## Abstract

The contactless inductive flow tomography (CIFT) is a measurement technique that allows reconstructing the flow of electrically conducting fluids by measuring the flow-induced perturbations of one or various applied magnetic fields and solving the underlying inverse problem. One of the most promising application fields of CIFT is the continuous casting of steel, for which the online monitoring of the flow in the mould would be highly desirable. In previous experiments at a small-scale model of continuous casting, CIFT has been applied to various industrially relevant problems, including the sudden changes of flow structures in case of argon injection and the influence of a magnetic stirrer at the submerged entry nozzle. The application of CIFT in the presence of electromagnetic brakes, which are widely used to stabilize the flow in the mould, has turned out to be more challenging due to the extreme dynamic range between the strong applied brake field and the weak flow-induced perturbations of the measuring field. In this paper, we present a gradiometric version of CIFT, relying on gradiometric field measurements, that is capable to overcome those problems and which seems, therefore, a promising candidate for applying CIFT in the steel casting industry.

This article is part of the themed issue ‘Supersensing through industrial process tomography’.

## 1. Introduction

Online monitoring of the flows of liquid metals or semiconductor melts has been a long-standing wish in a variety of metallurgical and crystal growth applications. Since optical measurement techniques are ruled out by the opaqueness of those melts, and the use of ultrasonic sensors or pressure and potential probes is often prevented by the elevated temperature, contactless methods for flow reconstruction would be highly desirable.

An approximate flow picture can be provided by the contactless inductive flow tomography (CIFT) [1]. CIFT relies on applying primary magnetic fields to the melt and measuring the flow-induced (secondary) magnetic field perturbations outside the fluid volume. While the inverse problem of flow reconstruction from induced magnetic fields is closely related to the corresponding problem of neural activity detection in magnetoencephalography (MEG) [2], the validity of CIFT can be stepwise enhanced by applying the primary field in various directions. As long as the typical time for velocity changes is larger than the switching times for the different applied fields, this allows one to collect more information about (basically) the same velocity field. Further to this, any prior information such as the divergence-free condition of the velocity and any available in- or outflow information can be easily implemented in the inverse-problem solver. The still remaining non-uniqueness is then circumvented by using appropriate regularization methods, such as Tikhonov regularization and the L-curve technique [3], in which a certain quadratic functional of the velocity, e.g. the kinetic energy or the squared curvature, is additionally minimized.

While the mathematical foundations of CIFT [4–6], and its first experimental verification on a propeller-driven three-dimensional flow [1] had been published some time ago, the last years have seen an increased interest in applying CIFT to industrially relevant problems such as Czochralski crystal growth [7] and continuous steel casting [8–14]. In various experiments on a small-scale model of continuous casting, CIFT has proved very useful for studying such phenomena as the sudden flow structure changes in case of argon injection [9] and the increased flow oscillations in the mould under the influence of magnetic stirring at the submerged entry nozzle (SEN) [10]. In contrast to these successful applications, the use of CIFT in the presence of electromagnetic brakes (EMBr), which are widely used in the industry to stabilize the flow in the mould, has turned out to be more challenging. The reason for that is the extreme dynamic range between the strong applied brake field and the weak flow-induced perturbations of the measuring field.

The aim of this paper is to present the theory, and first applications, of a new version of CIFT that uses gradiometric field measurements instead of absolute field measurements. We will show that this gradiometric version of CIFT (G-CIFT) is capable of overcoming a number of problems and represents, therefore, a promising candidate for applying CIFT in industrial steel casting.

In the next section, we motivate the use of CIFT by discussing some typical problems of continuous steel casting, and present the corresponding experimental facilities that are available at Helmholtz-Zentrum Dresden–Rossendorf. The paper continues with the basic mathematics for the forward and the inverse problem, both for the case of absolute field measurements and of gradiometric measurements. The feasibility of G-CIFT is then evidenced, numerically and experimentally, for the flow in a typical continuous casting slab geometry without an EMBr. We also present the results of the first test experiment with gradiometric sensors in the presence of EMBrs. We conclude by summarizing the advantages of G-CIFT and by delineating the prospects for its industrial implementation.

