# Contactless Inductive Flow Tomography for Real-Time Control of Electromagnetic Actuators in Metal Casting

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Contactless Inductive Flow Tomography

#### 2.2. Mini-LIMMCAST

^{2}. The SEN has an inner diameter of 12 mm and an outer diameter of 21 mm with two side ports directed downwards at an angle of 15°.

## 3. Compensation for the Impact of the EMBr on the CIFT Measurement System

#### 3.1. Krasnosel’skii–Pokrowski Model of Hysteresis

#### 3.2. Compensation for the Impact of the EMBr on the Flow-Induced Magnetic Field Measurements

#### 3.2.1. Compensation via the Numerical Model of Hysteresis

#### 3.2.2. Congruency-Based Compensation

## 4. Real-Time Reconstruction

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Experimental setup comprising CIFT coils and sensors as well as the electromagnetic brake (EMBr). Two excitation coils generate a primarily vertical magnetic field. Fourteen gradiometric coils, seven on each narrow side, are used to measure the flow-induced magnetic field. The EMBr generates a strong magnetic field below the submerged entry nozzle (SEN), perpendicular to the wide side of the mould. (

**a**) Photograph of the mould and CIFT coils. (

**b**) Sketch adopted from Schurmann et al. [4].

**Figure 3.**Measurement of the magnetic field for an experiment where the EMBr current ${I}_{B}$ is changed during the measurement for the sensors 1–7. The current was turned on at $t\approx $ 28 s to ${I}_{B}$ = 200 A and turned off at $t\approx 44\text{}\mathrm{s}$ to ${I}_{B}$ = 0 A.

**Figure 4.**Measured offset of the in-phase component of the magnetic field for an experiment where the EMBr current was cycled from 0 A to 600 A and back to 0 A in steps of $\mathsf{\Delta}{I}_{B}$ = 25 A. The plots show only measurements for sensors 1–7. (

**a**) plot over time; (

**b**) plot of offset over EMBr current where the hysteresis can be observed.

**Figure 6.**Generalized output of a single KP operator. (

**a**) Input function ${I}_{B}\left(t\right)$ in relation to parameters ${p}_{1}$, ${p}_{2}$ and a. (

**b**) Generalized output of the KP operator for the given input function. Colours represent the values of the input and the corresponding output of the KP operator.

**Figure 7.**Preisach plane. (

**a**) Discretization scheme of the Preisach plane. (

**b**) Evolution of the shape of the KP operators on the Preisach plane.

**Figure 8.**EMBr current profiles used for KP model training and validation. (

**a**) Current profile used for identification of model weights. (

**b**) Current profile used for validation of the model.

**Figure 9.**Plots of the absolute difference (red) between the measured in-phase magnetic field offset (green +) and the value predicted by the KP model of hysteresis (blue).

**Figure 10.**Congruency-based compensation for the impact of the EMBr current on the flow-induced magnetic field measured by sensors 1–7. (

**a**) EMBr current profile. (

**b**) Mean value and standard deviation of the magnetic field offset after the compensation for the periods between the current changes. (

**c**) Uncompensated in-phase component of the magnetic field. (

**d**) Compensated in-phase component of the magnetic field.

**Figure 11.**Results of congruency-based compensation for sensors 1–7, utilizing the numerical KP model of hysteresis for the experiment from Figure 10. (

**a**) Time plot of the compensated in-phase component of the magnetic field. (

**b**) Mean value and standard deviation of the compensated in-phase component of the magnetic field for periods in which the EMBr current is kept constant.

