Lorentz force velocimetry using a bulk HTS magnet system: proof-of-concept

This paper presents a proof-of-concept of the idea of using bulk high-temperature superconducting (HTS) materials as quasi-permanent magnets that would form, in the future, an integral part of an advanced Lorentz force velocimetry (LFV) system. The experiments, calculations and numerical simulations are performed in accordance with the fundamental theory of LFV, whereby a moving metal rod passes through a static magnetic field, in our case generated by the bulk HTSs. The bulk HTS magnet system (MS) consists of two Y–Ba–Cu–O samples in the form of bulk cylindrical discs, which are encapsulated in an aluminium holder and wrapped with styrofoam. The aluminium holder is designed to locate the bulk HTS magnets on either side of the metal rod. After field cooling magnetisation with an applied field of 1.5 T at 77 K, the bulk HTS MS provides a quasi-permanent magnetic field over 240 s, enabling Lorentz force measurements to be carried out with a constant velocity of the metal rod. Two sets of Lorentz force measurements with copper and aluminium rods with velocities ranging from approximately 54–81 mm s−1 were performed. The obtained results, which are validated using a numerical model developed in COMSOL Multiphysics, demonstrate the linear relationship between the Lorentz force and velocity of the moving conductor. Finally, the potential of generating very high magnetic fields using bulk HTS that would enable LFV in even weakly-conducting and slow-flowing fluids, e.g., glass melts, is discussed.


Introduction
Velocity or flow rate measurement of fluids is of great importance to industry because it improves the reliability of accounting and facilitates stricter control of production processes. In particular, there is a growing need for non-invasive measurements, in view of the hygienic requirements in the pharmaceutical and food industries [1]. Furthermore, there is a demand for a non-contact measurement method in several other industrial sectors (see table 1): (i) liquid metals for high-quality aluminium, cooper and steel production [2]; (ii) glass melts for the manufacturing of high-quality glass products [3,4]; (iii) molten salt energy or storage systems [5].
Currently-used flow measurement techniques (e.g. magneto-inductive) require an electrical contact to the wall of the channel directly to the fluid [6,7], or when using optical Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. methods, the wall should be transparent. In addition, the fluid itself can impose limitations, especially when it is opaque, hot or chemically aggressive.
To overcome this limitation, Lorentz force velocimetry (LFV) has been introduced and studied in recent years, extensively in the pursuit of the non-invasive measurements of the velocity or flow rate in opaque and aggressive fluids [2,[8][9][10][11][12][13][14]. The LFV working principle relies on measuring the force, namely the Lorentz force, that is generated by the relative motion of an electrically conductive medium through a transversely applied magnetic field. The theory of LFV was introduced in [8] and expressed through the scaling law: where the Lorentz force F L depends linearly on electrical conductivity σ, mean flow velocity u, magnetic flux density B and a characteristic volume V. By measuring the resultant reaction force for a prescribed magnetic flux density and known electrical conductivity, it is possible to estimate the mean flow velocity of the fluid. It should be stressed that equation (1) holds for stationary flow or for the steady state motion of a solid conductor of a constant cross-section [8,9,15]. Figure 1(a) illustrates the LFV principle: the primary magnetic field (shown by the B) is generated by a magnet system and when the moving conductor passes through this magnet system, the interaction between induced eddy currents and the primary magnetic field causes a retarding force to act on the moving conductor [8]. In accordance with Newton's third law, a force (shown by the F L vector) with a magnitude equal to the retarding force, but in the opposite direction, acts on the magnet system along the moving direction. It has been shown that LFV works well for liquid metals due to their relatively high electrical conductance (σ∼10 6 S m −1 ) [2,9,10]. However, in case of weakly-conducting and slow-flowing fluids, LFV treats the measurement of tiny forces down to 10 nN, depending on the fluid velocity and electrical conductivity (see table 1). The experimental