## 2. Continuous casting and its experimental modelling

Nowadays, more than 95% of the world's steel output is produced by means of continuous casting [15]. In this process (figure 1), liquid metal flows from a tundish through an SEN into a water-cooled copper mould, where a solid steel shell grows from the wall into the centre of the strand. The strand is pulled out of the mould supported by rolls and solidifies completely, promoted by secondary cooling.

A key determinant for the final quality of the produced steel is the flow structure in the mould [16]. For the casting of slabs, the most advantageous flow structure is the so-called double-roll flow, with two liquid-metal jets leaving the SEN under some angle, impinging on the narrow faces of the mould and then splitting up into smaller upper rolls and larger lower rolls. Unstable flows with jet oscillations, and single-roll flows where the jet bends sharply upward after leaving the SEN, can lead to several kinds of defects and an uneven solidification shell. In industry, EMBrs are widely used, supposing that a strong static magnetic field should have a stabilizing and decelerating effect on the jets. However, recent experimental [17] and numerical investigations [18] have revealed much more ambivalent effects of brake fields, which depend sensitively on the electrical conductance ratio of the liquid steel and the solid shell.

A basic problem for validating the effect of EMBrs, and other flow-influencing measures, is the lack of measurement techniques for liquid steel flows. So far the industry uses techniques like the nail board dip test and the evaluation of oscillation mark shapes [19], the strain gauge method [20] and mould flow control (MFC) sensors from AMEPA [21]. Unfortunately, all these are mainly limited to measurements near the mould wall or near the meniscus (MFC, nail board) or suffer from the restriction to very few measurement positions (strain gauge).

The LIMMCAST program at HZDR, including various test rigs (figure 2), strives to study systematically the flow structure in continuous casting, and the particular effects of EMBrs, by using low-temperature melting liquid metals for which various measurement techniques can be applied and bench-marked. For a wide variety of effects, ultrasonic Doppler velocimetry (UDV) has provided accurate flow fields within the mould. Complementary to this, CIFT has also been applied to various problems. In a few cases, UDV and CIFT have been applied simultaneously, showing remarkable agreement of the identified flow features [9]. While the results of CIFT are typically less accurate, its contactless character gives hope to make it applicable also in the high-temperature application of real steel casting.

## 3. Theoretical basis of contactless inductive flow tomography and gradiometric version of contactless inductive flow tomography

In the following, we outline the mathematical foundations of CIFT in general, and corroborate the particularities of G-CIFT. More details on how to deal with the non-uniqueness problem of the ill-posed inverse problem can be found in [4–6].

We consider the velocity field **v** of an electrically conductive fluid with conductivity *σ*. When exposed to a stationary magnetic field **B**, the movement induces the current density
3.1where we have replaced the electric field **E** by the (negative) gradient of the electric scalar potential *φ* (assuming quasi-static approximation). According to Biot–Savart's law [6,22], this current density **j** produces now a secondary magnetic field **b**(**r**):
3.2Here, d*V* ′ denotes the volume element and **r**′ the position vector in the volume. Accordingly, d*S*′ denotes a surface element and **n**(**s**′) represents the normal vector of the surface at the position **s**′. With the view on G-CIFT, we define also the difference (the ‘gradient’) of two magnetic fields at neighbouring positions with distance **h** as
3.3

Exploiting the divergence-free condition of **j** we can derive from (3.1) a Poisson equation for the electric potential:
3.4According to Green's theorem, the solution of this Poisson equation fulfills the boundary integral equation
3.5if insulating boundaries are assumed. In the case of one or more conducting boundary layers, one has to extend the integral equation system to model the induced currents through the conducting layers correctly [23].

