# A Study on the Model of Detecting the Liquid Level of Sealed Containers Based on Kirchhoff Approximation Theory

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## Abstract

**:**

## 1. Introduction

## 2. Echo Sound Pressure Calculation Model

#### 2.1. Sound Field of a Round Piston Transducer in a Solid

^{5}gm/cm

^{2}·s.

#### 2.2. Model of Calculating Echo Sound Pressure

## 3. Experimental Results

#### 3.1. Experimental Setup and Initial Conditions

#### 3.2. The Result of Model Simulation

#### 3.3. Calculation of Echo Sound Pressure

#### 3.4. Determining the Liquid Level by Echo Sound Pressure

#### 3.5. Results of Experiment

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The measurement principle: ${\mathrm{R}}_{\mathrm{wg}}$ represents the reflection coefficient at the inner surface above the liquid level; ${\mathrm{R}}_{\mathrm{wl}}$ is the reflection coefficient below the liquid level. ${\mathrm{R}}_{\mathrm{ws}}$ represents the reflection coefficient at the outer surface of the container wall. ${\mathrm{P}}_{\mathrm{g}}$ and ${\mathrm{P}}_{\mathrm{l}}$ are the sound pressure relating to the echoes reflected by the inner surface of the container.

**Figure 3.**The sound pressure distribution of a round piston transducer in aluminum; (

**a**) the 2D distribution in the XOZ plane; and (

**b**) the 3D distribution in the YOZ plane.

**Figure 4.**The sound pressure distribution of a round transducer with a radius of 10 mm in cross sections along the beam propagation direction; (

**a**,

**c**) the beam propagation distance z = 8 mm; (

**b**,

**d**) the beam propagation distance z = 20 mm.

**Figure 5.**The model for calculating the echo sound pressure by using the Kirchhoff approximation theory.

**Figure 6.**The calculation of the echo sound pressure received by the transducer.${\text{}\mathrm{R}}_{\mathrm{wg}}$ and ${\mathrm{R}}_{\mathrm{wl}}$ represent the reflection coefficients at two parts of the circle section, respectively.

**Figure 7.**Measurement system: (

**a**) TX/RX is a transducer with transmitting and receiving function; and (

**b**) calibration devices in the experiments.

**Figure 8.**The echo sound pressure versus the exceeding height above the liquid level. The wall thickness L = 50 mm; the abscissa axis is the exceeding height $\Delta \mathrm{d}$, as defined in Figure 6.

**Figure 10.**The echo sound pressure of three kinds of liquid media with different ultrasonic impedance calculated under different wall thicknesses; the abscissa axis h represents the scale value of the container height. (

**a**) L = 8 mm; (

**b**) L = 25 mm; (

**c**) L = 40 mm; and (

**d**) L = 50 mm.

**Figure 11.**(

**a**) The method for determining the liquid level: ${\mathrm{P}}_{\mathrm{max}}$ and ${\mathrm{P}}_{\mathrm{min}}$ were the echo sound pressure corresponding to the two critical positions respectively; ${\mathrm{h}}_{\mathrm{max}}$ and ${\mathrm{h}}_{\mathrm{min}}$ were scale values associated with ${\mathrm{P}}_{\mathrm{max}}$ and ${\mathrm{P}}_{\mathrm{min}}$; ${\mathrm{h}}_{\mathrm{m}}$ is the height of the measured liquid level; and (

**b**) is a sample of the changing characteristics of the echo sound pressure measured by using a transducer with the radius a = 10 mm, the wall thickness L = 25 mm, and the liquid was water.

**Figure 12.**The experimental results taking water as an example: (

**a**) the diameters of the transmitting beam cross section versus the wall thickness; (

**b**) the difference of the echo sound pressure between the critical positions versus the wall thickness; (

**c**) the measured liquid level versus the wall thickness; (

**d**) the errors versus the wall thickness.

Liquid | ρ (g/cm^{3}) | Zl (gm/cm^{2}·s) | Zg (gm/cm^{2}·s) | Zm (gm/cm^{2}·s) | Rwg | Rwl | Rws |
---|---|---|---|---|---|---|---|

water | 1 | 1.48 × 10^{5} | 0.0004 × 10^{5} | 17 × 10^{5} | 0.999 | 0.839 | 0.839 |

edible oil | 0.91 | 1.28 × 10^{5} | 0.0004 × 10^{5} | 17 × 10^{5} | 0.999 | 0.859 | 0.839 |

glycerin | 1.27 | 2.42 × 10^{5} | 0.0004 × 10^{5} | 17 × 10^{5} | 0.999 | 0.750 | 0.839 |

a (mm) | f (MHz) | L (mm) | N (mm) | d (mm) | P_{min} (Pa) | P_{max} (Pa) | $\overline{{h}_{m}}$ (mm) | $\left|\overline{\Delta \mathbf{E}}\right|$ (mm) |
---|---|---|---|---|---|---|---|---|

5 | 1 | 8 | 3.96 | 19.68 | 0.61 | 1.31 | 202.1 | 2.1 |

5 | 1 | 25 | 3.96 | 60.53 | 0.81 | 1.72 | 201.6 | 1.6 |

5 | 1 | 40 | 3.96 | 96.58 | 0.81 | 1.69 | 201.9 | 1.9 |

5 | 1 | 50 | 3.96 | 120.60 | 0.79 | 1.66 | 202.6 | 2.6 |

10 | 1 | 8 | 15.87 | 20 | 5.50 | 11.95 | 203.3 | 3.3 |

10 | 1 | 25 | 15.87 | 27.59 | 4.22 | 8.99 | 198.9 | 1.1 |

10 | 1 | 40 | 15.87 | 40.08 | 3.48 | 7.29 | 198.8 | 1.2 |

10 | 1 | 50 | 15.87 | 48.41 | 3.20 | 6.66 | 198.5 | 1.5 |

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**MDPI and ACS Style**

Zhang, B.; Song, W.-A.; Wei, Y.-J.; Zhang, D.-S.; Liu, W.-Y.
A Study on the Model of Detecting the Liquid Level of Sealed Containers Based on Kirchhoff Approximation Theory. *Sensors* **2017**, *17*, 1394.
https://doi.org/10.3390/s17061394

**AMA Style**

Zhang B, Song W-A, Wei Y-J, Zhang D-S, Liu W-Y.
A Study on the Model of Detecting the Liquid Level of Sealed Containers Based on Kirchhoff Approximation Theory. *Sensors*. 2017; 17(6):1394.
https://doi.org/10.3390/s17061394

**Chicago/Turabian Style**

Zhang, Bin, Wen-Ai Song, Yue-Juan Wei, Dong-Song Zhang, and Wen-Yi Liu.
2017. "A Study on the Model of Detecting the Liquid Level of Sealed Containers Based on Kirchhoff Approximation Theory" *Sensors* 17, no. 6: 1394.
https://doi.org/10.3390/s17061394