Numerical optimization of the magnet system for the Lorentz Force Velocimetry of electrolytes

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1 International Journal of Applied Electromagnetics and Mechanics 38 (2012) DOI /JAE IOS Press Numerical optimization of the magnet system for the Lorentz Force Velocimetry of electrolytes Artem Alferenok a,, Michael Werner b, Michael Gramss c, Ulrich Luedtke a and Bernd Halbedel b a Department of Electrothermal Energy Conversion, Ilmenau University of Technology, Ilmenau, Germany b Department of Inorganic-Nonmetallic Materials, Ilmenau University of Technology, Ilmenau, Germany c Department of Thermo- and Magnetofluiddynamics, Ilmenau University of Technology, Ilmenau, Germany Abstract. Lorentz Force Velocimetry is a contactless method for the flow rate measurement of electrically conducting fluids. This method is based on the interaction of the fluid flow with the transversal permanent magnetic field. The equal electromagnetic force acts on the fluid and on the magnet system. The flow rate can be determined by measuring of this electromagnetic force. The magnet system has been optimized to achieve maximal sensitivity at a given magnet system weight. The sensitivity was defined as the ratio between the Lorentz force and the magnet system weight. The numerical model was developed using COMSOL Multiphysics. Validation and verification of the numerical model has been performed. The magnet system was optimized using the optimization toolbox in MATLAB. Keywords: Electromagnetic forces, finite element methods, optimization, permanent magnets 1. Introduction Lorentz Force Velocimetry (LFV) is a contactless method to measure the flow rate of electrically conducting fluids [1]. The LFV working principle is based on the interaction of the fluid flow with the transversal constant magnetic field (Fig. 1). The fluid flows with velocity v across the magnetic field B which is generated by two identical sets of permanent magnets (PM) at rest. According to Faraday s law, because of the relative motion between the flow and the magnetic field B, the eddy current j arises in the flow. Interacting with the primary magnetic field B, the eddy current causes the Lorentz force F L, which brakes the flow. According to Newton s third law, the same force acts on the magnet system in opposite direction. The Lorentz force is proportional to the velocity v, electrical conductivity of the fluid σ, and squared magnetic flux density B [1]. Therefore, knowing these parameters it is possible to measure the flow rate by measuring the Lorentz force. Corresponding author: Artem Alferenok, Ilmenau University of Technology, Ehrenbergstr. 29, Ilmenau, 98693, Germany. Tel.: ; Fax: ; artem.alferenok@tu-ilmenau.de /12/$ IOS Press and the authors. All rights reserved

2 80 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 1. Sketch of the LFV: 1 electrolyte, 2 permanent magnets. The LFV is a well-developed technique to measure the flow rate of liquid metals [2]. The goal of our project is to develop LFV for electrolytes. The main problem is that the expected Lorentz force is approximately a million times smaller for electrolytes than for liquid metals for the same magnetic field distribution B, since the electrical conductivity of electrolytes is about 6 orders of magnitude smaller than the electrical conductivity of liquid metals. The expected value of the Lorentz force lies between 10 and 100 μn. If the channelcross-sectiondimensionsand the distancebetweenthe magnets arefixed, then there are only two ways to increase the Lorentz force. First, the velocity of the flow can be increased and, second, the magnetic field can be optimized, i.e. magnetization direction and magnet dimensions. The second possibility is the subject of this work. In work [3], the optimization of the magnetization directions of a permanent magnet array for an eddy current brake system was accomplished. Choi et al. designed a 2D finite element model and used the sequential linear programming (SLP) and the adjoint variable method (AVM) to maximize the braking force. The optimized magnetization pattern corresponds with the well-known Halbach-array structure [8]. Since the measurement system requires the magnet system (MS) weight to be less than 1 kg and the Lorentz force to be higher than 10 μn, the only way to generate a relatively high magnetic field is to use permanent magnets [4,11]. Because of its high magnetization, Nd-Fe-B was chosen as the material for permanent magnets. This investigation was performed for the following initial data: Electrolyte velocity v = 5m/s, electrolyte electrical conductivity σ = 4 S/m, electrolyte cross-section area S = 0.05 m 0.05 m, remanence of the permanent magnets B r = 1.09 T. The next sections are devoted to describing, verifying, and validating the numerical model (Section 2), as well as optimizing the magnet system (Section 3). Section 4 is devoted to conclusions. 2. Description of the numerical model The commercial software package COMSOL Multiphysics v.3.5a was used to simulate the 3D electromagnetic problem including the motion described above. The problem includes 3 subdomain types:

