# Analysis of a Symmetrical Ferrofluid Sloshing Vibration Energy Harvester

^{*}

## Abstract

**:**

^{2}is imparted, and the whole system is studied numerically using a level-set method to track the sharp interface between ferrofluid and air. The system is studied for two significant length scales of 0.1 m and 0.05 m while varying the four external magnets’ polarity arrangements. All of the system configuration dimensions are parametrized with the length scale to keep the system configuration invariant with the length scale. Finally, a frequency sweep is performed, encompassing the structure’s first modal frequency and impedance matching to obtain the system’s energy harvesting characteristics.

## 1. Introduction

^{2}acceleration.

## 2. System Set-Up

## 3. Governing Equations

## 4. Numerical Modeling and Validation

^{−4}) is reached. This convergence criterion is for both solution and the residual. The following segregated steps of dependent variables are solved in the following order:

- (a)
- Level set variable
- (b)
- Velocity and pressure
- (c)
- Magnetic scalar potential.

## 5. Results and Discussion

^{5}ohms to generate an impedance matching plot. This was performed for all the frequencies and both the tanks.

^{3.5}ohms, and for the vertical coil 10

^{2.75}ohms. Appendix B shows the time evolution of voltage, current, and external acceleration for this 2.05 Hz case.

#### 5.1. Analysis of Output from L = 10 cm Tank

^{5}ohms, with an increment of 10

^{0.25}ohms. This covers the total range of coil resistances shown in Table 4.

#### 5.2. Analysis of Output from L = 5 cm Tank

^{5}ohms, with an increment of 10

^{0.25}ohms.

## 6. Conclusions

^{2}. The tanks are then subjected to a frequency sweep for 16 different combinations of the four magnets by changing their polarity.

- (a)
- The maximum RMS voltages harvested reached almost 0.15 V
- (b)
- The maximum power after impedance matching was obtained from 10 cm tank, which is around a value of 20 µW or 2$\left(\frac{\mathsf{\mu}W}{g}\right)$.
- (c)
- For a 5cm tank, the highest power output was 0.7 µW, which is higher than what was obtained from the horizontal winding for L = 10 cm tank.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Surface Evolution for L = 10 cm and L = 5 cm Tank

## Appendix B. Waveform Plots for Acceleration, Voltage & Current

**Figure A32.**Voltage evolution for horizontal coil with 1000 turns for case 7 of magnet arrangement and 2.05 Hz frequency of excitation.

**Figure A33.**Current evolution for horizontal coil with 1000 turns for case 7 of magnet arrangement and 2.05 Hz frequency of excita-tion.

**Figure A34.**Voltage evolution for vertical coil with 1000 turns for case 7 of magnet arrangement and 2.05 Hz frequency of excitation.

**Figure A35.**Current evolution for vertical coil with 1000 turns for case 7 of magnet arrangement and 2.05 Hz frequency of excitation.

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**Figure 8.**Impedance matching for horizontal coil of 1000 turns on L = 10 cm tank for frequency = 2.05 Hz.

**Figure 9.**Impedance matching for Vertical coil of 1000 turns on L = 10 cm tank for frequency = 2.05 Hz.

**Figure 10.**Surface evolution and magnetic flux density arrow surface for case 1, near resonance frequency for L = 10 cm tank.

**Figure 11.**RMS voltage plots for Cases 1–16 while considering horizontally wound coil for L = 10 cm tank.

**Figure 12.**RMS voltage plots for Cases 1–16 while considering vertically wound coil for L = 10 cm tank.

**Figure 15.**Surface evolution and magnetic flux density arrow surface for case 1, near resonance frequency for L = 5 cm tank.

**Figure 16.**RMS voltage plots for Cases 1–16 while considering horizontally wound coil for L = 5 cm tank.

**Figure 17.**RMS voltage plots for Cases 1–16 while considering vertically wound coil for L = 5 cm tank.

Reference | Resonant Frequency (Hz) | Volume (cc) | Power (µW) | Power Density (µW/cc) | Acceleration (m/s^{2}) |
---|---|---|---|---|---|

[11] | 9 Hz | 44.23 | 0.85 | 0.02 | 3 |

[26] | 2.2 Hz | 1049 | 80,000 | 76.26 | 10 |

[27] | 2.1 Hz | 1000 | 80 | 0.08 | 0.3 |

Case | Magnet 1 | Magnet 2 | Magnet 3 | Magnet 4 |
---|---|---|---|---|

1 | South (k1 = −1) | North (k2 = −1) | South (k3 = −1) | North (k4 = −1) |

2 | South (k1 = −1) | North (k2 = −1) | South (k3 = −1) | South (k4 = 1) |

3 | South (k1 = −1) | North (k2 = −1) | North (k3 = 1) | North (k4 = −1) |

4 | South (k1 = −1) | North (k2 = −1) | North (k3 = 1) | South (k4 = 1) |

5 | South (k1 = −1) | South (k2 = 1) | South (k3 = −1) | North (k4 = −1) |

6 | South (k1 = −1) | South (k2 = 1) | South (k3 = −1) | South (k4 = 1) |

7 | South (k1 = −1) | South (k2 = 1) | North (k3 = 1) | North (k4 = −1) |

8 | South (k1 = −1) | South (k2 = 1) | North (k3 = 1) | South (k4 = 1) |

9 | North (k1 = 1) | North (k2 = −1) | South (k3 = −1) | North (k4 = −1) |

10 | North (k1 = 1) | North (k2 = −1) | South (k3 = −1) | South (k4 = 1) |

11 | North (k1 = 1) | North (k2 = −1) | North (k3 = 1) | North (k4 = −1) |

12 | North (k1 = 1) | North (k2 = −1) | North (k3 = 1) | South (k4 = 1) |

13 | North (k1 = 1) | South (k2 = 1) | South (k3 = −1) | North (k4 = −1) |

14 | North (k1 = 1) | South (k2 = 1) | South (k3 = −1) | South (k4 = 1) |

15 | North (k1 = 1) | South (k2 = 1) | North (k3 = 1) | North (k4 = −1) |

16 | North (k1 = 1) | South (k2 = 1) | North (k3 = 1) | South (k4 = 1) |

Property | Value |
---|---|

Viscosity | 12 [mPa·s] |

Density | 1420 [kg/m^{3}] |

Magnetic susceptibility | 3.52 |

Saturation magnetization | 65 [mT] |

Coil Number | Number of Turns | Resistance of Horizontal Coil (ohm) | Resistance of Vertical Coil (ohm) | Inductance of Horizontal Coil (H) | Inductance of Vertical Coil (H) |
---|---|---|---|---|---|

1 | 100 | 3.422 | 0.64 | 0.0083 | 0.0050 |

2 | 200 | 27.38 | 5.13 | 0.038 | 0.023 |

3 | 400 | 219.04 | 41.07 | 0.17 | 0.10 |

4 | 500 | 427.81 | 80.21 | 0.27 | 0.17 |

5 | 800 | 1752.30 | 328.55 | 0.75 | 0.47 |

6 | 1000 | 3422.47 | 641.71 | 1.20 | 0.77 |

7 | 1200 | 5914.02 | 1108.88 | 1.77 | 1.14 |

8 | 1500 | 11,550.83 | 2165.78 | 2.85 | 1.84 |

9 | 2000 | 27,379.74 | 5133.70 | 5.25 | 3.42 |

Coil Number | Number of Turns | Resistance of Horizontal Coil (ohm) | Resistance of Vertical Coil (ohm) | Inductance of Horizontal Coil (H) | Inductance of Vertical Coil (H) |
---|---|---|---|---|---|

1 | 100 | 6.84 | 1.28 | 0.0042 | 0.0025 |

2 | 200 | 54.76 | 10.27 | 0.019 | 0.012 |

3 | 400 | 438.07 | 82.14 | 0.084 | 0.053 |

4 | 500 | 855.62 | 160.43 | 0.13 | 0.086 |

5 | 800 | 3504.61 | 657.11 | 0.37 | 0.24 |

6 | 1000 | 6844.93 | 1283.43 | 0.60 | 0.39 |

7 | 1200 | 11828.05 | 2217.76 | 0.89 | 0.57 |

8 | 1500 | 23101.66 | 4331.56 | 1.43 | 0.92 |

9 | 2000 | 54759.48 | 10267.40 | 2.63 | 1.71 |

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**MDPI and ACS Style**

Anand, N.; Gould, R.
Analysis of a Symmetrical Ferrofluid Sloshing Vibration Energy Harvester. *Fluids* **2021**, *6*, 295.
https://doi.org/10.3390/fluids6080295

**AMA Style**

Anand N, Gould R.
Analysis of a Symmetrical Ferrofluid Sloshing Vibration Energy Harvester. *Fluids*. 2021; 6(8):295.
https://doi.org/10.3390/fluids6080295

**Chicago/Turabian Style**

Anand, Nadish, and Richard Gould.
2021. "Analysis of a Symmetrical Ferrofluid Sloshing Vibration Energy Harvester" *Fluids* 6, no. 8: 295.
https://doi.org/10.3390/fluids6080295