# Optimized Route Planning under the Effect of Hull and Propeller Fouling and Considering Ocean Currents

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{2}emissions was proposed while considering safety and environmental aspects. Speed loss is taken into consideration, pointing out its influence on fuel oil consumption. In Kytariolou and Themelis [8], a new weather routing tool was presented. The tool aimed to calculate the optimal path, in terms of fuel oil consumption, while considering weather forecast data and predefined safety criteria. It is well known that environmental factors affect fuel oil consumption and the safety of the voyage. In [9], the three-dimensional modified isochrone (3DMI) method was applied to solve the ship routing problem considering environmental elements such as wind, wave, and ocean currents. Optimization was performed following two different strategies—estimated time of arrival (ETA routing) and minimum fuel oil consumption (FUEL routing)—along with different constraints. Results show that when aiming to reduce fuel oil consumption, the optimal path requires more time and distance to be travelled in comparison with ETA routing. In addition, when safety constraints are also considered in the optimization process, for both cases, total fuel oil consumption and voyage duration are negatively impacted. Kurosawa et al. [10] developed a new routing system based on the A* algorithm, targeting coastal seas. The optimization considered the shortest distance, the minimum travelling time, and the minimum fuel oil consumption. Atmospheric and oceanic data were considered—whenever the data were updated, the optimal trajectory was recalculated. In addition, Shin et al. [11] proposed an improved A* algorithm to find the optimal path for economical ship operations. The improved method was based on an adaptive grid system, reducing exhaustive calculations required by the original A* algorithm. The validity of the new method was highlighted via a presented case study, where the original and improved methods were both utilized.

## 3. Weather Routing Tool

## 4. Calculation of the Main Engine’s Fuel Oil Consumption

_{tot}) is calculated as follows:

_{tot}= R

_{calm}+ R

_{wave}+ R

_{wind}

_{calm}, R

_{wave}, and R

_{wind}refer to calm water, added wave and wind components of the resistance, respectively. Having determined the ship’s total resistance, the brake power can be calculated as follows:

_{B}= R

_{tot}V

_{S}/(kη

_{S}η

_{0}η

_{H}η

_{R})

_{S}is the ship’s speed; k corresponds to the number of the propellers; η

_{S}is the shaft efficiency; η

_{0}is the propeller’s open water efficiency, which equals (K

_{T}J)/(K

_{Q}2π); η

_{H}is the hull efficiency, which equals (1 − t)/(1 − w); J is the advance coefficient; t is the trust deduction coefficient; and w is the wake coefficient, while η

_{R}is the propeller’s total rotational coefficient.

_{T}, K

_{Q}, J, and n are determined using the propeller’s open water diagram, which corresponds to calm water conditions. Firstly, the following quantity is calculated:

_{T}/J

^{2}= [T/(ρ n

^{2}D

^{4})]/(V/nD)

^{2}= T/(ρ V

^{2}D

^{2}) = CC

_{T}= CCJ

^{2}with the curve (K

_{T}− J) of the open water diagram of the propeller, the values of J, K

_{T}, K

_{Q}, and η

_{0}can be determined. Then, we have the following:

_{ad}/(JD)

_{ad}= V

_{s}(1 − w) is the advance speed of the propeller when operating in the ship’s wake, and V

_{s}is the ship’s speed.

_{B}) using Equations (2) and (4). Then, between every two points of interest (named x

_{i}and x

_{i+1}), the corresponding fuel oil consumption (FOC

_{i}) can be derived:

_{i}= t

_{i}SFOC(n

_{i},P

_{Bi})P

_{Bi}

_{i}is the sailing time from x

_{i}to x

_{i+1}while assuming constant weather conditions and ship speed.

## 5. Fouling Impact

_{f}= 0.044[(k

_{S}/L

_{WL})

^{3}− 10Re

^{(−1/3)}] + 0.000125

_{S}is the equivalent sand roughness height of the hull surface, L

_{WL}is the ship’s length at the waterline, Re is Reynolds number, and δC

_{f}is the empirical roughness allowance.

_{FL}= 1/2ρSV

_{S}

^{2}δC

_{f}

_{S}is the ship’s speed. The quantity of R

_{calm}in Equation (1) is now replaced with Equation (8).

_{calm_fouled}= R

_{calm}+ R

_{FL}

_{TS}= K

_{TM}− ΔK

_{T}

_{QS}= K

_{QM}− ΔK

_{Q}

_{TS}and K

_{QS}are the thrust and torque coefficients of the full-scale propeller incorporating the fouling impact; K

_{TM}and K

_{QM}are the thrust and torque coefficients without fouling, while according to [18]; and ΔK

_{T}and ΔK

_{Q}are derived by Equations (11) and (12), respectively:

_{T}= −0.3ΔC

_{D}(p/D)(c/D)Z

_{Q}= 0.25ΔC

_{D}(c/D)Z

_{D}is the difference in the drag coefficient:

_{D}= C

_{DM}− C

_{DS}

_{DM}= 2(1 + 2.0t

_{p}/c)[0.044/(Re

_{co})

^{1/6}− 5/(Re

_{co})

^{2/3}]

_{DS}= 2(1 + 2.0t

_{p}/c)[1.89 + 1.62log(c/k

_{p})]

^{−2.5}

_{p}is the maximum thickness, Z is the number of blades, p/D is the pitch ratio, Re

_{co}is the Reynolds number at x = 0.75R in the open water test based on the chord length and local velocity, and k

_{p}is the roughness of the propeller blade, which reflects the propeller’s fouling.

