predictions in terms of calibration constants. Finally, we report a series of model experiments using a room-temperature liquid metal and compare their results to our numerical predictions.
(a)
2. Theory The working principle of a LFF to be considered in this work is explained in figure 1. A magnetic field, created by permanent magnets (PM), acts upon a liquid metal flowing in a converging nozzle. The induced eddy current interacting with the magnetic field leads to the Lorentz force FL in the liquid metal. Simultaneously, the magnetic-field-generating system also experiences an equal but opposite force FM according to Newton’s third law. The principle of operation of a LFF consists of measuring the force FM acting upon the magnet system and determining the mass flow rate from the measured force. The relationship between the measured force FM and the unknown mass flow rate m˙ can be expressed in a scaling law [9, 10] as FM = cdB20 m, ˙
(b)
Figure 1. Sketch of a LFF showing an octagonal Halbach-array magnet system consisting of eight segments of PM in which magnetization directions are different as indicated by arrows. The inner surface of the magnet segments is inclined at the same angle as the angle of the conical part of the nozzle. F L is the Lorentz force acting upon the liquid metal, F M the force acting upon the magnets, and F L = −F M .
realistic flow measurement: how accurately is it currently possible to predict the calibration constant c using numerical simulation? How strongly does the calibration constant depend on whether the flow is laminar or turbulent? With this work we introduce a numerical multiphysics model, validated by an experimental prototype, to answer these questions. Here, we limit our discussion to the case of low magnetic Reynolds number Rem = μ0σ Ud, the so-called low Rem approximation, where μ0 is the magnetic permeability and U the typical flow velocity. Since even for non-ferrous molten metals and their alloys the value σ does not exceed 3 × 106 S m−1, with U ≈ 4 m s−1 and d = 5 × 10−3 m of the given work, the Rem is less than 10−1, i.e. we can neglect the influence of the induced magnetic field on the flow (also see [20]).
(1)
−1
where = σ /ρ (S m kg ) is the specific electrical conductivity, σ (S m−1) the electrical conductivity, ρ (kg m−3) the density of liquid metal, d (m) the typical length scale of the flow, B0 (T) a characteristic value of the magnetic field strength and c a dimensionless calibration constant whose determination is the central focus of this paper. In this regard, previous experimental calibration studies of a LFF with low melting point alloy have been conducted at Ilmenau University of Technology [18, 19]. However, there remain several critical questions to be answered for the application of a LFF to 2
2
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1 5 6
10
7
8
9 2 LM 4
3
Figure 2. Diagram of LFF calibration in a liquid metal laboratory: 1 is the liquid metal loop, 2 is the tank for liquid metal eutectic Ga68In20Sn12, 3 is the electromagnetic pump, 4 is the commercial flow meter, 5 is the flow adapter, 6 is the LFF prototype, 7 is the operating nozzle, 8 is the Halbach magnet system, 9 is the force sensor, 10 is the multivoltmeter 2700 DMM (Keithley Instruments). The arrows indicate the flow direction of liquid metal.
The goal of our work is to predict the calibration constant c of the LFF shown in figure 1. In order to be able to assess the quality of our numerical predictions, we perform a series of experiments whose set-up is described next. The LFF is based on the principle that a Lorentz force is induced in electrically conducting melt such as a liquid metal passing through the magnet system as shown in figure 1. This force acting in the melt is directed opposite to the flow and is proportional to the flow rate of the melt if the interaction parameter (Stuart number) N = Ha2 /Re = σ B20 d/ρU is less than 1. A force of exactly the same magnitude and with opposite direction acts on the magnet system according to Newton’s third law. We place a force sensor under the magnet system to measure the Lorentz force.