The total magnetic field **B** under the integrals of (3.2) and (3.5) is, in general, the sum of an externally applied (primary) magnetic field **B**_{0} and the induced (secondary) magnetic field **b**. In magnetohydrodynamics, the ratio between **b** and **B**_{0} is known to be proportional to the magnetic Reynolds number, defined as
3.6with *l* and *v* representing characteristic length and velocity scales of the fluid, respectively. For large values of Rm, and suitable flow topologies, it is even possible to achieve self-excitation of a magnetic field. In this case, one can obtain solutions of (3.2) and (3.5) even for **B**_{0}=0 [22,24].

In the most industrial applications, however, Rm is smaller than 1. Consider typical values for steel casting, with a mould width of 1 m, a typical flow velocity of 0.1 m *s*^{−1}, and a conductivity of the liquid steel of 7×10^{5} S m^{−1}, we arrive at Rm∼0.1. In such cases, it is justified to replace **B** by **B**_{0} under the integrals in equations (3.2) and (3.5), so that we arrive at a linear inverse problem for the determination of the velocity field **v** from the induced magnetic field **b** measured in the exterior of the fluid.

With the view on numerical implementations, we consider now *K* different external magnetic fields **B**_{0,k} to be applied to the fluid. Suppose, for each **B**_{0,k}, all measured induced magnetic field components to be collected into an *n*_{b}-dimensional vector with the entries . Accordingly, for G-CIFT, we can collect all measured field ‘gradients’ into an *n*_{b}-dimensional vector with the entries . In either case, the discretized electric potential at the surface *S* can be represented by an *n*_{φ}-dimensional vector with the entries , and the desired velocity **v** in the volume *V* is discretized as an *n*_{v}-dimensional vector with the entries *v*_{l}. Then, equations (3.2) and (3.5) can be cast into the matrix form:
3.7and
3.8If we have measured, instead of the absolute fields, the gradients, equation (3.7) should be replaced by the corresponding equation
3.9

Note that only the matrices **R**^{B0,k} (or, respectively, ) and **T**^{B0,k} depend on the applied magnetic field **B**_{0,k}, whereas the matrices **S** (or, respectively, ) and **U** depend on the geometry only.

The well-known peculiarity of inverting equation (3.8), which arises from the singularity of the matrix (**I**−**U**), can be circumvented by using the so-called deflation method [2], replacing (**I**−**U**) by the well-conditioned matrix (**I**−**U**)^{defl}. Inserting the solution *φ* of the (deflated) equation (3.8) into equation (3.7), or (3.9), we end up with a single linear relationship between the desired velocity field and the measured magnetic field
3.10or, respectively, the measured field gradients
3.11

For a given velocity field, equations (3.10) or (3.11) can now be used to solve the forward problem of determining the flow-induced magnetic fields or field gradients, respectively. Further details of the numerical solution of these equations, in particular concerning the accurate computation of the boundary integrals, can be found in [25].

The inverse problem, in turn, consists of estimating the velocity field from the induced magnetic field or its gradients. For this purpose, we have to solve the normal equations which arise from the minimization of the total functional of the velocity field
3.12which includes the functionals *F*_{B0,k}
3.13that represents the mean-squared residual deviation of the measured magnetic fields *b*^{B0,k}_{i,measured} from the fields *b*^{B0,k}_{i,modelled} modelled according to equation (3.10), or the corresponding functional for the field gradients
3.14according to equation (3.11), as well as the additional functionals
3.15
and
3.16where *F*_{div}[**v**] enforces the velocity field to be solenoidal, and the last functional *F*_{pen}[**v**] is the penalty functional which tries to minimize the kinetic energy.

The parameters *σ*_{B0,k}, or for G-CIFT, are the *a priori* errors for the measured induced magnetic fields or their gradients, respectively. *σ*_{div} is chosen very small in order to ensure the divergence-free condition. The parameter to play within the regularization procedure is *σ*_{pen}. Here we employ the so-called Tikhonov regularization [3]: by increasing the regularization parameter *σ*_{pen}, one obtains solutions of the inverse problem with increasing kinetic energy of the flow. The optimal solution is then found at the point of strongest curvature of Tikhonov's L-curve which is done by an automatic search described in [25].

