Lorentz Force Flowmeter for Liquid Aluminum: Laboratory Experiments and Plant Tests

The basic calibration equation for an LFF is expressed as follows:where F is the measured Lorentz force and q is the unknown volumetric flow rate. B is the amplitude of the magnetic field at an arbitrary fixed point in the liquid metal, T _M is the temperature of the magnet system, and c is a calibration factor that depends on the geometry of the channel as well as on the spatial structure of the magnetic field. In this equation, which is derived from Eq. [2], the force is in Newton (N). In the general case, the value of the electric conductivity σ depends on the temperature T _L of the liquid metal. Measurements of σ(T _L) in molten aluminum alloys (e.g., AlSi12)[15] show that this quantity varies by approximately 4.5 pct when the temperature varies in the range 973 K to 1073 K (700 °C to 800 °C). However, during the transport processes described in the introduction, the variation of the temperature is usually smaller and confined to the range 1023 K to 1053 K (750 °C to 780 °C); therefore, it follows that within the stated limits of operating temperature, the conductivity σ can be variable within 1.25 pct. If the temperature is permanently measured, which is often the case in production processes, then the variation of conductivity can be accounted for by introducing temperature dependence for conductivity into the measurement software. In industrial conditions, the magnet system of the LFF is greater than room temperature because the system is situated in the immediate vicinity of the channel in which the high-temperature liquid metal flows.

In our LFF prototypes, we use commercial rare-earth permanent magnets consisting of NdFeB. To characterize the dependence of their magnetization on temperature, we carried out a preliminary experiment in which a single block was exposed to different temperatures. We found that the magnetic induction decreased by 8.5 pct when the temperature of the magnets was increased from 293 K to 363 K (20 °C to 90 °C). The experiment was carried out with a magnet that has dimensions of 30 × 20 × 100 mm. An indication of the temperature was produced by a standard thermoresistor placed on the magnet surface. The temperature was registered by a multivoltmeter. For an accurate calibration, it is necessary to introduce a field dependence on temperature into the force calculation program covered by Eq. [3]. In particular, the dependence is calculated according to the following:(B ₀ is the magnetic field induction at the initial magnet temperature T _0M = 293 K (20 °C)). As follows from our preliminary experiment, the coefficient of temperature variation of the magnetic field is α = 1.116 × 10⁻³ (K⁻¹).

An industrial channel for liquid aluminum transportation on the external wall surface has a real temperature that does not exceed 323 K (50 °C). This is a consequence of the large thickness of 60 to 80 mm of the walls made of concrete, of their comparatively low thermal conductivity of about 0.5 W/m K, and of natural air convection around the channel. Taking this into account, we are led to the conclusions that the magnetic induction deviation does not exceed 1 to 1.5 pct. Nonetheless, to monitor a possible temperature variation in the industrial tests, we install a thermocouple on the surface of magnet system.

The relationship for the calibration constant c results from Eq. [3] as follows:The dimension of c is meter. Physically, c can be interpreted as a lengthscale characterizing the magnetic field nonhomogeneity along the flow direction. The calibration constant for a given LFF can be determined either by a numerical simulation or experiments in which the flow rate q is known. In this article, we determine c experimentally both for laboratory and industrial LFFs.

To determine c for our laboratory LFF described in Section IV–A, we measure the Lorentz force F directly for different volumetric flow rates q that are set up in a room-temperature, liquid-metal test channel. We choose the magnetic field induction B ₀ equal to its value on the flow axis. In Section IV–B, for the calibration of our industrial LFF, we use one of two tests at aluminum production in a plant in which the total mass of discharged aluminum is known. In this case, we also measure Lorentz force and use the cumulative weight of aluminum in the end of the process to compute its flow rate through the channel.

