# A Model-Based Analysis of Capacitive Flow Metering for Pneumatic Conveying Systems: A Comparison between Calibration-Based and Tomographic Approaches

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## Abstract

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## 1. Introduction

#### 1.1. Calibration-Based Approach

#### 1.2. ECT-Based Approach

- How does the number of electrodes influence the performance of the flow meter, or
- What is the potential benefit of the ECT-based approach with respect to the calibration-based approach.

- An analysis of the influence of the number of electrodes of the sensor,
- An analysis of different signal processing methods for capacitive flow metering and
- Reference measurement procedure to parametrize/validate the model for specific sensor evaluations.

## 2. Holistic Modeling of the Measurement Process

#### 2.1. Statistical Process Model

- A homogeneous mass concentration over the whole cross-section of the pipe corresponding to the dispersed and slug flow regimes and
- A dense lower phase with a certain height and a dispersed upper phase corresponding to flow regimes with a distinct material layer at the bottom of the pipe. Hereby, the mass concentration of the lower phase is not necessarily the bulk density of the material since the gas stream can aerate the transport good [30].

#### 2.2. Material Model

#### 2.3. Sensor Model, Noise Model and Sensor Calibration

#### 2.4. Estimation Algorithm for the Average Mass Concentration

#### 2.4.1. Calibration-Based Approach

#### 2.4.2. ECT-Based Approach

## 3. Laboratory Setup and Measurement Procedure for Model Validation

#### 3.1. Laboratory Test Rig and Measurement Setup

#### Emulation of Sensors with Different Numbers of Electrodes

#### 3.2. Measurement Experiments

## 4. Analysis and Comparison of Capacitive Flow Meters

#### 4.1. Setup and Procedure of the Analysis

#### 4.1.1. Validation Measurements on the Test Rig

^{4}random material inclusions are generated and simulated with the holistic model. The material inclusions are parametrized as it is depicted in Figure 5 and Equation (2) is used to generate the material distributions. The parameters and their respective distributions for the Monte Carlo simulation are summarized in Table 1.

#### 4.1.2. Simulation-Based Uncertainty Quantification for Pneumatic Conveying

#### 4.2. Analysis of the Influence of the Number of Electrodes

#### 4.2.1. Number of Electrodes: Measurement-Based Model Validation

#### 4.2.2. Number of Electrodes: Uncertainty Quantification for Pneumatic Conveying

#### 4.3. Analysis of Different ECT-Based Signal Processing Variants

#### 4.3.1. ECT Methods: Validation Measurements on Test Rig

#### 4.3.2. ECT Methods: Uncertainty Quantification for Pneumatic Conveying

#### 4.4. Summary and Outlook

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sensor Model, Noise Model and Sensor Calibration

#### Appendix A.1. Sensor Model

#### Appendix A.2. Noise Model

#### Appendix A.3. Sensor Calibration

## Appendix B. Calibration-Based Capacitive Flow Meters: Model-Based Parametrization of Empirical Functions

^{4}simulated capacitance ${C}_{{1}^{\u2033},{2}^{\u2033}}$ and the corresponding cross-sectional average mass concentrations ${\beta}_{\mathrm{s}}$ of the emulated sensor with two electrodes, which is illustrated in Figure 8. The simulated samples can be used to parametrize an empirical model, which describes the relationship between the mass concentration and the capacitive measurement on average. The sensor characteristic depicted in Figure A1 is given by a second order polynomial approximation. The dispersion of the simulated samples around the average trend represents the intrinsic uncertainty of the measurement system. The capacitive measurements can vary for the same cross-sectional average mass concentration due to the different flow regimes of pneumatically conveyed solids, between which cannot be distinguished with a single electrode pair. Such intrinsic uncertainties are a known property of capacitive sensors for distributed sensing with small numbers of electrodes [36].

**Figure A1.**Model based determination of the empirical sensor characteristics, which describe the relationship between the capacitance of a two electrode sensor and the cross-sectional average mass concentration.

## Appendix C. ECT-Based Flow Meters

#### Appendix C.1. Back Projection Type Estimators

#### Appendix C.2. Implementation of ECT-Based Algorithms for Pneumatic Conveying Processes

#### Appendix C.2.1. Formulation of a Prior Distribution

^{4}permittivity distribution samples. A prior distribution can then be obtained by computing a Gaussian summary statistic of the set of samples [39].

#### Appendix C.2.2. Summary Statistic of the Linearization Error

#### Appendix C.2.3. Training Data for Pneumatic Conveying Processes

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**Figure 1.**Flow regimes in horizontal pneumatic conveying processes [7].

**Figure 3.**Sketch of a holistic simulation model for capacitive mass concentration measurement in pneumatic conveying processes.

**Figure 4.**Holistic model of the measurement process of a capacitive mass concentration measurement system for pneumatic conveying systems.

**Figure 5.**Parametrization of flow patterns [26].

**Figure 6.**Exemplary random samples for both cross-sectional cases of horizontal pneumatic conveying flow patterns. The different mass concentrations result from the different aerated states of the transport good caused by the gas stream of the pneumatic conveying process.

**Figure 7.**Capacitive measurement system for the determination of flow parameters in a pneumatic conveying laboratory test rig, which uses two sensors with eight electrodes each.