The above equation system is valid for the case of different applied magnetic fields and a fully three-dimensional flow. It can, however, also be adapted to the essentially two-dimensional case of slab-casting, where only one (basically vertical) magnetic field *B*_{0,z} is applied. In this case, the dimensional reduction of the velocity structure from three to two dimensions can be realized by adding an additional functional
3.17which enforces the velocity to be two-dimensional by setting to zero the component *v*_{y} (which is parallel to the narrow faces of the mould). Optionally, the mean velocity at some inlet points *l*∈*M*_{inlet} can be prescribed by the functional
3.18

It should be noted that regularization based on some reasonable norm of the velocity is actually more than only a trick since in most cases the assumption of a rather smooth velocity field is sensible from a physical point of view. For the same reason, it is also clear that the method is not suited to detecting very small vortices in a strongly turbulent flow (see [5,6] for the corresponding uniqueness problem).

## 4. Applying the gradiometric version of contactless inductive flow tomography without electromagnetic brakes

In order to evaluate the quality of the velocity field that is reconstructed from the measured magnetic field gradients, we start from a numerically computed velocity field, solve the forward problem to determine the resulting magnetic field gradients, and try to reconstruct the pre-given velocity. This approach is similar to the procedure for the first application of CIFT to a model of continuous casting [8]. We used the same pre-given velocity field (illustrated in figure 3*a*) with a flow rate of 0.113 l s^{−1} and placed the excitation coil at a height of *z*=216 mm around the mould generating a horizontal magnetic field of about 1 mT.

Similar to [8], we start with a rather large number of 335 sensors along each narrow face of the mould as a reference case. This purely hypothetical number just results from an assumed sensor distance of 1 mm. The sensors are directed normal to the wall with a distance *r*_{x}=23 mm from the fluid and a distance *h*_{x} of 20 mm between the two sensing parts of the gradiometer. These distances correspond to the sensor configuration in the validation experiment, described later on. The calculated induced magnetic field is shown in figure 3*b*. From these 670 measurements, we reconstructed the velocity using the G-CIFT algorithm presented in §3. The result of this reconstruction is shown in figure 3*c*. When comparing with the original velocity it becomes evident that the jet and the double roll structure are reasonably reconstructed.

Figure 4*a* shows the L-curve of the reconstruction together with the empirical correlation coefficient and the normalized mean quadratic error between the original and the reconstructed velocity field. The maximum curvature of the L-curve is approximated by the maximum of the second derivative (lower panel of figure 4). It can be seen clearly that the best correlation and the lowest mean quadratic error is obtained close to the maximum of the second derivative. Both quantities turn out to be in the same order as in the case of absolute field measurements documented in [8].

Additionally to the L-curve, we show the reconstructed velocity for four different regularization parameters in figure 4*b*–*e* to demonstrate the effect of the regularization. The locations of the four regularization parameters on the L-curve are marked by circles and arrows in figure 4*a*. We start with a strong regularization in figure 4*b* for which the reconstructed velocity is nearly zero. For the two regularization parameters near the maximum of the second derivative shown in figure 4*b*,*c*, we get two very similar velocity fields. The jet and the double-roll on both sides are clearly visible and do not change very much. However, if we reduce the regularization parameter further, we get a very chaotic velocity field with additional vortices on the top and overestimated downward velocities near the narrow faces of the mold. This behaviour of the regularization is very similar to the version of CIFT that uses absolute magnetic field sensors. It yields a reasonable reconstruction of the flow which corresponds to complementary UDV measurements [8]. Even transient phenomena of the velocity field, which occur typically in two-phase flows, are correctly reconstructed as seen in comparison with UDV measurements described in [9]. If we add 5% noise (±10 nT) to the data from figure 3*b* and reconstruct the flow, the velocity shows almost no deviation from figure 3*c*.