Once the electrical conductivity is known and the calibration constant c has been determined, the time-dependent volumetric flow rate q(t) can be determined from the measured signals from the force-sensor F(t) as well as the measured temperature of the magnet T _M(t) by applying Eq. [3] as follows:From the volumetric flow rate in a time span from t = t ₀ to t = t ₁, one can compute the cumulative liquid metal volume as follows:

Combining Eq. [6] with the fluid density yields the following mass flow rate:From Eq. [7], the comulative mass within a given period of time is determined as follows;

Because of the applied magnetic field, a drag force (Lorentz force) occurs in the gap between the LFF magnet poles. Depending on the magnetohydrodynamic parameters, this force can exert some influence on the flow of the liquid metal. To estimate the corresponding drag coefficient, we suppose that an induced potential difference vBh (where v is flow velocity and h is a channel width in direction perpendicular to the magnetic field) generates the eddy electric currents as shown in Figure 1(b) and follows from Eq. [1]. A pressure difference Δp = k ₀σvB ² h is then associated with these currents (Eq. [2]). The factor k ₀ depends on the spatial decay of the magnetic field[14] and is estimated as k ₀ = 0.1. Using the general formula for drag coefficient, we obtain the following:where the nondimensional number N is called the interaction parameter and represents the ratio of Lorentz and inertia forces. In the case N > 1, one would have an appreciable pressure loss.[7] As a result, in open channels, the free surface can be inclined within the magnetic field area, which can lead to a deterioration of measurement accuracy. For the industrial tests in our work the condition, N << 1 is always satisfied (Table II) so that the free surface of flows is not distorted. The case N > 1 is acceptable for closed channels when a deformation of velocity profile does not influence the accuracy of measurement.

The experiment is carried out with a laboratory LFF and mentioned working fluid Ga^68 pctIn^20 pctSn^12 pct with the properties given in Table I. The aim of this experiment is to determine a procedure of calibration factor c for a well-defined flow for which the flow rate can be measured and monitored accurately.

The laboratory model of the LFF consists of a magnetic system with two permanent magnets installed on an iron yoke as shown in Figure 2(a), of a mechanical system for force transmission (not shown), and of electronic commercial scales with a measuring accuracy of 10⁻⁴ N. Both magnets are made of NdFeB and have dimensions of 30 × 20 × 100 mm and a magnetic field induction of 550 mT on the surface with a size of 30 × 100 mm. The magnetic gap of the system is 35 mm. The yoke of 20 mm in thickness provides an increase of magnetic field induction in the middle point of gap by 15 pct. The overall weight of the magnet system is 2.52 kg. The experimental channel is placed into the gap of the magnet system from earlier.

The distributions of the magnetic field along the channel height and in the direction of flow are shown in Figures 2(a) and (b). In vertical direction z within the height of channel, the nonhomogeneity of the field does not exceed 2.5 pct, whereas nonhomogeneity of field in flow direction x is strong. High nonhomogeneity along the flow is favorable to the production of intensive electric currents j as shown in Figure 1(b) and, consequently, to the generation of a large Lorentz force and the same force acting on magnet system.

The mechanical part of the LFF is a vertical metallic rod 500 mm in length linked with the magnetic system in its upper end and fixed in its middle point on a horizontal rotational axis. The point in which the force acts on the magnet system is at a distance of 305 mm from the rotational axis. The other end of the rod is linked with a balance weight for a stable equilibrium of the system at the initial instant of time of measurement. An additional horizontal rod of 175 mm in length is connected to the main vertical rod on the axis, and the other end is attached to the scales. The balance weight is chosen so that the imbalance of the mechanical system in the absence of measured force does not exceed 10⁻² N. This provides a high accuracy of force measurement because in a range of forces up to 10 N acting on the magnet system a deformation of scales sensor is less than 100 μm (i.e., imbalance of mechanical system at this deformation remains practically previous).

A sketch of the whole experimental setup is presented in Figure 3(a), where the closed liquid metal loop 1 made of Plexiglas with a constant cross section of h × d = 80 × 10 mm is shown. The whole loop has a length L ₁ = 830 mm and a width L ₂ = 330 mm. The total volume of the loop is 1860 cm³. The loop is filled by the eutectic alloy from a tank (2). An electromagnetic disk pump consisting of rotating permanent magnets (3) develops a pressure up to 1.0 bar and a volumetric flow rate of 1.0 L/s. An electronic controller for the pump motor allows a reverse pumping in the loop and registration of pump rotation speed to bring these data into accord with mean velocity. To prevent heating of the liquid metal by the pump, a cooling system is connected to the short branches (L ₂ = 330 mm) of the loop. Cooling water passes outside over these walls, which are made of copper.