3 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 81 Air, moving electrolyte, and permanent magnets at rest. It should be mentioned that in the numerical model the solid body motion was simulated and not the fluid flow. The reason for this assumption is that the turbulent flow profile, which is expectedto appearin presentproblem, is close to the constantor solid body profile. Furthermore, the computation time is much smaller for solid body motion than for turbulent flow Governing equations After reformulating, Maxwell s system of equations for any subdomain is given as follows [5]: (σv ( A) σ V )=0, (1) (μ 1 0 μ 1 r A M) σv ( A)+σ V =0. (2) This formulation, also called quasi-static approach, was successfully used in [6] to analyze changes of drag force on PM moving along a conducting plate with different spacial defects. Here, the magnetic field B is expressed as the curl of A, wherea is the magnetic vector potential. The electric field intensity E is expressed as the negative gradient of V, wherev is the electric scalar potential. Further, μ 0 =4π 10 7 H/m is the magnetic permeability in the air, and M is the magnetization of magnets. The boundary conditions are defined for the internal boundaries of the model, the so-called continuity condition Eq. (3), and for the external boundary, the magnetic insulation condition Eq. (4): n (H 1 H 2 )=0, n (J 1 J 2 )=0, (3) n A =0, V =0, (4) where H is the magnetic field strength, n is the normal unit vector on the boundary, and J is the eddy current density. Subscripts in Eq. (3) denote the field properties inside and outside the boundary. A-V formulation (1) (4) is a well-known method to solve the 3D electromagnetic problems including motional eddy currents [5]. We used an ungauged A-V formulation. The fundamental fieldequationis the Maxwell-Ampères law, expressedusingthe magnetic vectorpotentiala, the scalarelectric potential V, and an induced current density term of Lorentz type σ(v B). This formulation allows us to solve our problem in a fast and memory efficient way. Such a formulation is singular and there is an infinite number of possible solutions, but all solutions yield the same magnetic flux and current density field. In the given problem, the induced current distribution is stationary. The moving domain does not contain any magnetic sources that move along with it, is unbounded, and is invariant in the direction of the motion. It allows us to solve our problem using a stationary iterative solver. The Lorentz force density is the cross-product of the eddy current density and the magnetic flux density in the electrolyte. This can be expressed in terms of potential functions: f = σ[v ( A) V ] ( A). (5) 2.2. Verification of the numerical model concerning the magnetic field The magnetic flux density in any point in the vicinity of the permanent magnet, which is magnetized along one predefined direction, can be found using the analytical expressions [7 9]. The z-component of the magnetic flux density B z was determined numerically and analytically along the z-axis of the rectangular magnet, shown in Fig. 2 using following parameters: B r = 1.09 T, μ r = 1,

4 82 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 2. Rectangular permanent magnet. Fig. 3. Numerically and analytically calculated magnetic flux density along z-axis: x=a/2, y=b/2, z is varied from c to c+0.05 m(seefig.2). a = m, b = 0.02 m, c = m, where B r and μ r are remanence and relative permeability of the magnet, respectively. Figure 3 shows how the obtained B z depends on the distance from the middle point of the magnet surface along the z-axis. The analytically and numerically obtained results are in very good agreement. The difference is less than 2%. The z-component of the magnetic flux density in a point with coordinates x, y, andz outside of the

5 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 83 Fig. 4. Dry calibration experiment: 1 aluminum bar, 2 permanent magnet, 3 ironyoke. magnet is expressed analytically as follows [8]: B z (x, y, z) = μ 0 M 4π [ arctan 2 2 k=1 n=1 m=1 2 ( 1) k+n+m (x x n )(y y m ) ] (z z k )[(x x n ) 2 +(y y m ) 2 +(z z k ) 2 ] 1/2 Here, x n, y m,andz k are coordinates of the magnet boundaries Validation of the numerical model concerning the drag force, (6) Experiments using an aluminum bar were performed to validate the numerical model. These experiments followed the experiments described in [10]. The new experiments differ as follows: The velocity range was enlarged up to 0.2 m/s by using a new linear drive. The bar was moved with a constant velocity through the magnet system containing two permanent magnet blocks and an iron yoke. Each permanent magnet block was composed of small permanent magnet bars with dimensions 20 mm 20 mm 100 mm. The permanent magnet material was again Nd-Fe-B, but with slightly different properties compared to the permanent magnet material described in Section 1. The overall dimensions of the dry calibration experiment in the y-z plane are shown in millimeters in Fig. 4. The overall dimensions of the magnet blocks and iron yoke along the x axis are 140 mm and 160 mm, respectively. Following parameters constituted our input data: Electrical conductivity of the bar σ = S/m, PM remanence B r = 1.38 T, PM coercitivity H c = A/m, and a nonlinear BH-curve for steel