In order to validate the simulation, we measured the gradient of the magnetic field using one single Fluxgate gradiometer at 37 vertical positions by just changing the sensor position. For each position, two or three measurements were carried out. The position of the stopper rod, the vertical position of the meniscus and the liquid metal level in the tundish are recorded to assure reproducible flow conditions for each measurement. A comparison of the measured and the simulated gradient of the magnetic field is shown in figure 5*a*. The agreement is quite convincing.

With these experimental data at hand, the velocity reconstruction was carried out once again. By assuming, for the sake of simplicity, a symmetric flow configuration, we have mirrored the data measured at one narrow face to the opposite one. The reconstructed velocity field is shown in figure 5*b*. Since the measured gradients turn out to be very close to the simulated ones, it is not surprising that the reconstructed velocity field looks very similar to figure 3*c*. Additionally, we checked that reducing the number of magnetic field sensors to only eight sensors along each narrow face of the mould does not degrade the reconstruction quality. We can summarize, therefore, that in the absence of EMBrs G-CIFT works similarly well as the normal version of CIFT.

## 5. Applying gradiometric version of contactless inductive flow tomography in the presence of an electromagnetic brake

### (a) Why using pick-up coils?

Imagine a continuous casting process, with an EMBr field of about 300 mT being applied. Assume further a measuring field of some 3 mT to be used, from which typical flow-induced field perturbations of 30 nT result. The resulting dynamic range of 7 orders represents an enormous challenge for the measurement technique. In those applications, the use of Fluxgate sensors is not possible since they go into saturation at fields of a few millitesla. With the view on their nearly perfect linearity, we have decided to use pick-up coils for the magnetic field measurements. Since they detect only time-derivatives of fluxes (rather than fluxes), we have to switch from DC to AC excitation fields.

For our measurement, we used two different circular induction coil sensors, namely single coils and gradiometric coils (figure 6), made of a 25 μm baked enamel wire on a PTFE frame. The former consist of a single winding with 340 000 turns, whereas the latter are composed of two separate windings of 2×160 000 turns, wound in opposite directions [26]. Single coils pick up the absolute value of the time-derivative of the magnetic field, while gradiometric coils detect the time-derivative of the magnetic field gradient. The sensitivity is 510 VT^{−1} Hz^{−1} and 240 VT^{−1} Hz^{−1}, respectively.

### (b) Some sensitivity considerations

Using induction-coil sensors, no DC field whatsoever will induce a voltage in the sensor. Yet, any fluctuating inclination of the sensor in the DC field will generate a voltage because the flux comprised by the sensor varies in time. Typically, the magnetic field sensor is aligned orthogonal to the magnetic field of the excitation coil and positioned as close as possible to the mould wall (figure 2*b*,*c*). This set-up gives a maximum suppression of the excitation signal in the measurement chain because the projection of **B**_{0} onto the measurement vector of the sensor is small and the amplitude of the induced magnetic field is large. At the same time, however, this alignment is also problematic since even the slightest inclination of the sensor in the direction of the orthogonal magnetic field creates an artificial signal that might easily exceed the flow-induced magnetic field. In experiments and industrial applications, this effect cannot be excluded, even if special care is taken to ensure sufficient stiffness of the set-up.

