# A New Methodology-Based Sensorial System with Which to Determine the Volume of Liquid Contained in a Cylindrical Tank Subjected to Full Variations in Its Orientation

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed Capacitive Sensors

#### Measurement of the Wet Surface

## 3. Proposed Methodology

- -
- $\phi $ (rad): Inclination of the longitudinal axis of the cylinder with regard to the horizontal plane.
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- ${H}_{{\mathit{wet}}_{1}}$(m): Vertical distance between the water level and the first flat side of the cylinder (denoted by the subscript 1 to represent the first sensor).
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- ${H}_{{\mathit{wet}}_{2}}$(m): Measurement of the length of the axial sensor under wet conditions (denoted by the subscript 2 to represent the second sensor).
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- ${H}_{{\mathit{wet}}_{3}}$(m): The vertical position of the water level in relation to the second flat surface of the cylinder (denoted by the subscript 3 to signify the third sensor).
- -
- ${H}_{{\mathit{dry}}_{1}}\left(\mathrm{m}\right)=2R-{H}_{{\mathit{dry}}_{1}}$: The vertical distance between the dry area and the first flat side of the cylinder, where R corresponds to the radius of the cylinder.
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- ${H}_{{\mathit{dry}}_{2}}\left(\mathrm{m}\right)=L-{H}_{{\mathit{dry}}_{2}}$: The dry length of the axial sensor, where L corresponds to the length of the cylinder.
- -
- ${H}_{{\mathit{dry}}_{3}}\left(\mathrm{m}\right)=2R-{H}_{{\mathit{dry}}_{3}}$: The vertical distance between the dry area and the second flat surface of the cylinder.
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- $\u03f5\to +0$: A mathematical term that represents the smallest expected positive angle for the inclination of the longitudinal axis.

#### 3.1. Volume of Liquid Contained in a Cylindrical Tank of Infinite Length

- -
- ${G}_{1}\left(\mathrm{m}\right)=\frac{{H}_{{\mathit{wet}}_{1}}-R}{tan\phi}$ equals the point where the axial sensor detects the water level. ${H}_{{\mathit{wet}}_{1}}<R\Rightarrow {G}_{1}=0$, although the axial sensor does not detect the water level.
- -
- ${L}_{{\mathit{wet}}_{1}}\left(\mathrm{m}\right)=\frac{{H}_{{\mathit{wet}}_{1}}}{tan\phi}$ equals the maximum range of water over the circular contour of the cylinder.

- -
- The water-free surface, which is the horizontal surface that limits the upper part of the water contained and is located between 0 and $\frac{w}{tan\phi}+{G}_{1}$.
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- The cylindrical surface has an infinite length. This surface limits the bottom of the water contained and is located between $-\sqrt{{R}^{2}-{w}^{2}}$ and $\sqrt{{R}^{2}-{w}^{2}}$.
- -
- The only flat side of the cylinder is partially wet and is located between $R-{H}_{{\mathit{wet}}_{1}}$ and R.

#### 3.2. Partially Wet Flat Faces

#### 3.3. A Partially Wet Flat Face

#### 3.4. Horizontal Singularity

#### 3.5. One Totally Wet Face, While the Other Face Is Partially Wet

#### 3.6. One Totally Wet Face, While the Other Face Is Dry

#### 3.7. Redundant Measurements

## 4. Description of the Experimental Setup

#### 4.1. Capacitive Measurement

#### 4.2. Architecture of Instrumentation

^{®}-25 [47], is read by the NI9870 Module [48] by means of an RS-232 connection. The IMU is powered by an Isolated Module DC-DC Converter [49]. Both modules are integrated into a cRIO-9074 [50], which is connected to a PC Host [51] using an Ethernet connection, as illustrated in Figure 14.

## 5. Experimental Results

#### Relative Errors

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TEC | Tidal Energy Converter |

AUV | Autonomous Underwater Vehicles |

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**Figure 3.**Function $\zeta -sin\zeta $ for $0\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}\zeta \phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}2\pi $ (rad).