Industrial sector
Fluid type Material inspection Solid metal 10 7 1 1 0 2 Metallurgy Liquid metal 10 6 1 1 0 1 Chemistry Acid, base 10 2 1 1 0 −3 Glass/semiconductor Glass melts 10 1 10 −2 10 −6 Pharmacy/food Ultra pure water 10 −6 1 1 0 −11 investigation of LFV for salted water (as a model fluid) of varying electrical conductivity and flow velocity was reported in several past works [11][12][13][14]. This was realised by employing a high-precision force measurement system (FMS), i.e. electromagnetic force compensated weighing cell (Sartorius Lab Instruments), in a combination with an optimised Halbach array of permanent magnets (PMs) [16] of 1 kg. The PMs were placed at the minimum distance from the flowing salted water, thereby providing effective interaction of the magnetic flux density with the flow volume. However, in the case of aggressive fluids (i.e. glass melts) the realization of close magnetic interaction with the fluid is not possible due to the large isolation walls. Furthermore, the surface magnetic flux density of the strongest available NdFeB PMs is limited to about 0.5 T [17]. Hence, even in early works on LFV [11][12][13][14]18], the prospective idea to employ bulk HTS magnets has been proposed, because of their ability to act as much stronger trapped field magnets. Record trapped fields over 17 T using field cooling (FC) magnetisation was achieved in [19,20], whilst a maximum trapped field over 5 T was reported in [21] using compact and fast pulsed field magnetisation. Furthermore, bulk HTS magnets continue to be investigated as a replacement for conventional PMs in a variety of applications [22][23][24][25][26][27][28][29][30][31][32][33][34].
In contrast to PMs, the bulk HTS magnet has a characteristic feature of generating a conically-shaped trapped magnetic flux density, B T , distribution, that has a sharp field gradient [35]. Whether this feature adequately replicates the LFV theory [8] (the linear relationship between F L and the product σ·u·B 2 in equation (1)) is one question to be addressed by current experimental efforts. Hence, the main objective of this study is to examine the feasibility of bulk HTS magnets as a substitute for existing NdFeB PMs in LFV.
The paper is organized as follows: section 2 presents the experimental setup, including the FMS and the design of the bulk HTS MS. This includes the method of cooling and magnetising the bulks as trapped field magnets, as well as the magnetic field distribution from the magnets. In particular, section 2.4 gives details on the Lorentz force measurements. A numerical model in COMSOL Multiphysics is then introduced in section 3 to validate the experimental results and the theory of LFV as applied to this system. Finally, section 4 discusses the obtained experimental results and ends with a view to further developing the system practically in section 5.

Overview
The experiments were carried out using a 'dry calibration' setup [10,36], developed at the Department of Engineering, Techniche Universität Ilmenau, but upgraded with the bulk HTS MS and an appropriate load cell for the force measurements. The idea of the 'dry calibration' implies the replacement of a fluid flow with the controlled motion of a solid conductor (e.g. metal rod) of a fixed geometry [10]. This enables a fairly straightforward way of carrying out fundamental LFV measurements.  (5) aluminium rack. The experimental procedure is fully automated: an IBA-Automation environment controls the linear drive and is used for data acquisition [37]. The linear drive executes a repetitive motion of the metal rod with a prescribed constant velocity. At first, it moves downwards, then reverses its direction and moves back to its initial position. The cylindrical long metal rods have a length of 1000 mm and a diameter of 40 mm. Two types of metal rods were used: copper with an electrical conductivity of σ Cu =58.96± 0.20 MS m −1 and aluminium alloy (AlMgSi) with σ Al = 19.43±0.03 MS m −1 [37]. The mean velocities of the copper rod were u=[54; 64; 76; 81] mm s −1 . For the aluminium rod, F L measurements with only two velocities of u=[70; 81] mm s −1 were possible, due to its lower conductivity and limitations related to the resolution of the present LFV measurement system. Note, the metal rod starts its motion being z=+165 mm distant from the centre of the bulk HTS MS, whilst the distances between the surfaces of the bulk HTS and the metal rod is y=+12 mm on both sides, as shown in figure 1(a).

Experimental set up
A commercial load cell (Model PW6D, Hottinger Baldwin Messtechnik GmbH) combined with an analogue measuring amplifier (SOEMER Messtechnik GmbH) was used for the force measurements. The force is measured in terms of a voltage with the accuracy class C3. In-situ force-voltage calibration was performed using E2 class certified calibration masses of 5, 10, 20 and 100 g. Details on the force measurement calibration are provided in appendix A1.