6 84 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 5. Drag force in relation to the bar velocity as the material for the iron yoke. Figure 5 shows the numerically and experimentally obtained dependency of the drag force on the velocity of the bar. The results are in good agreement. The difference between the experimentally and numerically obtained drag force is less than 5%. This difference is caused by uncertainties of the physical properties used in our numerical model, namely σ, B r,andh c. Furthermore, it must be taken into account that there is some deflection of the velocity caused by the control system of the linear drive. In addition, positioning deviations affect the measurement result. 3. Optimization of the magnet system The optimization of the magnet system for LFV of electrolytes included following steps: First, selection of the magnet system components; second, optimization of the magnetization directions and of the magnet dimensions. The primary question was whether to use permanent magnets or solenoidal coils to generate the magnetic field. After preliminary calculations and a literature survey, PM were chosen, since the MS weight is restricted, and only PM can generate a relatively high magnetic field with an operating weight oflessthan1kg[11]. The next question to be answered was whether to use an iron yoke in LFV for electrolytes or not. To answer this question, two magnet systems were simulated: One with an iron yoke and another one built exactly alike but without an iron yoke. Results have shown that the magnet system without an iron yoke is much more efficient, since it has approximately 2 times higher sensitivity, which was defined as the ratio between the drag force and the MS weight [12]. Another important question was finding the optimal dimensions and arrangement of the permanent magnet system. To solve this problem, different optimization techniques can be used [13].

7 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 85 Table 1 Optimization results for different magnet systems Magnet System x 1 (m) x 2 (m) x 3 (m) ỹ max/m (N/kg) ỹ max (N) m (kg) 2 PM e-5 1.0e PM e e PM e-5 2.8e Fig. 6. Magnet system with 2 (a), 6 (b) and 10 (c) PM. Fig. 7. Magnetic field vector plot for each magnet system. Besides the magnet system including 2 PM, other magnet systems including 6 and 10 PM, which are also called Halbach-arrays [8], were investigated and compared (Fig. 6). Here, arrows show the magnetization direction. Figure 7 shows a magnetic field vector plot in the x y symmetry plane for each magnet system. Here, the permanent magnet dimensions correspond to the values given in Table Parametric study of the magnet system including two permanent magnets Considering the magnet system with 2 PM (Fig. 6a), design variables x 1, x 2, x 3 can be defined, which correspond to the length, width, and height of the permanent magnets, respectively. It is assumed that the magnets are identical, so that x 1, x 2,andx 3 of the first magnet are equal to x 1, x 2,andx 3 of the second magnet. Magnets are disposed symmetrically with respect to the channel and to each other. Additionally, design variable x 4 was defined, which corresponds to the gap between the magnets and the electrolyte. To better understand the problem, a parametric analysis was performed. Figure 8 shows drag force F versus design variables x i normalized by the characteristic length of the channel cross-section L. Each curve corresponds to the variation of a single design variable, while the other three variables are constant. Drag force grows nonlinearly by increasing x 1, x 2,andx 3 and sinks nonlinearly by increasing x 4. All curves in Fig. 8 tend to saturate after certain x i /L. This behavior can be better recognized by

8 86 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 8. Drag force F in relation to x 1/L, x 2/L, x 3/L, andx 4/L, wherel = 0.05 m is the characteristic length. Fig. 9. Ratio between the drag force and MS weight F/m in relation to x 1/L, x 2/L, x 3/L, andx 4/L, wherel = 0.05 m is the characteristic length. considering Fig. 9, which shows the ratio between the drag force and MS weight versus nondimensional design variables x i /L. All curves in Fig. 9, except F/m depending on x 4, have one peak, after that they decline. Looking at Figs 8 and 9 it is clear that: First, to increase the drag force, design variable x 4 must be minimized; second, optimal variables x 1, x 2,andx 3 must be found that satisfy the problem constraints and provide maximal drag force at a given MS weight. The gap between the electrolyte and