use as a characteristic length scale d of the flow in equation (1). To fit the shape of the channel, the inner shape of the magnets is also fabricated in the same manner. A force sensor with an operating range of 0–5 N measuring the Lorentz force is placed under the magnet system. Signal transmissions from the sensor and electronics are not shown. We put the LFF vertically in the liquid metal loop made of stainless steel pipe of 32 mm inner diameter and filled up by a eutectic alloy of composition Ga68In12Sn20 with melting temperature TM = 10.5 ◦ C, and possessing mass density ρ = 6.363 × 103 kg m−3, electrical conductivity σ = 3.31 × 106 S m−1 and kinematic viscosity υ = 4 × 10−7 m2 s−1, at room temperature. This loop is shown schematically in figure 2. An electromagnetic cylindrical pump consisting of a rotating cylinder with PM on its surface [19] develops a pressure of up to 2.0 bars and mass flow rate m˙ = 3.20 kg s−1. An electronic controller for the pump motor allows the registration of the pump rotation speed to bring these data into accord with mean velocity. The flow rate is measured using a commercial induction flowmeter intended for measurement in electrically conducting fluids at room temperature. A miniature force sensor with the linearity divergence of ± 0.15% at measurement and allowing the measurement of force less than 0.01 N is arranged under the magnet system. The LFF was mounted in the liquid metal loop with GaInSn. Circulation is produced by a magnetohydrodynamic (MHD) pump with rotating PM. More detailed description of the loop and measuring procedure are
3. Experimental set-up The flowmeter consists of a Halbach array [21] which is assembled using eight segments of NdFeB PM with special orientation of magnetization and a nozzle as shown in figure 1(a). Such combination of the magnetizations provides inside the pipe distributions of the magnetic field homogeneous in the horizontal plane and non-homogeneous along the vertical coordinate with magnetic induction maximum Bmax = B0 = 0.26 T. The vertical section of the magnet and the nozzle is shown schematically in figure 1(b). The nozzle consists of a conical part with an inclination angle of 11.5◦ and a cylindrical part. The diameter of the cylindrical part is 5 mm which we 3
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given in [18, 19]. To find the calibration factor c, according to equation (1), we measure in the experiment the mass flow rate m˙ using the loop flowmeter and Lorentz force using the force sensor.
model. This question is of considerable practical interest because in practical applications involving complex-shaped three-dimensional flow geometries and magnetic systems it is not always possible to perform turbulent computations. Another aspect reinforces the importance of laminar flow computation. In converging pipes such as those encountered in our experiments the turbulence is suppressed when the fluid approaches the narrow part of the pipe. Hence the quality of laminar flow computation might be better than initially expected. In our case the characteristic length is equal to the diameter of the pipe. At the same time, there is an effect of the turbulence suppression by the transverse magnetic field through the Joule dissipation in the turbulent perturbations due to the appearance of induced electrical currents in the perturbations [22–24]. Thus, we have no a priori knowledge of which is the correct model, laminar or turbulent, closest to the reality. Because of this, we have also included methodologically the laminar model in this paper for its comparison with the turbulent model and the experimental data. An additional point to emphasize is that the used methodology is also appropriate for higher induction of the magnetic field of the flowmeter when turbulent flow can become laminar under the action of the magnetic field. Kenjereˇs and Hanjalic [25] elaborated the implementation of the effects of the Lorentz force in turbulence closure models. The effects of the Lorentz force on the turbulent flow of the conductive fluid were analyzed within the framework of second-moment and eddy-viscosity closure models. Additional terms representing the MHD interactions in the transport equation for the turbulent stress tensor and energy dissipation rate were identified. These modifications become important when the magnetic interaction parameter becomes high. Since our problem is characterized by moderate values of N, we have decided not to adopt these modifications. For further studies on MHD turbulent closure, we refer to [26, 27]. The present problem is a three-dimensional multiphysics problem implying that the computational task is quite heavy. In order to efficiently compute and develop a generic program, the influence of grid design on the result of the Lorentz force is studied. Our preliminary tests have indicated that large values and gradients of the electromagnetic quantities were mainly encountered in the vicinities of the neck of the nozzle, where the fine grids should be set. Following these observations, the geometry of