# Optimal Design of Rotor Sails Based on Environmental Conditions and Cost

^{*}

## Abstract

**:**

_{2}emissions of ships. Previous studies focused on how rotor sails affect ship dynamics and energy consumption. In the present study, an optimization-based workflow was proposed to find the optimal design of a rotor sail based on given environmental conditions for a target ship. Since the performance of a rotor sail depends on both operational conditions and the design of the rotor sail, a two-level optimization problem was proposed to separate the optimization of operational conditions and rotor sail design. At the operational level, the spin ratio of a given rotor sail was optimized at each time step under different environmental conditions. Then, the design of the rotor sail was optimized on the design level considering the initial cost of rotor sails and the average operational cost of the ship depending on the environmental conditions and their probabilities. The reductions in energy consumption of ships having optimal rotor sail designs, considering 5-year, 10-year, 15-year, and 20-year investment plans were found to be 0.34%, 2.7%, 3.91%, and 4.29%, respectively. When more severe environmental conditions were assumed for the 10-year investment plan, the diameter of the rotor sail increased and the reduction in energy consumption increased from 2.7% to 4.06%.

## 1. Introduction

_{2}emissions have become an issue that needs to be studied further day by day. With their measures taken in many industries, CO

_{2}emission levels are planned to be reduced and the effects of global warming will be suppressed. Comprehensive studies are also being carried out for the purpose of decarbonizing the maritime industry. In particular, the measures to reduce CO

_{2}emissions determined by the International Maritime Organization (IMO) have been effective in this manner. IMO has announced regulations to achieve net-zero greenhouse gas emissions by 2050 [1]. Thus, studies related to the decarbonization of the shipping industry have gained momentum.

_{2}emissions [3]. Controllable-pitch propellers [4], reduction of the hull resistance through air cavities [5], and optimizing the hull form [6] are some of the other proposals related to ship design. Additionally, reducing ship speed [7,8,9] and optimizing the route [10,11] have been investigated to achieve a reduction in CO

_{2}emissions in shipping. In addition, different sailing systems have been proposed to use available wind power in the ocean as an additional energy source for ship propulsion. Rigid wind sails [12], rotor sails [13], kites [14], and suction wings are some of the proposed sailing systems. In the present study, wind-assistive propulsion systems were studied due to their applicability on new and existing ships, and rotor sails were chosen among them due to their large force generation capacity per unit projected area [15]. Lu and Ringsberg also compared rigid wind sails, rotor sails, and soft sails, and found that rotor sails generate larger thrust [16]. Kramer et al. compared rigid wind sails and rotor sails in terms of drift forces [17]. In another study [18], numerical kite and rotor sail models were built and their contributions to thrust power were compared for different routes.

## 2. Methods

_{G}is the longitudinal coordinate of the center of gravity, S

_{rudder}is the projected rudder area, and C

_{B}is the block coefficient of the ship.

_{m}is the sway speed, r is the yaw rate, U is the resultant speed, β is the drift angle, m is the mass of the ship, m

_{x,y}are added mass components, I

_{zG}is the inertia of the ship around the center of mass, and J

_{z}is an added inertia term. The acting forces were formulated component-wise and given on the right side of the equation. Indices of H, P, R, Wind, Wave, and Rotor correspond to forces related to hull hydrodynamic resistance and the propeller, rudder, hull–wind interaction, hull–wave interaction, and rotor sail, respectively. The formulation of forces related to all terms except rotor sail force generation were derived from previous studies [29,30]:

_{A}, as given in Equation (2).

_{A}, coefficients of lift, drag, and power (c

_{L}, c

_{D}, and c

_{P}), and the projected area of the sail, A. Apparent wind speed, V

_{A}, is determined by true wind speed, true wind direction, ship speed, u, and v

_{m}, and heading angle, ψ. Aerodynamic coefficients c

_{L}, c

_{D}, and c

_{P}are defined based on aspect ratio, AR, diameter ratio, D

_{e}/D, and spin ratio, SR, of the rotor sail, as given in Equations (3)–(5).

