# An Electromagnetic Time-Reversal Imaging Algorithm for Moisture Detection in Polymer Foam in an Industrial Microwave Drying System

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

#### 2.1. Scattering Model and Time-Reversal Imaging

#### 2.2. Dyadic Green Function of Multilayered Media

## 3. TRI-DORT Simulation Results

#### 3.1. Low-Contrast Media

#### 3.2. High-Contrast Media

#### 3.3. Single Frequency TRI

#### 3.4. Moderately Rough Surface

## 4. Experimental Results

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**A side view of the HEPHAISTOS microwave oven system. Main modules of the oven are represented by number tags 1, 2, 3, 4, 5. Tag 1 and Tag 5 represent the entrance of the wet foam and exit doors for the dry foam on the conveyor belt, respectively. Tags 2, 3, 4 indicate the three modular heating systems which are built into the hexagonal cavity with high power microwave heating sources and control system block.

**Figure 2.**Multilayer media with the wet spot(s) in the second layer illuminated by the open waveguide antennas.

**Figure 3.**The real and imaginary part of the total electric field inside three-layer media with dimensions 30 cm × 8 cm and ${\u03f5}_{r,1}=1.16$ for the dielectric layer at 8 GHz and 12 GHz. The observation point is in the middle of the first layer, i.e., $z=0\mathrm{cm}$ and $-25\mathrm{cm}\le y\le 25\mathrm{cm}$. Layer 2 is free space.

**Figure 5.**(

**Top**) Magnitude of the first four eigenvalues versus the frequencies and (

**second row**) reconstruction of one wet spot moisture case with TRI-DORT where the true location is marked by black dashed lines, (

**third row**) reconstruction using third eigenvalue and its associated eigenvector, (

**fourth row**) reconstruction using fourth eigenvalue and its associated eigenvector.

**Figure 6.**(

**Top**) Magnitude of the first four eigenvalues versus the frequencies, (

**middle**) reconstruction of the first wet spot moisture case with TRI using first dominant eigenvalue, and (

**bottom**) reconstruction of the second wet spot moisture case with TRI-DORT using second dominant eigenvalue.

**Figure 7.**The real and imaginary part of the total electric field inside three-layer media with dimensions 30 cm × 8 cm and ${\u03f5}_{r,1}=1.16$ for the dielectric layer at 8 GHz and 12 GHz. Layer 2 is PEC.

**Figure 8.**(

**Top**) Magnitude of the first four eigenvalues versus the frequencies, (

**second row**) reconstruction of the first wet spot moisture case with TRI-DORT using first dominant eigenvalue and (

**third row**) reconstruction of the second wet spot moisture case with TRI-DORT using second dominant eigenvalue when the layer 2 is a PEC plate, (

**fourth row**) reconstruction using MUDT imaging algorithm.

**Figure 11.**Prototype of MWT sensor array used in this study to generate measurement data. The MWT sensor configuration consists of 7 X-band open-ended waveguide antennas, indicated by Tag 1. The polymer foam is shown by Tag 2 and surrounded by absorbers as shown by Tag 3. The flexible PEC plate is shown by Tag 4. To acquire the measurement the solid switch, PC and VNA are used as shown by Tags 5, 6, and 7, respectively. This system is developed at KIT, Germany and will be integrated with the HEPHAISTOS technology in the final stage.

**Figure 12.**Experimental scattering response (in dB) in X-band of antenna 4 (middle antenna) for different moisture contents in the spherical wet spot of radius 1.5 cm.

**Figure 13.**(

**Top**) Magnitude of the first four eigenvalues versus the frequencies,(

**middle**) TRI-DORT reconstruction image of one PTFE Teflon, and (

**bottom**) SF-TRI-DORT at $8.07$ GHz.

**Figure 14.**(

**Top**) Magnitude of the first four eigenvalues versus the frequencies and (

**bottom**) TRI-DORT reconstruction image of one wet spot when the second layer is PEC plate.

**Table 1.**NRMS error values for compared analytical model and FEM in Figure 3.

$\mathbf{Re}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}8\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Im}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}8\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Re}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Im}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | |
---|---|---|---|---|

NRMS % | 1.76 | 2.06 | 2.11 | 2.74 |

M% | 0 | 30 | 36 |
---|---|---|---|

${\u03f5}_{r}$ | $1.16-0.01i$ | $1.69-0.1i$ | $1.81-0.16i$ |

**Table 3.**NRMS value for compared analytical model and FEM in Figure 7.

$\mathbf{Re}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}8\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Im}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}8\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Re}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | $\mathbf{Im}({\mathbf{E}}_{\mathbf{x}}),\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\mathbf{GHz}$ | |
---|---|---|---|---|

NRMS % | 2.42 | 1.89 | 1.94 | 2.15 |

**Table 4.**The difference between the first dominant eigenvalue and those belonging to the null space at four different frequencies.

f (GHz) | 8 | 9 | 10 | 11 |
---|---|---|---|---|

$\Delta \lambda \times {10}^{3}$ | 6.6654 | 2.029 | 2.4659 | 2.694 |

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**MDPI and ACS Style**

Omrani, A.; Yadav, R.; Link, G.; Lähivaara, T.; Vauhkonen, M.; Jelonnek, J.
An Electromagnetic Time-Reversal Imaging Algorithm for Moisture Detection in Polymer Foam in an Industrial Microwave Drying System. *Sensors* **2021**, *21*, 7409.
https://doi.org/10.3390/s21217409

**AMA Style**

Omrani A, Yadav R, Link G, Lähivaara T, Vauhkonen M, Jelonnek J.
An Electromagnetic Time-Reversal Imaging Algorithm for Moisture Detection in Polymer Foam in an Industrial Microwave Drying System. *Sensors*. 2021; 21(21):7409.
https://doi.org/10.3390/s21217409

**Chicago/Turabian Style**

Omrani, Adel, Rahul Yadav, Guido Link, Timo Lähivaara, Marko Vauhkonen, and John Jelonnek.
2021. "An Electromagnetic Time-Reversal Imaging Algorithm for Moisture Detection in Polymer Foam in an Industrial Microwave Drying System" *Sensors* 21, no. 21: 7409.
https://doi.org/10.3390/s21217409