Bulk high-temperature superconducting magnet system
The bulk HTS MS consists of two Y-Ba-Cu-O samples in the form of a cylindrical disc with a diameter of 46 mm and a thickness of 16 mm, provided by Adelwitz Technologiezentrum GmbH. The Y-Ba-Cu-O bulks were encapsulated in an aluminium holder and arranged opposite each other with a distance of 64 mm between. The aluminium holders were wrapped in styrofoam and radiation shielding tape, in order to reduce warming to room temperature. A G-10 plate of thickness of 10 mm was used to prevent heat transfer from the HTS MS to the load cell.
The bulk HTS MS must be magnetised before coupling with the force sensor. The magnetisation was carried out in the room temperature bore of a superconducting solenoid magnet (Cryogenics Ltd., London, UK) using FC magnetisation: at first, an applied field of B A =1.5 T was generated; then, the bulk HTS MS was cooled down to 77 K. Cooling was achieved by immersing the entire HTS MS in an open styrofoam container with liquid nitrogen (LN 2 ), which was refilled periodically, due to the evaporation of LN 2 . Afterwards, B A was ramped down at a ramp rate of 0.092 T min −1 . A certain magnetic flux density B T can be retained in the bulks, because of the trapped flux and resultant induced superconducting current. When B A reduced to zero, the B T (hereafter B T indicates the B y -component of B) distribution was measured using a scanning Hall probe arrangement to characterise the bulk HTS.
At first, B T was measured for each bulk HTS separately, then for the gap between the two magnetised bulks in the HTS MS. The peak trapped magnetic flux density B T (y) for each bulk HTS as a function of distance (along the y-axis) is shown in figure 2(a). A maximum field of 1.08 T (left) and 0.8 T (right) at the bulk surfaces was recorded for each bulk HTS, respectively. The magnetic flux distribution B T (x, y) in the gap of the bulk HTS MS is presented in figure 2(b). It is qualitatively consistent with data reported in [24,[38][39][40], where a similar bulk HTS face-to-face arrangement was employed. The relatively large gap between the bulk HTSs in this study was required to enable the unobstructed motion of the metal rod through the bulk HTS MS (see figure 1). In future studies, this gap should be minimised for the proper utilisation of the magnetic field, since the maximum trapped magnetic flux density exponentially decreases with increasing distance (see figure 2(a)). To achieve this, a novel cooling system for the bulk HTS MS must be developed.

Lorentz force measurements
When the bulk HTS MS is coupled with the load cell and mounted to the LFV setup, continuous cooling in a LN 2 bath is not possible, because of the mass and volume restriction of the load cell. Hence, it was necessary during the force measurements to extract the bulk HTS MS from its LN 2 bath. The bulk HTS MS holds a trapped field for a specific period of time until it begins to warm up to room temperature, during which LFV measurements for one assigned velocity are performed. In addition, the temperature of the system during LFV measurements was monitored with a Cernox sensor.
An example of the periodic rod motion L(t) for a velocity of u=54 mm s −1 and the corresponding time sequence of the Lorentz force measurements is depicted in figure 3(a). The experimental procedure is as follows: In the future, a more practical design of the bulk HTS MS with robust and long-term operation is required at temperatures T<77 K, hence cryogenic refrigeration in a compact and light-weight cryostat is an issue to be resolved. Figure 3(b) shows the time dependent data for B T (t) and T(t) (see the arrow, different scales are used) in the gap of the MS in the vicinity (y≈25 mm, equivalent to 7 mm from the bulk surface) of the one bulk HTS. B T (t) remains almost constant up to 240 s, and then gradually reduces to zero due to warming at room temperature and flux creep [41]. Although the direct temperature measurement of the bulk HTS during the F L (t) experiments was not performed, its influence on the HTS bulks is indirectly included in the measured B T (t) dependence. Still, in order to obtain a correct force signal, the temperature of the load cell was measured: T(t) drops at the beginning from room temperature down to 240 K and remains constant. Figure 3(c) shows an example of the measured Lorentz force F L (t) for the prescribed velocity of 54 mm s −1 . F L (t) exhibits a periodic step function behaviour with the anticipated attenuation due to the above effects. At the beginning, a steep decrease in the amplitude of F L (t) is attributed to the temperature drop, while gradual attenuation is consistent with the change in B T (t). Obtaining experimental data under the conditions of constant B T (t) and T(t) is important, since any changes strongly influence the F L measurements [14]. Therefore, the estimation of the Lorentz force was calculated for the second period of metal rod motion (see figure 3(c) and its inset), where B T (t) and T(t) are nearly constant.
The detailed F L (t) response of the metal rod for the second period is presented in the inset of figure 3(c). Initially, the force is zero, when the metal rod is not moving. Then the metal rod approaches the bulk HTS MS resulting in an increase in F L . Stationary motion of the metal rod through the bulk HTS MS results in a plateau-like F L (t). Afterwards, the metal rod reverses the direction, yielding an analogous but negative force. Admittedly, some asymmetry between the positive and negative force signals arises because of asymmetric up-and down-movement (i.e. along z-axis) and offaxis alignment (i.e. along x-axis and/or y-axis), as was stressed in [15].