the fluid is divided into three parts, where we set different mesh densities and keep the physical properties and boundary conditions unchanged; as seen in figure 3, the grids in the middle part (z ∈ [−5 mm, 16 mm]) are adopted with different mesh densities by setting the maximum element size. It should be noted that the higher mesh density covers the region where the magnetic field is localized in the axial direction, and the other two parts are meshed with relatively coarse meshes (maximum element size is 2.5 mm). We perform the computation and then obtain the integral force acting on the liquid metal (Fz), also shown in table 2; it indicates that Fz varies little by increasing the number of elements; table 1 also gives the order of our numerical estimation which is less than 1% in the range of grid investigation. In summary, our
4. Numerical method A multiphysics numerical model is built to compute the force acting in the fluid and consequently upon the magnetic-fieldgenerating system which is to be measured in the experiment. The numerical model is based on the Reynolds-averaged Navier–Stokes (RANS) method. The problem is solved by fully coupling Navier–Stokes equations and Maxwell equations: ∂u + (u · ∇ )u = −∇ p ∂t 1 (∇ × B) × B, ∇ · u = 0, (2) + υ∇ 2 u + μ0 ρ ∂B 1 (3) + (u · ∇ )B = (B · ∇ )u + ∇ 2 B, ∂t μ0 σ where u is the velocity field, t the time, p its pressure, υ the kinematic viscosity, μ0 the magnetic permeability and B the magnetic field strength. The magnetic field exerted by the Halbach array is computed with the relationship B = μ0(H + M), in which M = 106 A m−1 is the magnetization density. The magnetization direction of each magnet is specified as shown in figure 1(a). The equations are solved in two manners. First, the time derivative is set zero (∂/∂t = 0) and the laminar model is used. Second, in the k–ε turbulent model the logarithmic wall function with no-slip conditions on the walls is adopted, and the viscous stress closure is used in this model. The magnetic Reynolds number is small in our problem, but we adopt a full two-way coupling computation between the fluid flow and the magnetic field despite the fact that the magnetic field induced by the fluid flow is small, i.e. 10−3– 10−4 less than the applied magnetic field. On the other hand, the same results would have been obtained using a quasistatic approximation. However, since our numerical tool is also used for large-scale industrial applications where the finite value of the magnetic Reynolds number cannot be neglected, we have implemented the full MHD equations. It should be noted that for the turbulent computations we have effectively a turbulent viscosity which is several orders of magnitude higher than the molecular viscosity and the stiffness problem disappears. Thus, in order to develop a generic calibrating model, we solve the full MHD equations. In the present model, an artificially enhanced diffusion is adopted by setting a proper tuning parameter to overcome the stiffness of the PDE system. In order to assess the role of turbulence, we perform in this paper a comparison of numerical models for laminar and turbulent flows. We are aware that the flow in the experiment is turbulent and cannot be described by the laminar model. Nevertheless, we believe that the laminar flow computations are useful as a reference case because they permit a quantitative evaluation of the error that would be made if the computations were performed with the laminar rather than with the turbulent 4
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Table 1. Influence of the grid size on the results (grid sensitivity). Maximum size in the middle part (mm)
Number of elements
Number of degrees of freedom
Integral force Fz (mN) (Re = 4.50 × 104)
2.5 1.5 1.0 0.75
28 192 31 697 41 815 61 421
328 061 376 008 522 781 802 686
6.536 6.536 6.548 6.564
Table 2. Comparison of the numerical and experimental calibrating factors. Calibrating factor, c Numerical prediction with the turbulent model (full Halbach magnet)
8.361 × 10−2
One magnet segment changed
8.729 × 10−2 8.442 × 10−2 7.644 × 10−2
Experimental measurement Numerical prediction with the turbulent model Numerical prediction with the laminar model
(a)
Figure 3. Grid refinement within the fluid domain in order to investigate the numerical estimation order of the model. 1, 3 are the coarse mesh regions and 2 is the fine mesh region.
calibration results are not so sensitive to the grid sizes. The grid design of the third case given in table 1 is used throughout the paper, namely the number of meshes is 41 815 and the numbers of degrees of freedom are 465 895 (laminar model) and 522 781 (turbulence model), respectively. This model is realized with software package COMSOL multiphysics using the finite element method. The model is performed on an 8-core CPU 64-bit computer. (b)
5. Numerical results
Figure 4. Magnetic field distributions of the Halbach-array magnet system. (a) Vectors of the magnetic field in the horizontal plane at z = 6 mm; (b) magnetic field distribution in the z-direction at x = 0 and y = 0.