## 6. Case Study

#### 6.1. Examined Ship, Route, and Fouling Cases

- Case 1: Clean hull and propeller;
- Case 2: Hull fouling represented with k
_{s}= 5000 μm (mentioned as medium hull fouling) and clean propeller; - Case 3: Hull fouling represented with k
_{s}= 10,000 μm (mentioned as heavy hull fouling) and clean propeller; - Case 4: Hull fouling represented with k
_{s}= 5000 μm and propeller fouling represented with k_{p}= 1000 μm (mentioned as medium hull and propeller fouling); - Case 5: Hull fouling represented with k
_{s}= 10,000 μm and propeller fouling represented with k_{p}= 2000 μm (mentioned as heavy hull and propeller fouling).

#### 6.2. Fouling Impact

_{p}= 1000 μm and k

_{p}= 2000 μm, respectively) compared with the respective curves corresponding to the clean condition of the propeller.

- STW minus the currents’ velocity when the currents’ flow direction is the same as the ship’s direction.
- STW plus the currents’ velocity when the currents’ flow direction is the opposite of the ship’s direction.

#### 6.3. Optimisation Set Up

#### 6.4. Examination of the Impact of Fouling on the Orthodromic Path

#### 6.5. Examination of the Impact of Hull and Propeller Fouling on Optimized Routes

#### 6.6. Ignoring Current Effects

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Optimal routes with a constant speed of 18 kn for Cases 1 (clean hull), 2 (medium hull fouling), and 3 (heavy hull fouling).

**Figure 9.**Calm water resistance corresponding to optimum paths for different hull fouling conditions.

**Figure 10.**Added wave resistance corresponding to optimum paths for different hull fouling conditions.

**Figure 11.**Prevailing significant wave height corresponding to optimum paths for different hull fouling conditions.

**Figure 12.**Relative wave heading corresponding to optimum paths for different hull fouling conditions.

**Figure 13.**Fuel oil consumption per nautical mile for optimal paths found with and without the currents’ effect for clean hull conditions.

**Figure 15.**Fuel oil consumption per nautical mile for optimal paths with and without current effects under medium and heavy hull fouling conditions.

**Figure 16.**Prevailing current velocity along the optimal routes found with and without the currents’ effects under medium and heavy hull fouling conditions.

**Figure 17.**Added wave resistance along the optimal paths found with and without the currents’ effect under heavy hull fouling conditions.

**Figure 18.**Optimal paths with (red route) and without (blue path) the currents’ effect under heavy hull fouling conditions.

Parameter | Value |
---|---|

Length B.P. (m) | 244.80 |

Breadth (m) | 32.25 |

Depth (m) | 19.30 |

Draft Sc. (m) | 12.60 |

TEU | 4250 |

DWT (t) | 50,829 |

Main Engine (kW) | 37,046.4 |

Service Speed (m) | 24.5 |

Parameter | Value |
---|---|

Chord–Diameter ratio at 0.7R | 0.3019 |

Thickness–Diameter ratio at 0.7R | 0.0144 |

Number of Blades Z | 6 |

Diameter of ship propeller D (m) | 7.8 |

Mean Pitch–Diameter ratio p/D | 0.975 |

Propeller Revolutions n_{prop} (rps) | 12 |

Orthodrome | |||||
---|---|---|---|---|---|

Clean Hull (Case 1) | Medium Hull Fouling (Case 2) | Heavy Hull Fouling (Case 3) | Medium Hull and Propeller Fouling (Case 4) | Heavy Hull and Propeller Fouling (Case 5) | |

FOC (t) | 362.7 | 451.9 (+24.6%) | 548.7 (+51.2%) | 521.3 (+43.7%) | 655.2 (+80.6%) |

Voyage duration (h) | 133.5 | ||||

Distance (nm) | 2402.2 |

**Table 4.**Fuel oil consumption (FOC) for the optimal route found with a clean hull when investigating in other conditions.

No. Case | Optimal Path with Clean Hull (t) | FOC: Optimal Path with Optimisation for Each Case (t) | Increase % |
---|---|---|---|

1 | 347.5 | - | |

2 | 436.2 | 435.3 | 0.20 |

3 | 528.9 | 528.2 | 0.14 |

4 | 504.1 | 504.0 | 0.01 |

5 | 630.8 | 630.5 | 0.04 |

Orthodrome with/without Currents’ Effect | Loxodrome with/without Currents’ Effect | Optimal with/without Currents’ Effect | Ostensible Optimal Route | |
---|---|---|---|---|

FOC (t) | 548.7/549.7 | 544.6/550.1 | 528.2/540.6 | 546.9 |

Voyage duration (h) | 133.5 | 136.1 | 137/133.9 | 133.9 |

Distance (nm) | 2402.2 | 2449.1 | 2477.9/2410.8 | 2410.8 |

FOC/distance (t/nm) | 0.23 | 0.222/0.224 | 0.213/0.224 | 0.226 |

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**MDPI and ACS Style**

Kytariolou, A.; Themelis, N.
Optimized Route Planning under the Effect of Hull and Propeller Fouling and Considering Ocean Currents. *J. Mar. Sci. Eng.* **2023**, *11*, 828.
https://doi.org/10.3390/jmse11040828

**AMA Style**

Kytariolou A, Themelis N.
Optimized Route Planning under the Effect of Hull and Propeller Fouling and Considering Ocean Currents. *Journal of Marine Science and Engineering*. 2023; 11(4):828.
https://doi.org/10.3390/jmse11040828

**Chicago/Turabian Style**

Kytariolou, Ageliki, and Nikos Themelis.
2023. "Optimized Route Planning under the Effect of Hull and Propeller Fouling and Considering Ocean Currents" *Journal of Marine Science and Engineering* 11, no. 4: 828.
https://doi.org/10.3390/jmse11040828