A Vivès probe (5; Figure 3(a)) is installed in the middle point of the channel cross section at a distance of 410 mm from the cross-section inlet. A signal from the Vivès probe is fed into a multivoltmeter 2700 DMM (Keithley Instruments, Inc., Cleveland, OH) (6) and then to a computer for mean velocity calculation.

The signal of electronic scales (7) measuring a force on the magnet system is fed into the computer. The LFF prototype (8) is installed at a distance of 70 mm downstream from the Vivès probe (5). To examine the flow shape influence on readings of the LFF, we use a controlling gate (9), which is an obstacle partly blocking the cross section.

We apply the Vivès method[16] for a definition of the mean velocity to find the volumetric flow rate q shown in Eq. [3]. We reach this definition by measuring an averaged velocity v. To use this velocity for a definition of mean velocity, we check the homogeneity of the velocity profile over the cross section using an ultrasonic velocity profile (UVP) method.[17] For such an examination, an UVP sensor of 8-mm diameter is installed in the hole (9; Figure 3) from earlier onto the upper surface at a distance of 250 mm from the end of the test-section length L ₁. The measurements show that the vertical nonhomogeneity of the velocity profile is as small as 5 pct for both direct and reverse flows. The design of the Vivès probe (Figure 3(b)) is a combination of a small cylindrical permanent magnet of 4-mm diameter and length as well as four copper potential electrodes a, b, c, and d of 0.25 mm in diameter that measure two velocity components described by the following Eqs. [11] and [12]: where Δφ_ac and Δφ_bd are the electrical potential differences between sensitive points of electrodes, l _ac and l _bd are the distances between them, and B is the magnetic field induction generated by the permanent magnet. These relationships result from Ohm’s law (Eq. [1]). Of the components, v coincides with the mean flow, and the other component w is parallel to the vertical direction. Equations [11] and [12] may be simplified as follows: where c _v and c _w are calibration factors. These factors take into account the intensity and configuration of the magnetic field as well as distances between electrodes. For probe calibration, we use a flow created by a rotating annular channel of 415-mm diameter filled by the same eutectic alloy as in the experiment. At different rotation speeds, the probe immersed in the alloy gives linear dependences Eqs. [13] and [14] for corresponding components obtained by a probe orientation to the flow (Figure 3(b)). Calibration shows that the factors c _v and c _w are 7.48 × 10³ (m/s V) and 3.35 × 10³ (m/s V), respectively, if the potential difference is taken in volts and velocity components in m/s. Measurements by the probe in the test-section of the loop shows that the w component is 50-fold smaller than the v component. To define the volumetric flow rate, the signal Δφ_ac passing through the voltmeter is applied to a computer in which, according to Eq. [13] and knowledge of the area of flow cross section, the flow rate q = v × A (where A denotes the area of flow cross section and v denotes mean velocity) is computed. For instance, the electrical signal Δφ_ac = 10.1 × 10⁻⁵ volts corresponds to an average maximum velocity of (v _a )_max = 0.755 m/s. With regard to the realization of a turbulent regime in the experiment (Table II), a mean velocity is v = 0.625 m/s ((v _a )_max = 1.21 × v for turbulent regime, and calculating the flow rate gives q = 5 × 10⁻⁴ m³/s (0.5 L/s). A similar measurement in the GaInSn alloy flow is described in a recent article.[18]

To investigate the influence of the flow profile on the flow rate measurement, we obstruct the cross section to 0 pct, 50 pct, and 75 pct by inserting a solid obstacle that blocks the cross section while keeping the flow rate constant, as shown in Figure 4. In contrast to the present work, in Reference 5, a LFF has been used with two pairs of similar magnets separated by 50 mm from each other downstream (five characteristic sizes of flow) using an iron yoke on. In this case, generated Lorentz force attains a value of 10 N. The data in Figure 4 demonstrate the value of the force two times smaller than in Reference 5. The force acting on the magnet system is a linearly increasing function of the flow rate, as anticipated (Eq. [3]). Linearity of force is the result of flow rate q constancy (i.e., at a decrease of flow cross section, a proportional increase of velocity takes place). It is true if the magnetic field of the LFF accomplishes a high vertical homogeneity and covers the whole height of flow. This circumstance is important to designing a LFF intended for the measurements in the closed channels because it is desirable for a LFF with an output independent of the shape of the velocity profile. At first glance, in practice, one would expect that the influence of the incoming profile would be significant. It should be also noted the works,[19,20] where regimes of a generation of the large flow structures influencing a Lorentz force, are studied.

