# Design and Analysis of a Decoupling Buoyancy Wave Energy Converter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed WECDB Device

## 3. Dynamic Analysis of the WECDB Device

#### 3.1. Analysis of Active Mass Rising

_{BUOY}, drag force, weight, cable tension T

_{CF}, excitation force F

_{EX}, and radiation force F

_{RAD}.

_{EX}= f

_{e}A cos (ωt + Φ)

_{e}, A, ω, and Φ are the force excitation coefficient, wave amplitude, wave frequency, and wave phase, respectively. The coefficient (f

_{e}) was calculated using Equation (2):

_{ω}, ρ, and g are the radiation resistance, water density, and gravitational acceleration, respectively. The radiation force is given by Equation (3):

_{ω}, $\dot{Z}$, and$\ddot{Z}$ denote the aggregated mass, float vertical speed, and acceleration, respectively. The aggregated mass and radiation resistance of a cylindrical body are [21]:

_{ω}= 0.875·M

_{ω}= ω·ρ·(2·π/3)·a

^{3}·ε·e

^{−2·k·l}

_{SUB}) and the seawater density, as shown in Equation (5):

_{BUOY}= ρ·g·V

_{SUB}

_{fl}, Z

_{w}, and Z

_{FL}are the total float height, vertical displacement of the sea surface, and vertical displacement of the float. Figure 8 shows the displacements.

_{CF}is equal to the weight of the active mass (M

_{ACT}), but when a wave descends it is equal to zero. As previously explained, when a wave rises, the active mass is coupled to the float by the tension cable. As a consequence of this coupling, when a wave rises, the moving mass M is the mass of the float plus the mass of the active mass. However, when a wave descends, the active mass is decoupled from the float. Therefore, when a wave descends, the moving mass M is only the mass of the float.

#### 3.2. Analysis of Active Mass Descent

_{ACT}reaches the design height. The gravitationally stored energy can then be transformed into electrical energy. To this end, the ratchets are liberated. Then, the M

_{ACT}falls owing to gravitational force. As the mass moves inside a closed chamber, when the mass descends, the air pressure in the chamber increases. This increase in pressure, as well as the friction against the walls, reduces the M

_{ACT}speed. The M

_{ACT}movement is described by Equation (8), which, again, is Newton’s second law.

## 4. Case Study

^{3}. The active mass has a volume of 188 m

^{3}and a mass of 776,000 kg.

^{3}of air.

_{ACT}height and water level, respectively.

_{ACT}rises 3.3 m per cycle. This implies a stored energy of 24.8 MJ per cycle. Therefore, an average power of 3.1 MW is stored. The average incident power per unit width is [22]:

_{I}= (1/2)·ρ·g·v

_{g}·A

^{2}= 282.86 kW/m

_{g}denotes the group velocity. When the float diameter was 22.26 m and the wave period was 8 s, the incident energy was 50.4 MJ. The maximum extracted power for an axisymmetric device was [23]:

_{I}/k = 4.5 MW

^{−3}and β = 0.74. In order to make a fair comparison with the regular wave case previously studied, ω

_{0}will be chosen in such a way that both waves have the same energy:

_{ACT}reaches the design height (50 m in the case study), the gravitationally stored energy can be transformed into electrical energy. In this case, Equation (8) applies. To simulate the evolution of the system, another Simulink model was developed. This is illustrated in Figure 16. This model also sends to the workspace the variables whose evolution will be plotted.

## 5. Discussion

_{ACT}results in good performance. The hydrodynamic efficiency of this device for regular waves is 49.2% (24.8/50.4). This efficiency compares favorably with other heaving WEC systems [26]. Their hydrodynamic efficiencies (obtained by simulation) are mostly lower, varying between the 16% of a “single cylindrical body” and the 30% of the Danish Wave Energy Program System (30%). Only the Wavebob provides efficiencies similar to the device proposed here (between 40 and 51%) [27].

_{ACT}was unlocked, its height oscillated, as shown in Figure 17. This was due to the spring effect caused by the compression of air in the watertight chamber. This oscillation is dampened by the air and wall drag, and the energy generation in the turbine. All of these components are very small. The wall drag is small due to the low sliding friction factor. The air drag is low due to the relatively low mass speed (maximum value near 15 m/s). Regarding energy generation, during its first cycle, the average power was (a little lower than) 4 MW and the average power of the descending mass during its first cycle was 48 MW. This explains why the damping of the oscillation is very low. This oscillation was also reproduced in the watertight chamber air pressure, as shown in Figure 18.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 20.**Evolution of the available power during the active mass descent combining several devices.

Symbol | Quantity | Value |
---|---|---|

A | Wave amplitude | 3 m |

T | Wave period | 8 s |

ρ | Seawater density | 1026.7 kg/m^{3} |

g | Gravity acceleration | 9.807 m/s^{2} |

ω | Angular frequency | 0.7854 rad/s |

a | Float outer radius | 11.70 m |

S_{fl} | Float cross-section | 305.8 m^{2} |

H_{fl} | Float height | 5.2 m |

M_{fl} | Float mass | 4.893 × 10^{4} kg |

M_{Act} | Active mass | 7.666 × 10^{5} kg |

ε | Coefficient | 0.04 (dimensionless) |

Symbol | Quantity | Value |
---|---|---|

c_{a} | Viscous drag coefficient | 0.4 (dimensionless) |

S_{ActMass} | Surface of the active mass | 77.98 m^{2} |

P_{Ext} | Atmospheric air pressure (at sea level) | 101.325 kPa |

µ | Sliding friction factor | 0.04 (dimensionless) |

κ | Ratio of specific heats (air) | 1.4 (dimensionless) |

R | Perfect gas constant | 287.1 m^{2}/(s^{2}·K) |

Reference temperature at sea level | 288.15 K | |

Air density at reference conditions | 1.225 kg/m^{3} | |

S_{turb} | Cross-sectional area of the turbine | 0.2619 m^{2} |

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**MDPI and ACS Style**

Torres-Blanco, P.; Sánchez-Fernández, J.Á.
Design and Analysis of a Decoupling Buoyancy Wave Energy Converter. *J. Mar. Sci. Eng.* **2023**, *11*, 1496.
https://doi.org/10.3390/jmse11081496

**AMA Style**

Torres-Blanco P, Sánchez-Fernández JÁ.
Design and Analysis of a Decoupling Buoyancy Wave Energy Converter. *Journal of Marine Science and Engineering*. 2023; 11(8):1496.
https://doi.org/10.3390/jmse11081496

**Chicago/Turabian Style**

Torres-Blanco, Pablo, and José Ángel Sánchez-Fernández.
2023. "Design and Analysis of a Decoupling Buoyancy Wave Energy Converter" *Journal of Marine Science and Engineering* 11, no. 8: 1496.
https://doi.org/10.3390/jmse11081496