# Vibrational Responses of an Ultra-Large Cold-Water Pipe for Ocean Thermal Energy Conversion: A Numerical Approach

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## Abstract

**:**

## 1. Introduction

^{2}m

^{3}/s [7]. Such a substantial internal flow rate raises concerns about potential instability, which could lead to CWP failure. Floating OTEC plants are commonly located in tropical regions to harness the significant temperature gradient between deep and surface seawater. However, these regions are also susceptible to severe weather conditions, including typhoons and tsunamis. Consequently, the design of an ultra-large cold-water pipe must account for its ability to operate reliably under the influence of extreme wind, wave, and OTEC platform motion. Construction of a CWP accounts for 15% to 20% of the total capital cost, according to cost assessment studies for OTEC development [8]. Therefore, maximizing the design and performance of CWPs in OTEC projects is of economic importance.

## 2. Theoretical Formulas

- (1)
- The dynamic behavior of the pipe is considered to be in a two-dimensional plane.
- (2)
- Given that as the ratio of the pipe length to the diameter is sufficiently large, the cold-water conduit system can be examined using the Euler–Bernoulli theory.
- (3)
- The internal flow is homogeneous and unidirectional.
- (4)
- The frictional force between the pipe and fluid is neglected.
- (5)
- The pipe’s cross-sectional area remains unchanged.
- (6)
- The effect of platform motion on the pipe is in the axial direction.
- (7)
- The weight of the pipe is uniformly distributed.

**Figure 2.**Schematic vibration of a cold-water pipe conveying liquid flow under marine environmental conditions.

Nomenclature | Description |
---|---|

EI | Bending stiffness (N/m^{2}) |

L | Pipe length (m) |

m_{a} | Added mass (kg/m) |

m_{f} | Mass of the internal flow per unit length (kg/m) |

m_{r} | Mass of the pipe per unit length (kg/m) |

T(z) | Axial equivalent tension (N) |

U | Velocity of the internal flow (m/s) |

w(z,t) | Transverse displacement of the pipe (m) |

A_{i} | Internal cross-sectional area (m^{2}) |

A_{0} | External cross-sectional area (m^{2}) |

${\rho}_{r}$ | Density of the pipe (kg/m^{3}) |

${\rho}_{f}$ | Density of the seawater (kg/m^{3}) |

u | External flow velocity (m/s) |

t | Time of vibration (s) |

C_{a} | Added mass coefficient |

C_{d} | Adapted drag coefficient |

$\Omega $ | Circular frequency (rad/s) |

$\sigma $ | Structural damping coefficient |

g | Gravitational acceleration (m/s^{2}) |

T_{wc} | Weight of the clump (N) |

T_{d} | Dry weight (N) |

- (a)
- clamped-clamped boundary conditions (C-C)

- (b)
- clamped-clump weight boundary conditions (C-W)

- (c)
- clamped-free boundary conditions (C-F)

- (d)
- simply supported-simply supported boundary conditions (S-S)

- (e)
- simply supported-clump weight boundary conditions (S-W)

- (f)
- simply supported-free boundary conditions (S-F)

## 3. Proposed Vibration Model Using GITT Method

#### 3.1. Eigenfunctions and Eigenvalues

- (a)
- C-C

- (b)
- C-W

- (c)
- C-F

- (d)
- S-S

- (e)
- S-W

- (f)
- S-F

#### 3.2. Transformed Governing Equation

#### 3.3. Variation in the Fundamental Frequency

## 4. Results and Discussion

#### 4.1. Convergence and Accuracy

#### 4.2. Parametric Study

#### 4.2.1. Effects of the Boundary Condition

#### 4.2.2. Effects of Internal Flow

_{i}equals 1 for the C-C, C-W, and C-F boundary conditions and 0.5 for the S-S, S-W, and S-F boundary conditions.

#### 4.2.3. Effects of External Flow

#### 4.2.4. Effects of the Clump Weight

## 5. Conclusions

- (1)
- The eigenfunctions and eigenvalues were calculated for the C-W and S-W boundary conditions using the GITT for the first time.
- (2)
- The boundary conditions had a significant effect on the convergence of the transverse displacement, in that different boundary conditions changed the eigenfunctions and eigenvalues of the displacement function.
- (3)
- The first-mode natural frequency of the pipe decreased as the internal flow velocity increased under the C-C, C-F, and S-S boundary conditions but remained constant under the C-W, S-W, and S-F boundary conditions. The first-mode natural frequency is important as it likely to be associated with the critical velocity during the operation of a CWP.
- (4)
- The increase in the transverse displacement with an increasing external flow velocity showed a proportional relationship, and peak displacement of the pipe under the C-W boundary condition was smaller compared with the other boundary conditions.
- (5)
- By setting the weight of the clump at the bottom, the transverse displacement and the first-mode natural frequency of the pipe were adjusted, and the effect was better with the S-W boundary condition. The results of this research can be an important reference for improving the stability and safety of an ultra-large CWP by adjusting the clump weight.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The GITT solutions with different truncation orders of NW for the transverse displacement profiles ($x$ = 0.5).

**Figure 5.**The time histories of dimensionless transverse displacement of the pipe ($x$ = 0.5) using the GITT, Galerkin method and Fourier series expansion technique.

**Figure 6.**Variation in the dimensionless transverse displacement with the time ($\tau $) under different boundary conditions, (

**a**) C-C, (

**b**) C-W, (

**c**) C-F, (

**d**) S-S, (

**e**) S-W, (

**f**) S-F.

**Figure 7.**Variation in the dimensionless transverse displacement with the time ($\tau $) under different boundary conditions (

**a**) C-C, (

**b**) C-F, (

**c**) S-S.

**Figure 8.**The vibration of the first-mode natural frequency with respect to the dimensionless internal flow velocity under different boundary conditions (

**a**) first-mode frequency; (

**b**) first-mode frequency ratio.

**Figure 9.**The dimensionless transverse displacement and the first-mode natural frequency with respect to the dimensionless internal flow velocity: under boundary conditions (

**a**) C-W, (

**b**) S-W, (

**c**) S-F, and (

**d**) FFTs of the motions.

**Figure 10.**The dimensionless transverse displacement and the first-mode natural frequency analysis of the pipe vibration with different boundary conditions (

**a**) transverse displacement; (

**b**) frequency.

**Figure 12.**The variation in the transverse displacement with respect to the dimensionless internal flow velocity under the C-W boundary condition, (

**a**) $\mu =0$, (

**b**) $\mu =0.5$.

**Figure 13.**The variation in the dimensionless transverse displacement with respect to the dimensionless external flow velocity under the C-W boundary condition, (

**a**) $\upsilon =0$, (

**b**) $\mu =0.001$.

**Figure 14.**The variation in the dimensionless transverse displacement with respect to the dimensionless internal flow velocity under the S-W boundary condition, (

**a**) $\mu =0$, (

**b**) $\mu =0.5$.

**Figure 15.**The variation in the dimensionless transverse displacement with respect to the dimensionless external flow velocity under the S-W boundary condition, (

**a**) $\upsilon =0$, (

**b**) $\mu =0.001$.

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

Length (m) | 1000 |

Density of the pipe (kg/m3) | 1206 |

Density of the seawater (kg/m3) | 1025 |

Inner diameter (m) | 1.5 |

Outer diameter (m) | 1.6 |

Section area (m^{2}) | 0.243 |

Dry weight (N/m) | $2.88\times {10}^{3}$ |

Wet weight (N/m) | $4.32\times {10}^{2}$ |

Young’s modulus (Pa) | $1.38\times {10}^{10}$ |

Circular frequency (rad/s) | 110 |

Hysteretic damping loss factor | 0.016 |

Additional mass coefficient | 1.0 |

**Table 3.**The first-mode natural frequency and the dimensionless critical velocity under different boundary conditions.

Boundary Condition | Frequency (Hz) | Critical Velocity (${\mathit{\upsilon}}_{\mathit{c}}$) |
---|---|---|

C-C | 8.246 | 0.6830 |

C-W | 58.785 | - |

C-F | 8.160 | 0.6811 |

S-S | 9.246 | 0.5510 |

S-W | 27.336 | - |

S-F | 7.146 | - |

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**MDPI and ACS Style**

Tan, J.; Zhang, Y.; Zhang, L.; Duan, Q.; An, C.; Duan, M.
Vibrational Responses of an Ultra-Large Cold-Water Pipe for Ocean Thermal Energy Conversion: A Numerical Approach. *J. Mar. Sci. Eng.* **2023**, *11*, 2093.
https://doi.org/10.3390/jmse11112093

**AMA Style**

Tan J, Zhang Y, Zhang L, Duan Q, An C, Duan M.
Vibrational Responses of an Ultra-Large Cold-Water Pipe for Ocean Thermal Energy Conversion: A Numerical Approach. *Journal of Marine Science and Engineering*. 2023; 11(11):2093.
https://doi.org/10.3390/jmse11112093

**Chicago/Turabian Style**

Tan, Jian, Yulong Zhang, Li Zhang, Qingfeng Duan, Chen An, and Menglan Duan.
2023. "Vibrational Responses of an Ultra-Large Cold-Water Pipe for Ocean Thermal Energy Conversion: A Numerical Approach" *Journal of Marine Science and Engineering* 11, no. 11: 2093.
https://doi.org/10.3390/jmse11112093