The results of the direct measurements of force as a function of F(q) shown in Figure 4 give a value of a force coefficient k = F/q = 9.612 × 10³ N s/m³ that should be put into Eq. [5] to compute the calibration factor c. Using the electrical conductivity σ of the working fluid and the magnetic field induction B ₀ listed in Tables I and II, we obtain the following numerical value:for the calibration coefficient of our laboratory LFF. The numerical value is c = 1.605 × 10⁻² m. This fact reflects an important scaling property of LFF because it defines a characteristic half-length of magnetic field nonhomogeneity in flow direction.

In this subsection, we describe the measurements performed using an industrial LFF prototype shown in Figure 5 within two production processes in a plant of a secondary aluminum manufacturer. All measurements are conducted in an open channel 1 intended to transport molten aluminum 2 from a rotary furnace. The channel made of a concrete is supported by a nonmagnetic steel frame. The LFF prototype comprises a V-shaped magnet system with an opening angle of 60 deg, a mechanical system based on a pendulum principle, electronic scales with an accuracy of 10⁻⁴ N, and computer to plot the measured Lorentz force as a function of time.

The form and dimensions of the magnet system are in agreement with a geometry of the production channel (Figures 5(a) and (b)). Each magnet pole 3 consists of 16 NdFeB magnets with sizes of 30 × 30 × 100 mm. The poles are linked to each other by an iron yoke 4. Each pole area facing the lateral channel wall is a × b = 100 × 260 mm (where a coincides with aluminum flow direction). The thickness of the poles is 60 mm. The distance between the low edges of the poles is 300 mm, whereas the upper edges are 550 mm apart (Figure 5(a)). The magnetic induction on the magnet surface is 450 mT. The magnetic field B at a height of 50 mm from the channel bottom in the middle plane is 13.5 mT. The magnet system is linked to a counterweight (5) by a steel rod (6) that can turn around the axis (7). The force F is measured by commercial scales (8), and a digital signal from scales is fed to the computer (9) to compute the volumetric flow rate, mass flow rate, and accumulated volume and mass of aluminum.

The magnetic field in this LFF at first glance is represented as nonhomogeneous along the height for such a magnet system design. However, a special measurement of magnetic induction inside the channel directly before the industrial tests demonstrates a magnetic field homogeneous enough along the channel height of 0…125 mm with a deviation of 1.13 pct. A height of flow can vary between 65 mm and 80 mm. Such homogeneity is a result of the affinity of the horizontal part of the iron yoke to the lower edges of magnet poles as shown in Figure 5(a), which weakens this area transversal component of the magnetic field that provides equalization of the field along the vertical. In regard to the field distributions in two other directions, they are also identical for both industrial tests. Thus, for both tests used, factor c reflects magnetic field nonhomogeneity in flow direction. In the industrial flow meter with a large distance between poles, the magnetic field is 30 times less than in the laboratory case. However, the much bigger flow rate in industrial tests takes place; therefore, this point provides a reliable measurement of Lorentz force in industry (Table II).

The positions of the main channel and the installed LFF from test to test do not vary. We emphasize that the channel and LFF are separated from each other with a gap of 10 mm between the magnet poles and the channel wall outer surface. The LFF is installed on a concrete foundation. Because of this, a channel position relative to LFF is rigorously fixed.

The weight of magnet system is 40 kg. The counterweight, also about 40 kg, provides equilibrium of the mechanical system. In the same manner as the laboratory experiment, we properly select a mass of counterweight to realize an initial unbalance of the mechanical system equal to 10⁻² N.