**Figure 12.**Reconstruction of the time dependant flow field at $t=$ 17 s. (

**a**) Three-second time average of the velocity field obtained from the numerical simulation. (

**b**) Reconstructed velocity field for the optimal regularization parameter selected via the L-curve method.

**Figure 13.**The L-curve for the Reconstruction of the time dependant flow field at $t=$ 17 s. Annotated (a)–(f) are the regularization parameters for which the velocity field is shown in Figure 14. ${\lambda}_{opt}$ is marked by a red triangle and annotated with (e). (a) $\lambda =$ 1.000 ×${10}^{-8}$. (b) $\lambda =$ 1.000 ×${10}^{-9}$. (c) $\lambda =$ 1.000 ×${10}^{-10}$. (d) $\lambda =$ 2.125 ×${10}^{-11}$. (e) ${\lambda}_{opt}=$ 1.896 ×${10}^{-11}$. (f) $\lambda =$ 1.000 ×${10}^{-12}$.

**Figure 14.**Velocity reconstructions for the regularization parameters annotated in Figure 13. (

**a**) $\lambda =$ 1.000 ×${10}^{-8}$. (

**b**) $\lambda =$ 1.000 ×${10}^{-9}$. (

**c**) $\lambda =$ 1.000 ×${10}^{-10}$. (

**d**) $\lambda =$ 2.125 ×${10}^{-11}$. (

**e**) ${\lambda}_{opt}=$ 1.896 ×${10}^{-11}$, the optimal regularization parameter selected via the L-curve method. (

**f**) $\lambda =$ 1.000 ×${10}^{-12}$.

**Figure 15.**Time evolution plots of error, correlation and optimal regularization parameter. (

**a**) Error and correlation plot of reconstructions for optimally selected regularization parameter ${\lambda}_{opt}$. (

**b**) Temporal evolution of the optimal regularization parameter ${\lambda}_{opt}$.

**Figure 16.**Absolute difference of error and correlation between the velocity fields reconstructed by the standard and the real-time algorithm.

**Figure 17.**(

**a**) Three-second time average of the velocity field obtained from the numerical simulation at $t=$ 40 s. (

**b**) Reconstructed velocity field. (

**c**) Time plot of error and correlation for velocities close to the narrow face of the mould. $\left|x\right|\ge $ 100 mm. y = 0 mm, $z\ge $ 450 mm.

**Figure 18.**Comparison of the jet impingement point obtained from the numerical simulations and the CIFT real-time reconstruction for the same numerical velocity field.

**Table 1.**Reconstruction parameters $\lambda $ and corresponding values of error and correlation for the velocity reconstructions in Figure 14. Reconstruction (e) is done with the optimal regularization parameter calculated by the L-curve method.

Reconstruction | $\mathit{\lambda}$ | Correlation | Error |
---|---|---|---|

(a) | 1.000 × ${10}^{-8}$ | 0.65378 | 0.57316 |

(b) | 1.000 × ${10}^{-9}$ | 0.69040 | 0.52493 |

(c) | 1.000 × ${10}^{-10}$ | 0.70435 | 0.50469 |

(d) | 2.125 × ${10}^{-11}$ | 0.69651 | 0.52070 |

(e) | 1.896 × ${\mathbf{10}}^{-\mathbf{11}}$ | 0.69601 | 0.52175 |

(f) | 1.000 × ${10}^{-12}$ | 0.69084 | 0.53272 |

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**MDPI and ACS Style**

Glavinić, I.; Galindo, V.; Stefani, F.; Eckert, S.; Wondrak, T.
Contactless Inductive Flow Tomography for Real-Time Control of Electromagnetic Actuators in Metal Casting. *Sensors* **2022**, *22*, 4155.
https://doi.org/10.3390/s22114155

**AMA Style**

Glavinić I, Galindo V, Stefani F, Eckert S, Wondrak T.
Contactless Inductive Flow Tomography for Real-Time Control of Electromagnetic Actuators in Metal Casting. *Sensors*. 2022; 22(11):4155.
https://doi.org/10.3390/s22114155

**Chicago/Turabian Style**

Glavinić, Ivan, Vladimir Galindo, Frank Stefani, Sven Eckert, and Thomas Wondrak.
2022. "Contactless Inductive Flow Tomography for Real-Time Control of Electromagnetic Actuators in Metal Casting" *Sensors* 22, no. 11: 4155.
https://doi.org/10.3390/s22114155