Numerical model
Numerical modelling is a powerful tool to validate and interpret experimental results and is useful for predicting the performance of bulk superconductors in practical applications [42]. To validate the experimental results in this work, a fully 3D model based on the finite-element method was implemented using the AC/DC module of COMSOL Multiphysics 5.3a. The geometry of the model is shown in figure 3, with the same dimensions as the experimental setup described in section 2. The electromagnetic properties of the bulk HTS magnets and the metal rod are implemented using the AC/DC module's magnetic and electric fields interface, satisfying Ampere's law: and current conservation, such that The bulk HTS magnets are assumed to be fully magnetised, utilising the 'external current density' node to assume a current density of constant J c [43]. The value of J c for each magnet is determined from the experimental trapped field measurements of the bulks (see figure 2(a)), where B T,1 =1.08 T and B T,2 =0.8 T, and the following equation based on the critical state model presented by Bean [35,44] and application of the Biot-Savart law [43]: where B T is the peak trapped magnetic flux density at the centre of the top surface of a c-axis oriented, single-grain bulk superconductor, μ 0 is the permeability of free space, J c is the critical current density of the superconducting material, and a is the sample radius. k is the correction factor to the simple Bean (slab) approximation due to the finite thickness, t, of a disc-shaped bulk superconductor: This results in an average, in-field J c for the two bulks of J c,1 =9.9 × 10 7 A m −2 and J c,2 =7.33 × 10 7 A m −2 . The resultant magnetic flux density within the rod cross-section across the centre of the bulk HTS MS is shown in figure 4(b). The calculated magnetic flux density is consistent with the experimental results shown in figure 2(b). The movement of the metal rod through the magnets is simulated by applying a velocity (Lorentz term) condition to Finally, the force is calculated by The calculated force for the two rods for different rod velocities is compared with the experiments in the following section (see figure 5(a)).

Results and discussion
With the aim of scaling the force and velocity according to equation (   velocity and the electrical conductivity of the moving metal rods, which is consistent with theory (see equation (1)) [8].
Additionally, the F L values obtained with same-sized copper and aluminium rods for one prescribed velocity, e.g. u=81 mms −1 , scales as σ Cu /σ Al . Unfortunately, in the case of aluminium, the induced Lorentz force approaches the resolution limit of the current LFV setup and therefore can result in significant measurement errors. Furthermore, it is shown that the experimental and simulation results agree well, validating the numerical model (time independent, 3D) as a fast and accurate tool to predict the LFV performance. Thus, the expected F L with respect to the electrical conductivity of relevant fluids, e.g. solid metals, acids/base and glass melts (see table 1), can be estimated in accordance with equation (1). It should be noted that the F L values are valid only for our particular design of the proof-ofprinciple LFV setup (see figure 1(b)) and may vary for each individual system differently: in particular, the spatial distribution of the magnetic flux density within the interaction volume of the moving conductor. Figure 5(b) shows the expected F L as a function of the electrical conductivity. With use of the present proof-of-principle LFV setup (the bulk HTS trap B T =0.94 T, i.e., the average of 1.08/0.8 T), the generated forces for weakly-conducting and slow-flowing fluids are in the range of μN and below. In particular, F L ∼10 −6 N for acids/bases and F L ∼10 −7 N for glass melts. It is immediately clear that the use of the bulk HTS MS with higher trapped fields of 3 and 5 T enhance the resultant F L over an order of magnitude and thereby the F L -resolution. This provides evidence that the bulk HTS MS is feasible and serves as a starting point for the future development of a new LFV prototype with improved performance.
Still, for practical LFV application to weakly-conducting and slow-flowing fluids, a high-precision force measurement in combination with the bulk HTS magnets is required. Measurements of such forces can be achieved by replacing the simplified load cell with a torsion balance based system for high-precision force measurement (TFMS), proposed in [45,46]. In order to raise the issue of the limiting total mass (which is always an issue for high-precision force measurements), the bulk HTS MS and TFMS can be merged within an integrated cryostat. This idea was patented by the authors [47] and the relevant portable LFV prototype is currently under construction.

Conclusions
The current work demonstrates the applicability of a bulk HTS magnet system (MS) to Lorentz force velocimetry (LFV) using an experimental setup, where the Lorentz force, acting on the moving metal rod, is measured by a load cell that carries the bulk HTS MS. The bulk HTS MS offers higher magnetic field than the previously used NdFeB PM-based system, with a similar mass, despite it requiring appropriate cooling and magnetisation. However, the magnetic field of the bulk HTS magnets strongly decreases with a distance, implying that proximity of the bulk HTS MS to the moving metal rod is important for optimal LFV performance.
The obtained experimental and numerical simulation results agree well, exhibiting the linear relationship between the Lorentz force and product of the electrical conductivity and velocity, in accordance with LFV theory [8]. Hence, these results serve as a starting point for the future development of a new LFV prototype with improved performance.
In particular, the LFV prototype currently under construction consists of bulk HTS magnets that are combined with the high-precision TFMS and encapsulated in an integrated cryostat. All these features enable a further extension of the LFV application to a number of industrial weaklyconducting and slow-flowing fluids, e.g. glass, molten salts melts and/or acid and lyes.
Finally, a simple, time independent 3D numerical model has been developed as a fast and accurate tool to predict the LFV performance, but a time dependent, dynamic model should be developed in the future that can consider more detailed superconducting properties, including different magnetisation process and flux creep effects.