As shown in Figure 5(b) the eddy currents j result in the vicinity of the edges of the magnetic field. Because the channel walls are nonconducting, the electric currents flow wholly inside the fluid. We emphasize that the aluminum oxide film on the upper surface of flow is motionless and plays a similar role as a nonconducting wall (i.e., eddy currents are fully closed in the liquid aluminum as sketched in Figure 5(b)). These currents are considerable as they are closed in the fluid in which the magnetic field is absent and an electromotive force is not generated. Therefore, the confined magnetic field provides a measurement of the Lorentz force by LFF at even a small induction of magnetic field.

The operating temperature of liquid aluminum in both tests is approximately 1053 K (780 °C). Despite this, the temperature of the external surface of the channel walls near the LFF prototype does not exceed 323 K (50 °C); it makes cooling the magnet system unnecessary. To take into account the variation of the magnetic field induction in consequence of the magnet temperature changes (Eqs. [4] and [5]), we place a thermocouple on the magnet surface for a temperature correction of the processing data.

Two tests are executed. Test 1 is used as a base to define factor c (Eq. [5]), using the known alloy mass received in this production cycle. It should be noted that this factor keeps a constant value if the magnetic field along the flow height changes within 1.13 pct (see previous sections). In the case of growth of the flow rate, an increase in the top level of flow adds a force according to Eq. [3].

We note that, in another case of nonhomogeneous magnetic field along the height, lifting of the flow level reaches an area of weaker magnetic field and gives a smaller increase in the Lorentz force (i.e., the factor c seems underestimated). Therefore, we give a special attention to the vertical distribution of the magnetic field, the precise position of LFF, and inspection of the upper levels of the aluminum flows in the operating channel at a distance of the LFF installation (3.5 m from the channel inlet) in the beginning and during both tests. It is important because, each time after aluminum transportation, the position of the channel outlet, which influences the height level of flow in the channel, is technically compelled to be disassembled for cleaning and reconditioning between the two transportation processes.

The results of the measurements are shown in Figure 6 where we plot the force acting on the LFF as a function of time. In both experiments reported, similar amounts of aluminum are discharged. Measurement 1 is characterized by a slow discharge, whereas in measurement 2, the liquid metal is discharged with a higher flow rate. It is clearly shown that the higher flow rate in curve 2 results in a higher Lorentz force than in curve 1. Moreover, the surfaces under curves 1 and 2 visually appear virtually equal in agreement with the fact that the cumulative masses discharged in tests 1 and 2 are similar.

Strictly speaking, the calibration Eq. [3] is only valid if the cross section of the flow is constant; in which case, the flow rate q is related to the mean velocity v and the cross section A by q = vA, where A is a constant. In the open channel flow typical of secondary aluminum production, however, the cross section changes because of level fluctuations of the liquid metal. Hence, the calibration constant weakly depends on the level of the liquid metal and thereby on q. In what follows, we assume that this dependence is so weak that c can be regarded as a constant. We then can determine c from the cumulative mass or volume.

The procedure is as follows. We carry out the measurements of Lorentz force F within the process shown in Figure 6, curve 1, and compute an average value F _av during a process time Δt = 1119 seconds by integrating F(t). We find that F _av = 8.26 × 10⁻² N. From weighing the discharged aluminum, we know its total mass M = 8350 kg. From these two quantities and the tabulated value ρ = 2370 (Table I) for the density, we obtain the average volume flow rate as follows:where q = 3.148 × 10⁻³ m³/s. In the same manner as the laboratory experiment, we calculate a force coefficient k = F _av/q = 26.24 N s/m³. Following Eq. [15], we use the values of k, of the electrical conductivity σ, and of magnetic field induction B from Table I to obtain the desired calibration factor as follows:This figure for c is received through the direct measurement of alloy weight within basic test 1 and now is the characteristic of our LFF, and we apply it for the next measurements of flow in test 2. In fact, the flow meter in this position becomes an instrument for measurement in the next cycles of aluminum flow rate. Curve 2 in Figure 6 is a result of the measurement of volumetric flow rate in test 2 by using of obtained factor c written in Eq. [17].