The magnetic field distributions are presented in figure 4. The arrows denoting the values of magnetic field induction are shown in figure 4(a) both in the magnet material and the inner domain of the magnet system at z = 6 mm. The distribution of the magnetic field in the horizontal plane inside the pipe cross-section within a liquid metal flow domain (d 5 mm, z = 6 mm) demonstrates a non-homogeneity of the magnetic field no more than 2.2%. The Cartesian coordinate system is chosen so that the magnetic lines across the flow are in the y-direction whereas the streamwise coordinate coincides with the z-direction. The magnetic field distribution along the
z-direction, shown in figure 4(b), is highly non-homogeneous and appears as a Gaussian-like curve. When it is considered that the coordinate z = 0 lies in the lower plane of the magnet system, the vertical distribution of the magnetic field has a maximum located at the coordinate of z = 6 mm, where the conical-shaped part of the pipe changes to cylindrical. The value of this maximum Bmax = B0 = 0.26 T is used 5
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force in the flow. This force is a braking force and is directed opposite to the main flow. The theoretical analysis of such an eddy current distribution is elaborated in [20]. Figure 6 shows the distribution of the Lorentz force density in the x–z-planes as well as in the y–z-planes. Here we focus only on the z-component of the Lorentz force. We emphasize here that the Lorentz force acting upon the liquid metal is the opposite of streamwise, as if the magnetic field is an invisible obstacle for the flow. Due to the inhomogeneous distribution of electric currents, the Lorentz force is spatially inhomogeneous. Figure 6(a) shows the contours of the Fz distribution in the x–z-plane perpendicular to the magnetic field. It is seen that the force has a tendency to be greatest in the vicinity of the wall in the cross-sections of the conical part; the reason is that the flow of the electric current in the vicinity of the conical wall (see figure 5) creates an additional force component Fz near the wall. Figure 6(b) demonstrates the Fz force distribution in the y–z-plane. In this plane a concentration of the force appears as two zones with the largest magnitude of Fz = 9 × 104 N m−3 near the walls in the y-axis direction and decays to the two sides in the z-axis direction. In a lower area of the cylindrical part the negative force with a value of 0.2 (Fz)max is observed. Appearance of the negative force is linked with the retention of a high enough magnetic field value (see figure 4(b), z = −3 mm) and the change of electric current sign here. When a magnetic field acts upon a pipe flow, according to equation (2), the Lorentz force influences the velocity distribution. Figure 7 shows the velocity profiles for laminar and turbulent models, at three cross-sections (z = 16 mm, z = 6 mm and z = 0). In the simulation the velocity varies from zero starting from the wall in the laminar model, whereas in the turbulent model, the velocity is computed from the overlap regions between the boundary layer and the flow core via a logarithmic wall function neglecting the velocity profile within the boundary layer [29]. In the conical part (z = 16 mm, figure 7(a)) the laminar flow is subjected to the action of a large enough magnetic field (see the value in figure 4(b)) and there is a tendency for the profile to transform
Figure 5. Eddy current distributions in the x–z-plane in the flow at the flow rate of m˙ = 0.445 kg s−1. The arrows denote the vectors of current densities generated by the interaction of the flow with the magnetic field. Turbulent flow, Re = 4.50 × 104, Ha = 46.9.
in equation (2). For the numerical model we take the same physical properties of the fluid used in the experiments. The main results presented below are considered for the flow rate of m˙ = 0.445 kg s−1. Taking the diameter d = 5 mm of the cylindrical part of the pipe as a characteristic scale we obtain the mean velocity of u0 = 3.6 m s−1, the Reynolds number of Re = (u0d)/ν √= 450 × 104 and the Hartmann number of Ha = B0 d σ /ρν = 46.9. From ordinary hydrodynamics it is known that in a narrowed section of a nozzle a suppression of turbulence takes place. For instance, the investigation of turbulent flow in a bifurcated nozzle applied in continuous casting by Hershey et al [28] confirms this point. Figure 5 shows that the electric currents generated by the interaction of turbulent flow and applied magnetic field form two eddy current loops in the x–z-plane with opposite directions closing within the fluid domain due to the nonconductivity of the wall. The interaction creates a Lorentz
(a)
(b)
Figure 6. Lorentz force density distribution in the flow at the flow rate of m˙ = 0.445 kg s−1. Contours of the z-component of the Lorentz force F z in (a) x–z-plane and (b) y–z-plane. Turbulent flow, Re = 4.50 × 104, Ha = 46.9. 6
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(a)
Figure 8. Comparison of the Lorentz force with dependence on the flow rate between numerical laminar and turbulent models and experiment. Turbulent flow, Ha = 46.9. Experiment is conducted at Re = 2.04 × 104, 3.31 × 104, 3.82 × 104, 4.50 × 104 and 4.58 × 104.