**Figure 8.**Combining measurements to emulate sensor designs with different numbers of electrodes. Adjacent electrodes are combined to emulate a sensor assembly with four electrodes and a sensor with two electrodes is emulated by combining three electrodes each.

**Figure 9.**Sketch and a photo of the material holder, which is used to carry out measurement experiments with stationary material distributions [28].

**Figure 10.**Relationship between the relative signal change off the measured capacitances and the mass concentration within the sensor. Additionally, the corresponding filling height h of the material holder is shown.

**Figure 11.**Comparison between the holistic model and measurement experiments for sensors with different numbers of electrodes.

**Figure 12.**Model based analysis of the RMSE for sensor with different number of electrodes. The analyses are performed over all flow regimes covered by the stochastic process model.

**Figure 14.**Model based analysis of the RMSE for ECT-based flow meters with different estimators. The analyses are performed over all flow regimes covered by the stochastic process model.

**Table 1.**Parameters and their respective distributions for the Monte Carlo simulation for model validation.

Parameter | Distribution |
---|---|

h | $\mathcal{U}(0\mathrm{m},2r)$ |

${h}_{i}$ | $\mathcal{U}(h-0.05r,h+0.05r)$ |

${\beta}_{\mathrm{s},\mathrm{l}}$ | $\mathcal{U}(0.975{\rho}_{\mathrm{bulk}},1.025{\rho}_{\mathrm{bulk}})$ |

**Table 2.**Parameters and their respective distributions for the Monte Carlo simulation for uncertainty quantification for both cross-sectional cases of the flow regimes.

Cross-Sectional Case 1 | Cross-Sectional Case 2 | ||
---|---|---|---|

Parameter | Distribution | Parameter | Distribution |

${\beta}_{\mathrm{s},\mathrm{l}}$ | $\mathcal{U}(0\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3},{\rho}_{\mathrm{bulk}})$ | ||

${\beta}_{\mathrm{s},\mathrm{u}}$ | $\mathcal{U}(0\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3},{\beta}_{\mathrm{s},\mathrm{l}})$ | ||

${\beta}_{\mathrm{s}}$ | $\mathcal{U}(0\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3},{\rho}_{\mathrm{bulk}})$ | h | $\mathcal{U}(0\mathrm{m},2r)$ |

${h}_{i}$ | $\mathcal{U}(h-0.2r,h+0.2r)$ | ||

$\gamma $ | $\mathcal{U}(2r,20r)$ |

**Table 3.**Quantitative comparison between the average trend of the simulated estimates ${\widehat{\beta}}_{\mathrm{s},\mathrm{sim}}$ and the estimates ${\widehat{\beta}}_{\mathrm{s},\mathrm{mean}}$ of the measurement experiments (see Figure 11) for sensors with different numbers of electrodes.

Approach | RMS $({\widehat{\mathit{\beta}}}_{\mathbf{s},\mathbf{meas}}-{\widehat{\mathit{\beta}}}_{\mathbf{s},\mathbf{sim}})/{\mathit{\rho}}_{\mathbf{bulk}}$ |
---|---|

% | |

2 elec. cal.-based | 0.94 |

4 elec. ECT-based | 1.07 |

8 elec. ECT-based | 0.71 |

**Table 4.**Quantitative comparison between the average trend of the simulated estimates ${\widehat{\beta}}_{\mathrm{s},\mathrm{sim}}$ and the estimates ${\widehat{\beta}}_{\mathrm{s},\mathrm{mean}}$ of the measurement experiments (see Figure 13) for different ECT-based estimation algorithms.

Approach | RMS $({\widehat{\mathit{\beta}}}_{\mathbf{s},\mathbf{meas}}-{\widehat{\mathit{\beta}}}_{\mathbf{s},\mathbf{sim}})/{\mathit{\rho}}_{\mathbf{bulk}}$ |
---|---|

% | |

MAP | 1.08 |

MAP enh. er. | 0.71 |

OSOA | 1.33 |

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## Share and Cite

**MDPI and ACS Style**

Suppan, T.; Neumayer, M.; Bretterklieber, T.; Puttinger, S.; Wegleiter, H.
A Model-Based Analysis of Capacitive Flow Metering for Pneumatic Conveying Systems: A Comparison between Calibration-Based and Tomographic Approaches. *Sensors* **2022**, *22*, 856.
https://doi.org/10.3390/s22030856

**AMA Style**

Suppan T, Neumayer M, Bretterklieber T, Puttinger S, Wegleiter H.
A Model-Based Analysis of Capacitive Flow Metering for Pneumatic Conveying Systems: A Comparison between Calibration-Based and Tomographic Approaches. *Sensors*. 2022; 22(3):856.
https://doi.org/10.3390/s22030856

**Chicago/Turabian Style**

Suppan, Thomas, Markus Neumayer, Thomas Bretterklieber, Stefan Puttinger, and Hannes Wegleiter.
2022. "A Model-Based Analysis of Capacitive Flow Metering for Pneumatic Conveying Systems: A Comparison between Calibration-Based and Tomographic Approaches" *Sensors* 22, no. 3: 856.
https://doi.org/10.3390/s22030856