Curves 1 and 2 in Figure 6 distinctly indicate the beginning and end of two processes with a good accuracy. In the beginning of test 2, a sharp maximum resulting from a sudden flow from the furnace is observed. Fluctuations of flow feeding from the furnace generate a fluctuating force measured by LFF. Furthermore, the flow is turbulent, which is observed by inspecting the industrial Reynolds number given in Table II. We note that the LFF also measures the large-scale perturbations in the flow with a frequency in the range of 0.3 to 0.6 Hz as shown by the inset in Figure 6. Mechanical channel vibrations are excluded because of the special support of the LFF (operating channel and LFF are not connected mechanically). Therefore, the cause of such fluctuations can be linked to large-scale intensive turbulent pulsations generated by jet flow that comes into the channel from the furnace. The oscillations of a free surface of flow can serve as an additional contribution to fluctuations. However, the mean force signal directly indicates the mean flow rate. The ratio of amplitude of force oscillated by large-scale turbulent pulsations to a mean force f′/F is less than 5 pct. Slow fluctuations of the mean force during the process sometimes can reach 12 pct (see both curves in Figure 6). We note that observed fluctuations play no essential role in receiving an accumulated mass during the process (Figures 7 and 8).

To show the role of natural oscillations of LFF and channel in pulsating signal, we estimate their frequency. The magnet system with an arm length of L = 0.35 m and a mass of M = 80 kg possesses an inertia moment I = 10.17 kg m² that leads to a natural frequency of oscillations f′ _0/mag = 1.17 Hz. These oscillations do not coincide with measured oscillations (Figure 6). Estimations of inertia moment for the channel of M = 1250 kg in mass and L = 10 m in length give a value of inertia moment of I = 5218 kg m². These parameters provide the natural frequency of oscillations f′ _0/ch = 0.55 Hz. This last frequency coincides in value with the legitimated frequency but is related to the channel oscillations in the vertical direction. In this connection, it should be noted that the LFF measures streamwise velocity oscillations and the mean velocity in this direction (i.e., does not measure oscillations in vertical direction).

In addition to the raw data shown in Figure 6 by curves 1 and 2, Figures 7 and 8 show the instantaneous mass flow rates m(t) (labeled 1) and cumulative mass M (labeled 2) calculated according to Eqs. [8] and [9]. For the calculation, we use the density for alloy AlSi12 given in Table I. The curve 2 of the cumulative mass grows with time. The final value M (Δt = 1200 seconds) provides the total amount of aluminum that is passed through the channel. Curve 2 has the highest slope in the range between t ₁ = 400 seconds and t ₂ = 650 seconds in which the flow rate is maximum. In this test, the accumulated mass M adds up to 8350 kg.

Figure 8 demonstrates the dependences for test 2 as the same for test 1 in Figure 7. This process distinctive by its duration is also represented in the terms of mass flow rate (curve 1) and cumulative mass (curve 2) as functions of time. Curve 1 faithfully reproduces the behavior of curve 2 for the flow rate shown in Figure 6. In the construction of curves in Figure 6 and Figure 8, we use Eqs. [6], [8,] and [9] as well as calibration factor c (Eq. [17]). We emphasize that q here at a smaller process duration is threefold higher than q for test 1. The accumulated mass reaches 8930 kg.

We perform the estimations of relative error of measurements for industrial tests. A correction of the magnetic field induction with a temperature variation is not conducted by virtue of the fact that temperature changes in a small range during the transportation process. This variation is approximately 280 K (7 °C). Accounting for a change in the magnetic field with a temperature reported in this work, we find that error is |Δ_B(T)| = 0.85 pct. Measurements of the Lorentz force performed by scales have an error of |Δ_F| = 0.05 pct. Taking into account a possible variation of the molten alloy temperature of 303 K (30 °C)—relatively operating temperature T = 1053 K (780 °C)—we estimate the error of alloy electrical conductivity as |Δ_σ(T)| = 0.92 pct. The change in alloy temperature at 303 K (30 °C) leads to alloy density variation. An estimation of error for the density of the given alloy is |Δ_ρ(T)| = 0.37 pct. Control measurement of weight (mass) in the basic test 1 with an accuracy of 10 kg gives an error of |Δ_M| = 0.15 pct. Thus, the relative error of flow rate measurement by our LFF is expressed as follows:This estimation can be reduced to |ξ| = 0.57 pct if introducing a correction of magnetic field induction (Eq. [4]) by measuring the temperature of the magnet system, and the correction of electrical conductivity can be reduced by measuring the molten alloy temperature at times, using corresponding software.