6. Comparison between numerical and experimental results (b)
Using a eutectic alloy in the composition of GaInSn with properties presented above we carried out an experiment with the described LFF to verify the correctness of the numerical method to define the calibration factor c for the given flow geometry and magnet system (figure 2). This configuration mimics a certain realistic industrial application, for instance, a nozzle flow in a continuous casting process. In the magnet system (see figures 1 and 4(a)), a soft iron segment is taken instead of a segment of PM, which has a cylindrical hole in order to organize a magnet system support and force measurement. The numerical model is also adapted to this modification (table 2). The numerical calculation and laboratory measurements of the modified magnet system testify that the intensity of the magnetic field decreases a little from 0.26 to 0.25 T, but it preserves all the physical intrinsic features of the LFF. For comparison, the numerical simulation of the case with eight segments of magnets, full Halbach magnet, is also performed. Since the dependence between the Lorentz force and the flow rate is linear, it is possible to use a large number of measurements to reach a precise value of factor c. Table 2 gives the calibrating constants in cases of numerical predictions and experimental measurement for the full Halbach magnetic array (eight-segment) and one segment changed, respectively. In figure 8 the experimental and numerical data of the Lorentz force with dependence on the flow rate for laminar and turbulent regimes are presented. One can see that the forces in laminar and turbulent flows having linear dependence are different, i.e. the force in the turbulent regime is approximately 10% higher than that in the laminar regime. We emphasize that the experiment and turbulent model data coincide with an accuracy of 3.4% (see table 1). Both results, being in good agreement with each other, indicate a possibility of numerical turbulent model application as a tool for preliminary calibration of a LFF manufactured for the industry. The
(c)
Figure 7. Velocity profiles in the plane perpendicular to the magnetic field for laminar and turbulent models and three vertical coordinates (for the positions see figure 1(b)). Turbulent flow, Re = 4.50 × 104, Ha = 46.9.
into an M-shape. In the turbulent regime the velocity profile is flatter due to turbulence transfer processes in the flow. Input of flow into the cross-section (z = 6 mm, figure 7(b)) where the conical part changes to cylindrical is characterized essentially by increases of non-homogeneity of the velocity profiles in both laminar and turbulent regimes, because here the electric current and magnetic field and, consequently, the braking Lorentz force, reach maxima (figures 5 and 6). In the cross-section coinciding with the lower border of the magnet (z = 0, figure 7(c)) the velocity profile in the laminar regime is still non-homogeneous, whereas in the turbulent regime the profile again becomes flat under the action of turbulent transfer processes in spite of the presence of a high value of magnetic field in this cross-section (figure 4(b)). By solving equation (3), also with equation (2), the predictions of the calibrating factors are cnl = 7.644 × 10−2 for the laminar flow model, and cnt = 8.442 × 10−2 for the turbulent flow model (also see table 2). 7
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Research Foundation (Deutsche Forschungsgemeinschaft) in the framework of the Research Training Group (Graduiertenkolleg) ‘Lorentz force velocimetry and Lorentz force eddy current testing’.
industry often brings a requirement of 1–1.5% accuracy of measurement of the liquid metal flow rate; therefore the presented result is encouraging because there is a possibility to reach this requirement accuracy if the dependences of the magnetic field and the electrical conductivity of liquid metal on temperature are to be used. These dependences are linear with temperature at least in a range of 293–333 K. During the experiment, the temperature is changed within a range of 275–279 K.