The advantage of using gradiometric coils is now evidenced in a numerical experiment that simulates the effect of such inclinations (figure 7*a*). Consider one of each sensors to be positioned near a rectangular coil located in the *xy*-plane with the assumed dimensions 340×240 mm^{2}, driven by a current of 40 A through 12 turns. Additionally, a 300 mT homogeneous static magnetic field, simulating an EMBr, is applied in the negative *y*-direction, thereby creating no magnetic flux through the sensor. The assumption of a uniform magnetic field is justified as long as (i) the dimensions of the gradiometric sensors are smaller than typical dimensions of an assumed EMBr and (ii) the sensor is not placed close to the yokes where significant stray fields with spatially declining amplitude would be present. The magnetic centre of the sensor is located at (0.05 m;0;0.2 m) and faces the negative *x*-direction. This alignment serves as reference case. Now assume the sensor to be inclined either around the *y*-axis, or additionally around the *z*-axis by an angle *α*. The resulting change in the measured magnetic field is documented in figure 7*b*.

If the inclination only happens around the *y*-axis, both sensor types show a small additional signal of, e.g. 10 nT at 0.1^{°}, with the single coil performing slightly better. Yet, if the inclination is also in direction of the strong homogeneous field, the measurement error grows to 5.2 μT at 0.001^{°} for the single coil, while the gradiometric coil shows no difference at all because the induced voltages cancel each other out. Note that the induced voltage in the single coil, caused by the changing enclosed flux from the EMBr, is only generated as long as the sensor rotates around the *z*-axis and would return to zero as soon as the rotation stops. This illustrates that gradiometric coil have a substantial advantage in these kind of set-ups. This is not limited to set-ups with EMBrs, but can be relevant in any scenario having approximately homogeneous far-fields, e.g. magnetic pumps [27] or strong AC currents in some distance. In any case, the gradiometric sensor should be placed close to the melt where the gradient of the induced field becomes maximal.

### (c) First test measurements

In a recent publication [28] concerning CIFT for an EMBr influenced flow at the Mini-LIMMCAST facility, we had evidenced significant flow fluctuations to be present at low frequencies *f*≤3 Hz. Under the influence of the strong static brake field, the resulting field fluctuations superimpose the signal of actual interest that results from the (more or less) steady flow under the influence of the 3 Hz measuring field. To circumvent this effect, an excitation with higher frequencies should be applied, which enables successful magnetic field measurements [28].

In another measurement campaign, we saw in addition the induced magnetic field to be frequency-dependent as a result of a changing liquid metal level in the mould and significant eddy-currents (figure 8). To separate both effects, the signal from the eddy-currents must be measured separately, and the result must be subtracted from the total induced magnetic field in order to obtain the actual flow-induced magnetic field. Therefore, we conducted calibration measurements where the liquid metal level in the mould is changed manually and the resulting change in the induced magnetic field is measured. This was done by lifting the drain hoses attached to either side of the mould (figure 2*b*, lower right corner), then completely filling the mould and finally lowering the hoses again which allows the liquid metal to drain. During this procedure, a small force is applied to the set-up, resulting in some ill-defined inclinations of the sensors, as mentioned earlier.

Figure 9 shows the induced magnetic fields for a single and a gradiometric coil in the set-up with the EMBr when the level is changed. The hoses were manually held up and the mould was filled. Then the measurement was started, and after at least 10 s the hoses were lowered again. One can see that a measurement is possible for the single coil when the EMBr is switched off, even though there are some variations on the initial baseline. Measuring while the EMBr is switched on seems impossible due to severe signal fluctuations that result from the inclinations caused by slightest changes in hoses' position. The gradiometric coils in contrast show a very clear baseline for either case, and signal variations are hardly visible. This confirms our sensitivity considerations from figure 7.

Using these measurements one can correct, for example, the average-induced magnetic field from an experiment, as can be seen in figure 10. To test this, we took a comparably badly impaired measurement with a large frequency of 20 Hz. When we subtract the data obtained from the measurement in figure 9 from these ‘high-frequency’ (HF) measurements, then the resulting green curve agrees with the ‘low-frequency’ (LF) (2 Hz) reference measurements for the gradiometric sensor, but not for the absolute sensor. For the single coil, an almost constant offset remains for the three uppermost sensors, while the other four sensors show good agreement with the reference. The gradiometric coil gives better agreement between the corrected measurement and the reference measurement, where only a small offset (20 nT) for the uppermost sensor remains. This is negligible because the regularization of the inverse problem can handle such small measurement errors. All in all, this demonstrates that corrected CIFT measurements should be possible using high-frequency and gradiometric coils.