9 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 87 Fig. 10. Calibration coefficient c in relation to x 1/L, x 2/L, x 3/L, andx 4/L, wherel = 0.05 m is the characteristic length. the magnets was fixed at x 4 = 3 mm. It includes the wall thickness of the channel and air gap. To compare different magnet systems it is reasonable to use a nondimensional parameter, such as the calibration coefficient c, which is defined as follows: c = F σ B 2 0 L V. (7) Here, F is the drag force, B 0 is the characteristic magnetic field in the middle point between the magnets, and V = L 2 v is the specific volumetric flow rate in the channel. Figure 10 displays the calibration coefficient c versus x i /L. It is notable that c decreases nonlinearly by increasing x i /L. Furthermore, saturation appears after certain x i. It is important to mention that for the magnet system used in the dry calibration experiment (see Section 2, C and Fig. 5) the calibration coefficient is equal to c = For the magnet system without an iron yoke this coefficient can be approximately 6 times higher (see Fig. 10). Here, only values of c corresponding to a drag force higher than 10 μn were considered. In accordance with the previous discussion, it is obvious that the calibration coefficient should be maximized Optimization algorithm In this subsection, the optimization algorithm will be described for the case of a magnet system with two permanent magnets (see Fig. 6a). From the previous investigation it was found that design variable x 4 must be as small as possible, whereas variables x 1, x 2,andx 3 must be optimized to obtain maximal drag force at a given MS weight. The design variable x 4 was fixed at 3 mm due to design conditions. The other design variables are restricted using lower and upper boundaries, forming a so-called design space. The objective function y = y(x 1,x 2,x 3 ) is the drag (or Lorentz) force, which depends on the design variables and must be maximized at the limited weight of the permanent magnets, so that the

10 88 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 11. Approximated drag force as a function of length x 1 and height x 3 of PM in case of magnet system with 2 PM, x 2 = 17.5 mm, x 4 = 3 mm, B r = 1.09 T. equality constraint g = g(x 1,x 2,x 3 ) is fulfilled. In general, the problem can be defined as follows [13]: Maximize y Subject to g =0 x min i x i x max i,i=1, 2, 3. (8) The main difficulty in Eq. (8) is to find an objective function. Since there are only three design variables, one of them can be kept constant, e.g. x 2. Then the discrete function y(x 1,x 3 ) can be obtained using the numerical model. Afterwards, the polynomial approximation ỹ of y can be found using the surface fitting tool in MATLAB. Then the problem can be formulated as follows: Maximize ỹ(x 1,x 3 )=p 00 + p 10 x 1 + p 01 x 3 + p 20 x p 11x 1 x 3 + p 02 x p 30 x p 21 x 2 1x 3 + p 12 x 1 x p 03 x p 40 x p 31 x 3 1x 3 + p 22 x 2 1x p 13 x 1 x p 04x p 50x p 41x 4 1 x 3 + p 32 x 3 1 x2 3 + p 23x 2 1 x3 3 + p 14x 1 x p 05x 5 3 Subject to 2 ρ x 1 x 2 x 3 m =0, x 2 = const, x min i x i x max i,i=1, 3. (9) Here, p ij are coefficients of the polynomial, ρ = 7500 kg/m 3 is the density of the permanent magnets, and m is the magnet system weight. The nonlinear problem with equality constraint (9) can be solved using the function fmincon in MATLAB. Thus, for any one x 2, only one maximum of ỹ can be found which satisfies the equality constraint g. Figure 11 shows an example of the approximated function ỹ(x 1,x 3 ) for x 2 = 17.5 mm. For this case, following input data were used: B r = 1.09 T, v = 5m/s,and σ = 4 S/m. The points in Fig. 11 correspond to the numerically obtained drag force, whereas the surface corresponds to the polynomial approximation. This approximation is very accurate. The difference between y and ỹ is less than 1%. To find the maximal Lorentz force for a given magnet system weight using the optimization toolbox in MATLAB, it is convenient to use a polynomial as an objective function. In this case, the optimization

11 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 89 Fig. 12. Maximal drag force ỹ max in relation to the thickness of PM x 2 for different MS weight m in case of MS with 2 PM, x 4 = 3 mm, B r = 1.09 T, ρ = 7500 kg/m 3, x 1 and x 3 are optimized to obtain maximal drag force ỹ max by fixed MS weight m. procedure is performed very quickly (less than 2 seconds). For the FEA-evaluation of the objective function, the optimization procedure is performed much more slowly (several hours). Moreover, the solution obtained using the polynomial is much more accurate. It was proved by comparing these two approaches and by finding the optimal dimensions for the magnet system with two permanent magnets. To obtain the Lorentz force polynomial for one fixed design variable, it was necessary to carry out about 50 FEA-evaluations. One FEA-evaluation takes approximately 2 minutes. It is relatively time consuming, but we were able to obtain very accurate results using this polynomial Optimization of the permanent magnet dimensions Using the algorithm described above, it is possible to obtain the maximal drag force ỹ max for fixed m and x 2. Figure 12 shows the dependencies of the maximal drag force ỹ max on x 2 for different m for the magnet system with two PM. The drag force reaches its maximal value at different x 2 for every MS weight m. Thisfigure also indicates a nonlinear behavior of the problem. Moreover, by increasing the MS weight the optimal x 2 increases. The maximal drag force in relation to the MS weight for each considered magnet system is represented in Fig. 13. The maximal drag force ỹ max achieved by each magnet system depends nearly linearly on the MS weight m. To achieve the required drag force 10 μn the MS weight of the MS with 6 and 10 PM is significantly less than for the MS with 2 PM. Finally, Fig. 14 depicts the maximal drag force ratio to the MS weight in relation to the MS weight for each considered MS. It indicates monotonic decay of ỹ max /m by increasing m for the magnet system with 2 PM. At a MS weight of less than 320 g the required drag force is not achievable for the MS with 2 PM. Therefore, the corresponding curve in Fig. 14 starts at m = 0.32 kg.