References [1] Shercliff J A 1962 The Theory of Electromagnetic Flow Measurement (Cambridge: Cambridge University Press) [2] Faraday M 1832 Experimental researches in electricity Phil. Trans. R. Soc. 15 175 [3] Shimizu T, Takeshima N and Jimbo N 2000 A numerical study on Faraday-type electromagnetic flowmeter in liquid metal system (I) J. Nucl. Sci. Tech. 37 1038–48 [4] Shimizu T and Takeshima N 2001 A numerical study on Faraday-type electromagnetic flowmeter in liquid metal system (II) J. Nucl. Sci. Tech. 38 19–29 [5] Arnberg B T, Britton C L and Seidl W F 1974 Discharge coefficient correlations for circular-arc Venturi flowmeters at critical (sonic) flow J. Fluids Eng. 96 111–24 [6] Langdon R M 1985 Resonator sensors—a review J. Phys. E: Sci. Instrum. 18 103–15 [7] Griffin O M and Hall M S 1991 Review—vortex shedding lock-on and flow control in bluff body wakes J. Fluids Eng. 113 526–37 [8] Franklin D L, Schlegel W and Rushmer R F 1961 Blood flow measured by Doppler frequency shift of back-scattered ultrasound Science 134 564–5 [9] Thess A, Votyakov E V and Kolesnikov Y 2006 Lorentz force velocimetry Phys. Rev. Lett. 96 164501 [10] Thess A, Votyakov E, Knaepen B and Zikanov O 2007 Theory of the Lorentz force flowmeter New J. Phys. 9 299 [11] Cervantes M, Enstr¨om C, Kelvesj¨o H and Ohlsson H 2000 Method and apparatus for non-contact measuring of metal bed parameters US Patent 6538433 B1 [12] Bucenieks I 2005 Modelling of rotary inductive electromagnetic flowmeter for liquid metals flow control Proc. 8th Int. Symp. on Magnetic Suspension Technology (Dresden, Germany, Sept) ed G Fuchs, L Schultz, O de Haas and H-J Schneider-Muntau pp 204–8 [13] Stefani F, Gundrum T and Gerbeth G 2004 Contactless inductive flow tomography Phys. Rev. E 70 056306 [14] Priede J, Buchenau D and Gerbeth G 2011 Contactless electromagnetic phase-shift flowmeter for liquid metals Meas. Sci. Technol. 22 055402 [15] Vir´e A, Knaeppen B and Thess A 2010 Lorentz force velocimetry based on time-of-flight measurements Phys. Fluids 22 125101 [16] Thess A, Kolesnikov Y, Karcher C and Votyakov E 2006 Lorentz force velocimetry—a contactless technique for flow measurement in high-temperature melts Proc. 5th Int. Symp. Electromagnetic Processing Materials (Sendai, Japan) pp 731–4 [17] Thess A, Kolesnikov Y and Karcher C 2008 Method and arrangement for the contactless inspection of moving electrically conductive substance US Patent 0252287 A1 [18] Karcher C, Kolesnikov Y, Minchenya V and Thess A 2009 Flow rate measurement in liquid aluminum using Lorentz force velocimetry Proc. 6th Int. Symp. Electromag. Processing Materials (Dresden, Germany) pp 419–22 [19] Kolesnikov Y, Karcher C and Thess A 2011 Lorentz force flowmeter for liquid aluminum: laboratory experiments and plant tests Metall. Mater. Trans. B 42 441–50 [20] Moreau R 1990 Magnetohydrodynamics (Dordrecht: Kluwer) p 35 [21] Halbach K 1980 Design of permanent multipole magnets with oriented rare earth cobalt material Nucl. Instrum. Methods 169 1–10
7. Conclusions A multiphysics numerical model was developed to find the calibration factor of the LFF. This model has the ability that it can be applied to devices with a complex-shaped channel for contactless electromagnetic flow measurement. We used a magnetic system of a special configuration with eight magnet elements providing a homogeneous magnetic field in the flow cross-section and a high inhomogeneity of the field along the flow. The numerical results show that replacing one magnet element in the magnet system by an identical ferrous element (in the place where the force sensor is installed) decreases the magnetic field induction B0 from 0.26 to 0.25 T, and the corresponding force (proportional to B20 ) is reduced to 92%. The missing magnet element also increases the horizontal inhomogeneity of the magnetic field. These changes must be compensated for by a calibration factor which we found in the given study. We showed in this study that the Lorentz force linearly varies with the flow presuming that the interaction parameter N < 1. If we increase the diameter of the nozzle or decrease the flow velocity, so presuming that N 1, then such a linear relationship cannot exist because of essential deformation of the velocity field, leading to the formation of thin boundary layers parallel to the magnetic field, which gives a nonlinearity. The numerical simulation when the turbulent model is used gives a reasonable agreement with the results of the experiments performed with the LFF prototype. The difference between the calibration factors is 3.4%. When the laminar model is used the difference reaches 14.2%, i.e. we can conclude that the numerical turbulent model used adequately depicts the real flow in the given LFF prototype and is rewarding. In closing it may be said that the calibration factor is a function of Re, Ha and the geometry or typical scale length d of flow, c(Re, Ha, d), that follows from equation (3). In the case of higher magnetohydrodynamics, the magnetic interaction number N 1, and it should be crosschecked with other full coupling numerical models, i.e. MHD models, using direct numerical simulation [30] or large eddy simulations [27] to compute the calibrating function.
Acknowledgments We are grateful to the Bundesministerium f¨ur Bildung und Forschung (BMBF) for financial support in the framework of the ForMaT program under grant number 03FO2202. AT and YK also acknowledge financial support from the German 8
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