### (d) First flow measurements with the electromagnetic brake switched on

Finally, we validate the flow measurement for the EMBr switched on with gradiometric sensors and an excitation frequency of 8 Hz. The same experimental configuration was used for UDV measurements in [17], where well-expressed transitions of the flow structure from the left to the right sides of the mould had been observed. Our aim here is to get some comparable indications for such flow transitions from magnetic field measurements. Insulating boundary conditions were used, and the EMBr generated a 300 mT field.

While it is known that there exists a conducting boundary between the liquid steel and the strand shell, it is assumed [29] that there exists no electrically conducting interface between the mould plates and the strand shell through the flux powder. Isolated spots of electrical contact can nevertheless occur. The strand shell can be modelled as a (better) conducting boundary layer using an extended integral equation system [23]. The case regarded here serves as an illustration only because the conducting boundary would lead to a more stable flow [17].

The induced magnetic field gradients for all sensors are shown in figure 11. In figure 11*b*, three separate sections can be identified: (i) the time interval for 10 s<*t*<20 *s* where the oscillating flow starts do develop, (ii) a part between 20 and 50 s, and (iii) a segment between 60 s and 130 s that is similar to the first part. Time intervals (ii) and (iii) can be seen in more detail in figure 11*c*,*d*.

When we take the average for both of the last two segments (we omit the first segment here, since it is too short), we obtain figure 12*a*. It can be seen that both averages are quite similar, but appear reversed, meaning that a typical asymmetrical flow for these conditions has swapped sides. UDV data from a different run show a similar result in figure 12*b*, where the vertical flow component close to the wall of both narrow faces is depicted. Here, blue indicates an upward flow, red a downward flow. After roughly 10 s a transition occurs and the flow also swaps sides. This gives us confidence that our measurement is indeed correct. The correction of the induced magnetic field is still to be done, together with the actual G-CIFT flow reconstruction.

## 6. Conclusion

In this paper, we have outlined the basic process and the importance of continuous casting, as well as its need for an online flow monitoring system. Furthermore, we have delineated the fundamentals of G-CIFT, an extension of CIFT that uses gradiometric field measurements instead of absolute ones. A numerical and experimental example for slab-casting confirms our implementation to be working, at least for set-ups without an EMBr.

Moreover, we illustrated the advantages of gradiometric sensors in G-CIFT applications where a rather homogeneous, yet strong magnetic brake field is present. This enables us to measure induced magnetic fields in scenarios where absolute sensors failed. Additionally, it was depicted that for an excitation frequency of 8 Hz, the induced magnetic field appears reasonable when compared with UDV data.

Still to be accomplished is the G-CIFT flow reconstruction in the presence of an EMBr and a thorough study of the flow in the mould under various conditions, like for different boundary conductivities, vertical positions of the EMBr and different EMBr field amplitudes.

Presently, work is going on to test various versions of CIFT at a real casting facility. Yet, the remaining obstacles for its successful implementation, including the oscillation of the copper mould and the presence of (magnetic) Nickel coatings, are significant.

## Data accessibility

Supporting experimental data are available under .

## Authors' contributions

All authors contributed to developing the velocity reconstruction method. M.R. drafted the §§5 and 6, conducted the experiments and analysed the data. T. W. implemented the numerical routines, ran the simulations and compared the results with experimental data in §4. F.S. drafted the §§1–3. All authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

Financial support from the Helmholtz-Alliance LIMTECH is gratefully acknowledged.

## Acknowledgements

We would like to thank Klaus Timmel and Michael Röder for valuable discussions and technical support.

## Footnotes

One contribution of 10 to a theme issue ‘Super-sensing through industrial process tomography’.

- Accepted February 2, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.