12 90 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry Fig. 13. Maximal drag force ỹ max in relation to the MS weight m in case of MS with 2, 6, and 10 PM, respectively (see PM arrangements in Fig. 6). Fig. 14. Maximal drag force ratio to the MS weight ỹ max/m in relation to the MS weight m in case of MS with 2, 6, and 10 PM, respectively. Looking at Figs 13 and 14, it becomes obvious that the optimal magnet system weight in the case of a magnet system with 2 PM is 0.32 kg, since this ratio corresponds to the drag force equal to 10 μnandto the highest sensitivity. Corresponding design variables are given in Table 1. The calibration coefficient in this case is equal to c = Considering the magnet systems with 6 and 10 PM (see Figs 6b and 6c, respectively), the same

13 A. Alferenok et al. / Numerical optimization of the magnet system for the Lorentz Force Velocimetry 91 procedure was used to obtain optimal PM dimensions. One important assumption was made: All magnets have the same dimensions along x, y,andzaxis, meaning that again only three design variables were considered. The magnet system with 10 PM is more efficient than the other two magnet systems, since it provides a higher sensitivity at the same magnet system weight. It should be mentioned that the optimal design variables given in Table 1 remain valid also for other liquids, such as liquid metals, and for other flow rates. In this case, only the absolute values of the drag force andits ratio to the MSweightare changed. Theonly conditions, which needto be fulfilled are: (1) the cross-section dimensions of the channel and the distance between the magnets must be the same as in the present case; (2) the velocity profile in the channel must be turbulent, i.e. close to the solid body profile; (3) PM properties must be the same as in the present case. 4. Conclusion The numerical model for the LFV of electrolytes has been developed, verified, and validated. The optimization of the magnet system was done to obtain the maximal drag force for the given magnet system weight. Further work is necessary for the case of a higher number of design variables. For instance, a magnet system with 6 or 10 permanent magnets could principally have different design variables for each magnet. In this case, different optimization algorithms should be used. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft within the framework of the Research Training Group (RTG) Lorentz Force Velocimetry and Eddy Current Testing under Grant GRK 1567/1. The authors are grateful to Andrè Thess, Rico Klein, Bojana Petkovi`c, and Cindy Karcher for assistance in preparing this article. References [1] A. Thess, E.V. Votyakov and Y. Kolesnikov, Lorentz Force Velocimetry, Physical Review Letters 96 (2006), [2] A. Thess, Y. Kolesnikov, Ch. Karcher and E. Votyakov, Lorentz Force Velocimetry A contactless technique for flow measurement in high-temperature melts, Proceedings of 5th International Symposium on Electromagnetic Processing of Materials, 2006, pp [3] J.S. Choi and J. Yoo, Optimal design method for magnetization directions of a permanent magnet array, Journal of Magnetism and Magnetic Materials 322 (2010), [4] F. Herlach and N. Miura, High Magnetic Fields: Science and Technology 1 World Scientific Publishing Co. Pte. Ltd, [5] User s Guide for COMSOL Multiphysics v. 3.5a. [6] M. Ziolkowski and H. Brauer, Fast computation technique of forces acting on moving permanent magnet, IEEE Trans Magn 46(8) (2010), [7] X. Gou, Y. Yang and X. Zheng, Analytic expression of magnetic field distribution of rectangular permanent magnets, Applied Mathematics and Mechanics 25(3) (2004), [8] E.P. Furlani, Permanent magnet and electromechanical devices. Materials, analysis, and applications, Academic Press, [9] A.N. Mladenovi`c and S.R. Aleksi`c, Determination of magnetic field for different shaped permanent magnets, 7th Int. Symposium on Electromagentic Compatibility and Electromagnetic Ecology, 2007, pp [10] V. Minchenya, Ch. Karcher, Yu. Kolesnikov and A. Thess, Dry calibration of the Lorentz force flowmeter, Magnetohydrodynamics 45(4) (2009),

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