A universal noncontact flowmeter for liquids

APPLIED PHYSICS LETTERS 100, 194103 (2012) A universal noncontact flowmeter for liquids André Wegfrass,a) Christian Diethold,a) Michael Werner,a) Thomas Fröhlich,a) Bernd Halbedel,a) Falko Hilbrunner,a) Christian Resagk,a) and André Thessb) Department of Mechanical Engineering, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany (Received 10 February 2012; accepted 15 April 2012; published online 10 May 2012) Lorentz force velocimetry (LFV) is a noncontact electromagnetic flow measurement technique for liquid metals that is currently used in fundamental research and metallurgy. Up to now, the application of LFV was limited to the narrow class of liquids whose electrical conductivity is of the order 106 S/m. Here, we demonstrate that LFV can be applied to liquids with conductivities up to six orders of magnitude smaller than in liquid metals. We further argue that this range can be extended to 103 S/m under industrial and to 106 S/m under laboratory conditions making LFV C 2012 American Institute of Physics. applicable to most liquids of practical interest. V [http://dx.doi.org/10.1063/1.4714899] In spite of remarkable advances in flow measurement,1–4 it is currently impossible to measure the flow rate of tap water in a copper pipe without mechanical contact between the flowmeter and the pipe. Whereas, metering of tap water does not represent a grand scientific challenge, there are a variety of less mundane situations where noncontact flow measurement through opaque walls or in opaque liquids would be highly desirable. Such applications include flow metering of chemicals, food, beverages, blood, aqueous solutions in the pharmaceutical industry, molten salts in solar thermal power plants,5 and high temperature reactors6 as well as glass melts for high-precision optics.7 Here, we demonstrate that Lorentz force velocimetry (LFV)8 which has been developed for a narrow field of application—namely flow measurement of liquid metals in metallurgy9—can be transformed into a universal noncontact flow measurement technique using ideas of Henry Cavendish10 and successors11–14 for the measurement of Newton’s gravitational constant G. Our simple flowmeter to be described below, whose measurement uncertainty is considerably larger than in state-of-the-art G-measurements, shows already that LFV can be applied to liquids with electrical conductivities from 106 S/m down to 1 S/m. After having presented our experimental results, we shall further argue using order-of-magnitude estimates that this range can be extended to 103 S/m under industrial and to 106 S/m under laboratory conditions making LFV applicable to most liquids of practical interest. Finally, at the end of the present work we will comment on the speculative historical question whether Michael Faraday’s unsuccessful attempt to measure the velocity of Thames River using electromagnetic induction15 could have been turned into a success if the principle of LFV was known to him. A noncontact flowmeter is a device that is neither in mechanical contact with the liquid nor with the wall of the pipe in which the liquid flows. Such flowmeters are highly desirable in applications where pipe walls are at temperatures above 1000  C like in glassmaking7 and metallurgy. Noncontact flowmeters are equally useful when walls are contaminated like in the processing of radioactive materials, when pipes are strongly vibrating or in cases when portable flowmeters are to be developed. If the liquid and the wall of the pipe are transparent and the liquid contains tracer particles, laser-based optical flow measurement techniques1,16 represent an effective and efficient tool to perform noncontact measurements. However, if either the wall or the liquid are opaque as is often the case in food production, chemical engineering, glass making, and metallurgy, very few possibilities for noncontact flow measurement exist. To accurately define our terminology, it should be noticed that ultrasonic Doppler velocimetry2 does not formally belong to the family of noncontact flow measurement methods since it requires mechanical contact between the transducer and the wall of the pipe. Inductive flowmeters do not belong to the category of noncontact flowmeters3 either because they require contact between the sensing electrodes and the liquid or between the electrodes and the wall of the pipe. LFV is an electromagnetic noncontact flow measurement technique that was invented in the 1960s (Ref. 3) and that has been recently developed to the state,8,17,18 where its application in aluminium production9 and steelmaking19 is imminent. The basic principle of LFV is explained in Figure 1(a). When an electrically conducting fluid, for instance liquid aluminium or salt water, is exposed to the action of a magnetic field produced by a permanent magnet, eddy currents are induced in the fluid.20 The eddy currents and the magnetic field create a Lorentz force which brakes the fluid. Due to Newton’s third law, a force with equal strength but opposite direction must then act upon the magnet. It was shown in Ref. 8 that the Lorentz force F can be written in the form a) Author contributions: B.H., F.H., T.F., C.R., and A.T. designed the research; A.W., C.D., and M.W. performed the research and analysed the data; A.T. wrote the manuscript and all authors commented on it. b) Author to whom correspondence should be addressed. Electronic mail: [email protected]. 0003-6951/2012/100(19)/194103/4/$30.00 F ¼ crqB2 L: (1) Here, c is a dimensionless calibration constant that depends on the shape of the magnet system and the flow geometry, r 100, 194103-1 C 2012 American Institute of Physics V Downloaded 10 May 2012 to 141.24.80.51. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions 194103-2 Wegfrass et al. FIG. 1. Schematic of the noncontact Lorentz force flowmeter used in the experiments: (a) A turbulent flow of salt water (with Reynolds numbers roughly between 3.2  104 and 1.3  105) is exposed to the magnetic field generated by a lightweight permanent-magnet system hanging on a fourwire pendulum. The displacement of the pendulum is measured using an interferometer as sketched in (b): 1 – He-Ne-Laser, 2 – beam splitter, 3 – reference corner cube reflector, 4 – photo detector, and 5 – measurement corner cube reflector. is the electrical conductivity of the liquid, q is the volume flux, B is the magnetic flux density at a certain reference point in the fluid, and L is the hydraulic diameter of a pipe. LFV consists in measuring the force F and deducing the unknown flow rate q from this measurement. The following numerical example illustrates why the application of LFV to liquid metals is straightforward, whereas its application to weakly conducting fluids like glass melts, food, and chemicals has remained an unresolved challenge. Assume that c ¼ 1, q ¼ 103 m3/s, B ¼ 0.1 T, L ¼ 0.1 m, and that the mass of the permanent magnet is 1 kg. Then, Eq. (1) predicts that F ¼ 1 N for a liquid metal with r ¼ 106 S/m, whereas F ¼ 1 lN for a glass melt with r ¼ 1 S/m and F ¼ 5.5 pN for ultrapure water with r ¼ 5.5  106 S/m. Hence, the ratio between the useful Lorentz force F and the unavoidable dead weight FG of the magnet (FG ¼ mg) is F/FG  101 for liquid metals, F/FG  107 Appl. Phys. Lett. 100, 194103 (2012) for glass melts and food, and F/FG  5.5  1013 for ultrapure water. These figures demonstrate that in order to apply LFV to weakly conducting fluids one must measure tiny Lorentz forces acting upon comparatively heavy permanent magnets—a task that requires force measurements with similar accuracy as those used in the experimental determination of Newton’s gravitational constant G.10–14 We demonstrate the feasibility of LFV in weakly conducting liquids by setting up the experiment sketched in Figure 1. A turbulent flow of salt water21 with electrical conductivities in the range 2.3 S/m  r  6.2 S/m is set up in a channel with rectangular cross section whose walls are made of glass. The height of the channel is 50 mm and the width of the channel is 30 mm. The wall thickness is 2 mm and hence the cross section of the liquid is 26  46 mm2. The flow is exposed to a non-uniform magnetic field created by a magnet system consisting of two identical blocks of NdFeB permanent magnets held together by an aluminium frame. The size of the permanent magnets is 30 mm  30 mm  70 mm, the gap between the poles is 32 mm, and the total mass of the magnet system is m ¼ 1.286 kg and thus FG ¼ 12.62 N. Numerical computations of Lorentz forces acting upon magnet systems with different shapes and yokes interacting with a translating electrically conducting solid bar have indicated that a lightweight non-ferromagnetic yoke made of aluminium is superior to an iron yoke. The superiority is expressed by the fact that F/FG—which is the key optimisation parameter for magnet systems in LFV—is higher for the aluminium yoke than for an iron yoke. The maximum of the transverse component of the magnetic field in the vertical symmetry plane (halfway) between the two magnet poles is 0.410 T. Details of the magnetic field distribution are provided as supplementary material.28 The magnet system is attached to an aluminium frame by four tungsten wires with a diameter of 125 lm and is free to oscillate as a pendulum as sketched in Figure 1(b). The restoring force due to the stiffness of the wires is much weaker than the restoring gravity force and is therefore neglected. It is straightforward to show that the Lorentz force F displaces the pendulum by x ¼ k1F, where the “spring constant” of the pendulum k ¼ (22.94 6 1.21  102) N/m (at the 2sigma uncertainty level22) is determined by the equation k ¼ mg/l, where l ¼ 0.550 m is the length of our pendulum and g ¼ 9.810151 m/s2 is the acceleration of gravity in the laboratory where the experiment was performed (latitude 50 400 51.600 N, longitude 10 560 3.8400 E, height 474 m). The value of g is taken from the Gravity Information System of the Physikalisch-Technische Bundesanstalt Braunschweig (Germany). The measured frequency of free oscillation of the pendulum 0.682 Hz was found to be in agreement with the value (0.672 6 3.53  104) Hz computed from the values of l and g given above. The horizontal displacement x of the magnet under the action of the flow is measured using a commercial laser interferometer23 with a corner cube mirror fixed to the pendulum. This interferometer has a resolution of 1 nm. All parts of the experiment are attached to a nonferromagnetic granite block which is in turn embedded in a box filled with sand in order to suppress vibrations coming from the environment. The mean velocity of the salt water flow is determined using an ultrasonic flowmeter. Downloaded 10 May 2012 to 141.24.80.51. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions 194103-3 Wegfrass et al. Appl. Phys. Lett. 100, 194103 (2012) 40 0.3 30 0.25 25 F [µN] c 0.35 35 0 1 2 3 4 5 Q [l/s] 20 15 10 5 0 0 0.5 1 1.5 2 2.5 u [m/s] 3 3.5 4 FIG. 2. Results of measurements and simulations: Lorentz force as a function of the volumetric flow rate of salt water with electrical conductivities 2.3 S/m (red), 4.0 S/m (blue), and 6.2 S/m (green) as obtained from the experiments (symbols) and from numerical simulations of a steadily translating solid body (lines). The inset shows the experimental values of the calibration constant (dots) and the numerical value (horizontal line) as a function of the volume flux in litres per second. The calibration constant c is determined as an average over all shown measurements. For fixed volume flux, the Lorentz force is proportional to the electrical conductivity. The results of the measurements are plotted in Figure 2. The experimental data show that the Lorentz force is an almost linear function of the mean velocity u, as expected. The data also confirm that the slope of the function F(u) increases with increasing electrical conductivity of the salt water. This demonstrates the feasibility of contactless electromagnetic flow measurement in a weakly conducting liquid. Notice that other electromagnetic flow measurement methods involve either mechanical contact such as the traditional inductive flowmeter3 or have insufficient sensitivity for electrolytes such as eddy current flowmeters17,24 or contactless electromagnetic flow tomography.25 The validity of the experimental data is confirmed by the results of numerical simulations plotted as linear functions in Figure 2. Given the fact that the exact velocity profile is unknown in the present experiment, the numerical simulations are carried out under the assumption that a solid body with the same electrical conductivity as the salt water undergoes a translational motion with a velocity equal to the mean velocity of the liquid. The numerical simulation includes a three-dimensional solution of the full magnetostatic problem as well as an eddy current computation. Based on Eq. (1), it is possible to determine the calibration constant of our Lorentz force flowmeter from the experimental data as cexp ¼ 0.2863 6 0.1020. In calculating cexp, we use the reference values B ¼ 0.383 T and L ¼ 0.03 m. The numerical results are in good agreement with the experiments which is exemplified by the corresponding value cnum ¼ 0.2840 of the numerical calibration constant. The validity of the calibration formula (1) permits us to draw several far reaching conclusions about the potential applicability of LFV to weakly conducting liquids other than salt water. The present flowmeter prototype is simple and inexpensive in comparison to pendulum experiments measuring G,14 yet it already allows measurements with a resolu- tion of approximately F/FG ¼ 107. If our displacement measurement system was upgraded to that used in Ref. 14, the resolution would increase by two orders of magnitude. Recent design studies and numerical simulations26 indicate that a further optimization of the magnet system combined with flattening the cross section of the channel would increase the product cB2 by another order of magnitude. These measures would increase the resolution to F/FG ¼ 1010 and allow a further extension of the applicability of LFV from r ¼ 1 S/m to r ¼ 1 mS/m thereby covering the vast majority of all industrially relevant liquids including slugs, slurries, and other complex systems. More sophisticated improvements such as lightweight magnet systems based on high-temperature superconductors and replacement of the four-wire pendulum by a torsional pendulum27 as in Cavendish’s original experiments10 and in Ref. 11 could increase the resolution by another three orders of magnitude, i.e., to F/FG ¼ 1013 under laboratory condition and would allow measurements for conductivities as small as r ¼ 1 lS/m. In 1832, Michael Faraday15 attempted to determine the velocity of the Thames River near Waterloo Bridge by measuring the electric potential difference induced by its flow across Earth’s magnetic field lines. It has been speculated8 that this famous but unsuccessful experiment could have been turned into a success using Henry Cavendish’s force measurement technology if Faraday was aware of the LFVprinciple. Faraday could have placed a horseshoe magnet over the water and measured the Lorentz force acting upon this magnet. Such measurement would have required a resolution of the order F/FG ¼ 108 which is roughly equal to that in Henry Cavendish’s experiment10 in 1798. Our work demonstrates that such measurement is more than a mere theoretical possibility and could be actually carried out. It would be an intriguing and challenging task to actually perform the hypothetic “Faraday-Cavendish experiment” at Waterloo Bridge on the occasion of the 200th anniversary of Faraday’s original experiment to be celebrated in 2032. The authors gratefully acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft) for the Research Training Group (Graduiertenkolleg) “Lorentz force velocimetry and Lorentz force eddy current testing.” We thank Gerd Jäger, Jörg Schumacher, Ulrich Lüdtke, and Artem Alferenok for useful discussions. 1 C. Tropea, A. L. Yarin, and J. F. Foss, Handbook of Experimental Fluid Mechanics (Springer-Verlag, GmbH, 2007). 2 R. C. Baker, An Introductory Guide to Flow Measurement (Wiley Verlag, 2002). 3 J. A. Shercliff, Electromagnetic Flow Measurement (Cambridge University Press, 1962). 4 D. Bonn, S. Rodts, M. Groenink, S. Rafai, N. Shahidzadeh-Bonn, and P. Coussot, Annu. Rev. Fluid Mech. 40, 209–233 (2008). 5 U. Herrmann, B. 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Mech. 38, 79–92 (2012). 27 C. Diethold, internal report (unpublished). 28 See supplementary material at http://dx.doi.org/10.1063/1.4714899 for more details about the magnetic field distribution between the two magnet poles. 20 Downloaded 10 May 2012 to 141.24.80.51. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions