# A Power Demand Analytical Model of Self-Propelled Vessels

## Abstract

**:**

## 1. Introduction

## 2. State of the Art of Vessel Performance Modeling

## 3. The Open-Water Propeller

## 4. The Full-Scale Vessel

## 5. Performance Evaluation

_{synth}), synthetic draft (Draft

_{synth}), synthetic rate of propeller rotation (n

_{synth}), and synthetic shaft power demand (P

_{S,synth}) with the biofouling coefficient already applied.

## 6. Conclusions

- −
- $\mathsf{\rho}$ is the mass density of the water;
- −
- $\mathsf{\nu}$ is the kinematic viscosity of the water;
- −
- $\mathrm{D}$ is the propeller diameter;
- −
- ${\mathrm{K}}_{\mathrm{Qo}}$, ${\mathrm{J}}_{\mathrm{oq}}$, ${\mathrm{k}}_{\mathrm{q}}$, ${\mathrm{K}}_{\mathrm{To}}$, ${\mathrm{J}}_{\mathrm{ot}}$ and ${\mathrm{k}}_{\mathrm{t}}$ are the open-water characteristics of the scaled model propeller;
- −
- ${\mathrm{P}}_{0.7}/\mathrm{D}$ is the pitch ratio at the blade section $\mathrm{r}/\mathrm{R}=0.7$;
- −
- ${\mathrm{t}}_{0.7}$ is the propeller maximum blade thickness at the blade section $\mathrm{r}/\mathrm{R}=0.7$;
- −
- ${\mathrm{c}}_{0.7}$ is the propeller blade chord length at the blade section $\mathrm{r}/\mathrm{R}=0.7$;
- −
- $\mathrm{Z}$ is the number of propeller blades;
- −
- ${\mathsf{\eta}}_{\mathrm{S}}$ is the shaft efficiency;
- −
- ${\mathsf{\eta}}_{\mathrm{R}}$ is the relative rotative efficiency;
- −
- $\mathrm{t}$ is the thrust deduction fraction;
- −
- ${\mathrm{k}}_{\mathrm{p}}$ is the blade roughness;
- −
- $\mathrm{n}$ are the propeller revolutions;
- −
- ${\mathrm{R}}_{\mathrm{T}}$ is the vessel towing resistance.

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Propeller DTMB 3376 open-water characteristics. Data from [54].

**Table A1.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to DTMB 3376 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.4575 |

${\mathrm{J}}_{\mathrm{ot}}$ | 1.181 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.5122 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0701 |

${\mathrm{J}}_{\mathrm{oq}}$ | 1.2327 |

${\mathrm{k}}_{\mathrm{q}}$ | 0.7298 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999987 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999920 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.999949 |

**Table A2.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to DTMB 3377 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.3880 |

${\mathrm{J}}_{\mathrm{ot}}$ | 1.0000 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.7115 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0492 |

${\mathrm{J}}_{\mathrm{oq}}$ | 1.0331 |

${\mathrm{k}}_{\mathrm{q}}$ | 0.2231 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999844 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999824 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.997424 |

**Figure A3.**Propeller DTMB 3378 open-water characteristics. Data from [54].

**Table A3.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to DTMB 3378 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.4872 |

${\mathrm{J}}_{\mathrm{ot}}$ | 1.1486 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.4307 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0721 |

${\mathrm{J}}_{\mathrm{oq}}$ | 1.2082 |

${\mathrm{k}}_{\mathrm{q}}$ | 0.6835 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999947 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999999 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.999854 |

**Table A4.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to DTMB 3379 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.4407 |

${\mathrm{J}}_{\mathrm{ot}}$ | 1.1165 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.5654 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0625 |

${\mathrm{J}}_{\mathrm{oq}}$ | 1.1732 |

${\mathrm{k}}_{\mathrm{q}}$ | 0.9210 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999746 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999544 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.999566 |

**Figure A5.**Propeller DTMB 3380 open-water characteristics. Data from [54].

**Table A5.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to DTMB 3380 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.4791 |

${\mathrm{J}}_{\mathrm{ot}}$ | 1.1879 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.4017 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0757 |

${\mathrm{J}}_{\mathrm{oq}}$ | 1.2654 |

${\mathrm{k}}_{\mathrm{q}}$ | 0.4292 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999892 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999967 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.999796 |

**Figure A6.**Propeller KP458 open-water characteristics. Data from [51].

**Table A6.**Fitting parameters and R

^{2}scores regressing Equations (9)–(11) to KP458 open-water propeller data.

${\mathrm{K}}_{\mathrm{To}}$ | 0.3174 |

${\mathrm{J}}_{\mathrm{ot}}$ | 0.8107 |

${\mathrm{k}}_{\mathrm{t}}$ | 0.4702 |

${\mathrm{K}}_{\mathrm{Qo}}$ | 0.0314 |

${\mathrm{J}}_{\mathrm{oq}}$ | 0.8765 |

${\mathrm{k}}_{\mathrm{q}}$ | 1.1326 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{T}}\right)$ | 0.999969 |

${\mathrm{R}}^{2}\left({\mathrm{K}}_{\mathrm{Q}}\right)$ | 0.999787 |

${\mathrm{R}}^{2}\left({\mathsf{\eta}}_{\mathrm{o}}\right)$ | 0.998864 |

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**Figure 1.**Shaft Power (${\mathrm{P}}_{\mathrm{S}}$) from Equation (23) and Series 60 Models 4221, 4280, 4281, 4282 data.

**Figure 2.**Flowchart diagram of the proposed process to obtain the time evolution of the performance of the vessel.

Model | Propeller | ${\mathbf{R}}^{2}\left({\mathbf{P}}_{\mathbf{S}}\right)$ |
---|---|---|

4210 | 3378 | 0.999868 |

4213 | 3379 | 0.999692 |

4214 | 3377 | 0.999994 |

4215 | 3378 | 0.999971 |

4218 | 3380 | 0.999655 |

4221 | 3376 | 0.999626 |

4256 | 3380 | 0.999651 |

4260 | 3377 | 0.999994 |

4272 | 3378 | 0.999919 |

4280 | 3376 | 0.999326 |

4281 | 3376 | 0.999254 |

4282 | 3376 | 0.999674 |

KVLCC2 | KP458 | 0.999865 |

$\mathbf{W}\mathbf{N}$ | $\mathbf{W}\mathbf{N}-2\left[\frac{\mathbf{W}\mathbf{N}}{2}\right]$ | $60+40\cdot \left(\mathbf{W}\mathbf{N}-2\left[\frac{\mathbf{W}\mathbf{N}}{2}\right]\right)$ |
---|---|---|

1 | 1 | 100 |

2 | 0 | 60 |

3 | 1 | 100 |

4 | 0 | 60 |

$\vdots $ | $\vdots $ | $\vdots $ |

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**MDPI and ACS Style**

Zamora, J.
A Power Demand Analytical Model of Self-Propelled Vessels. *J. Mar. Sci. Eng.* **2021**, *9*, 1450.
https://doi.org/10.3390/jmse9121450

**AMA Style**

Zamora J.
A Power Demand Analytical Model of Self-Propelled Vessels. *Journal of Marine Science and Engineering*. 2021; 9(12):1450.
https://doi.org/10.3390/jmse9121450

**Chicago/Turabian Style**

Zamora, Javier.
2021. "A Power Demand Analytical Model of Self-Propelled Vessels" *Journal of Marine Science and Engineering* 9, no. 12: 1450.
https://doi.org/10.3390/jmse9121450