**Figure 10.**Lateral view of the cylinder when one face is totally wet and the other is partially wet.

**Figure 13.**Main parts of the prototype designed. (

**a**) Outer part of the cylinder cover (1); (

**b**) inner part of the cylinder cover (1); (

**c**) aluminum ring piece (8); (

**d**) rubber gasket (2); (

**e**) metallic plate (2); (

**f**) metallic plate fitted in the cover; (

**g**) inner view of the cover with layer of vinyl (4); (

**h**) Inertial Measurement Unit (IMU); (

**i**) positions of the rod and the IMU.

**Figure 15.**Circuit designed to accommodate the capacitive sensor of the reading range of the covers.

**Figure 16.**Experimental results for a volume of 6 l for different angles. (

**a**) Voltage signal; (

**b**) wet height measurement; (

**c**) determined volume; (

**d**) filtered volume.

**Figure 17.**Experimental results for a volume of 10 l for different angles. (

**a**) Voltage signal; (

**b**) wet height measurement; (

**c**) determined volume; (

**d**) filtered volume.

**Figure 18.**Experimental results for the filling process. (

**a**) Voltage signal; (

**b**) wet height measurement; (

**c**) determined volume.

**Figure 19.**Experimental results for the emptying process. (

**a**) Voltage signal; (

**b**) wet height measurement; (

**c**) determined volume.

Case | ${\mathit{H}}_{{\mathit{wet}}_{1}}$ | ${\mathit{H}}_{{\mathit{wet}}_{2}}$ | ${\mathit{H}}_{{\mathit{wet}}_{3}}$ | Redundant Measurement |
---|---|---|---|---|

Section 3.2 | $\left(0,R\right)$ | 0 | $\left(0,R\right)$ | ${H}_{{\mathit{wet}}_{3}}={H}_{{\mathit{wet}}_{1}}-L\xb7tan\phi $ |

Section 3.2 | $\left[R,2R\right)$ | $\left(0,L\right)$ | $\left(0,R\right)$ | ${H}_{{\mathit{wet}}_{3}}={H}_{{\mathit{wet}}_{1}}-L\xb7tan\phi $ |

${H}_{{\mathit{wet}}_{2}}=\frac{{H}_{{\mathit{wet}}_{1}}-R}{tan\phi}$ | ||||

Section 3.2 | $\left(R,2R\right)$ | L | $\left[R,2R\right)$ | ${H}_{{\mathit{wet}}_{3}}={H}_{{\mathit{wet}}_{1}}-L\xb7tan\phi $ |

Section 3.3 | $\left[R,2R\right)$ | $\left(0,L\right)$ | 0 | ${H}_{{\mathit{wet}}_{2}}=\frac{{H}_{{\mathit{wet}}_{1}}-R}{tan\phi}$ |

Section 3.5 | $2R$ | $\left(\frac{R}{tan\phi},L\right)$ | $\left(0,R\right)$ | ${H}_{{\mathit{wet}}_{3}}=\left(H-L\right)\xb7tan\phi +2R$ |

$\Rightarrow H={H}_{{\mathit{wet}}_{2}}-\frac{R}{tan\phi}$ |

Case | ${\mathit{H}}_{{\mathit{wet}}_{1}}$ | ${\mathit{H}}_{{\mathit{wet}}_{2}}$ | ${\mathit{H}}_{{\mathit{wet}}_{3}}$ | Redundant Measurement |
---|---|---|---|---|

Section 3.4 | $\left(0,R\right)$ | 0 | $\left(0,R\right)$ | ${H}_{{\mathit{wet}}_{3}}\approx {H}_{{\mathit{wet}}_{1}}$ |

Section 3.4 | $\left[R,2R\right)$ | L | $\left[R,2R\right)$ | ${H}_{{\mathit{wet}}_{3}}\approx {H}_{{\mathit{wet}}_{1}}$ |

Circular Sensor Capacitance 1 | Completely Dry | ${C}_{1D}=0.51$ nF |

Completely Wet | ${C}_{1W}=11.89$ nF | |

Circular Sensor Capacitance 2 | Completely Dry | ${C}_{2D}=0.13$ nF |

Completely Wet | ${C}_{2W}=0.62$ nF | |

Circular Sensor Capacitance 3 | Completely Dry | ${C}_{3D}=0.44$ nF |

Completely Wet | ${C}_{3W}=12.45$ nF |

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## Share and Cite

**MDPI and ACS Style**

del Horno, L.; Segura, E.; Somolinos, J.A.; Morales, R.
A New Methodology-Based Sensorial System with Which to Determine the Volume of Liquid Contained in a Cylindrical Tank Subjected to Full Variations in Its Orientation. *J. Mar. Sci. Eng.* **2023**, *11*, 2316.
https://doi.org/10.3390/jmse11122316

**AMA Style**

del Horno L, Segura E, Somolinos JA, Morales R.
A New Methodology-Based Sensorial System with Which to Determine the Volume of Liquid Contained in a Cylindrical Tank Subjected to Full Variations in Its Orientation. *Journal of Marine Science and Engineering*. 2023; 11(12):2316.
https://doi.org/10.3390/jmse11122316

**Chicago/Turabian Style**

del Horno, Leticia, Eva Segura, José A. Somolinos, and Rafael Morales.
2023. "A New Methodology-Based Sensorial System with Which to Determine the Volume of Liquid Contained in a Cylindrical Tank Subjected to Full Variations in Its Orientation" *Journal of Marine Science and Engineering* 11, no. 12: 2316.
https://doi.org/10.3390/jmse11122316