_{L}, c

_{D}, and c

_{P}

_{,}and depends on both operational (spin ratio) and design parameters (height, aspect ratio, and diameter ratio) as they can be understood from the approximation formulas of lift, drag, and power coefficients. For this reason, the optimization problem was considered as a two-level problem. The design of the rotor sail was defined considering height, H, aspect ratio, AR = H/D, and diameter ratio, D

_{e}/D. Then, several scenarios with different wind speeds, wave heights, wave periods, and directions of wind/waves were considered. The operational parameter of the rotor sail, spin ratio (SR), was optimized during the simulation to maximize the power reduction due to the rotor sail for given design parameters and each scenario in the operational optimization level. Then, the performance of the rotor sail was evaluated considering the percent reduction in average power for each scenario and the probability of each scenario. On the upper level of optimization, the design parameters of the rotor sail were determined to minimize the cost, considering different investment plans. The overall workflow of the optimization problem is given in Figure 2.

_{e}/D were obtained for varying spin ratios, SRs, by using Equations (6)–(8). The range of spin ratio, SR, was determined for each scenario between ±5, considering the clockwise and counterclockwise rotation of the rotor sail. Then, lift and drag forces were calculated within the range of the spin ratio and transformed to the ship reference frame to find thrust and side forces generated by the rotor sail. To determine the spin ratio during simulations, the net power reduction was calculated as the difference between the thrust power generated by the rotor sail and the required power of the rotor sail, as given in Equation (9). Then, the optimal spin ratio was found by maximizing the net power reduction at each time step for all scenarios.

_{e}/D. The lower and upper limits of the design variables were determined as 10 and 40 m for height, 4 and 8 for aspect ratio, and 1.5 and 8 for diameter ratio based on the limits considered by a previous study [22] and existing rotor sails in the market.

_{initial}, and total energy consumption as given in Equation (10), where p

_{(i,j)}, P

_{prop}, and P

_{rotor}are probability, average propeller power (kW), and rotor sail power (kW) for a scenario having ith Beaufort scale and jth true wind/wave direction, h is specific fuel consumption (0.160 kg/kWh), c

_{fuel}is the price of fuel (0.6 USD/kg), w is working rate (0.5), and t is the interested time interval in hours.

_{initial}, was expected to increase. The initial cost of a rotor sail, c

_{initial}, was predicted based on the surface area, including lateral area, A

_{lateral}, and top plate area, A

_{top}, as well as volume of the cylinder, V, as given in Equation (11). The coefficients of a and b were assumed to be 2500 USD/m

^{2}and 800 USD/m

^{3}, respectively.

## 3. Results

_{2}emissions and the introduction of new policies such as carbon taxes pushes businesses to seek solutions for existing ships. New technologies or methods have been proposed to reduce the CO

_{2}emissions of existing and new ships. Therefore, it is necessary to find a solution for different investment periods. Optimal designs that minimize the total cost for the investment periods of 5, 10, 15, and 20 years were found, as shown in Figure 7. The optimal design for the 5-year investment is found to have quite a small size compared to the size of optimal designs for longer investment periods. When the investment period increases, optimal designs change by increasing the height and decreasing the aspect ratio. Although the ratio of end plate diameter and cylinder diameter, D

_{e}/D, significantly affects the aerodynamic coefficients, it was found that all optimal designs have a D

_{e}/D of 1.5, which is the lower bound. When the size of the optimal rotor sail increased together with the investment period, it was found that the required capital also increased and average power and operational cost decreased. A single rotor sail reached an approximately 4% reduction in average power for the cases of optimal design considering the 15-year and 20-year investment periods.

## 4. Discussion

_{2}emissions of large ships by using available wind energy and generating additional thrust. Many studies have been previously conducted to predict how much of a reduction in average power and CO

_{2}emissions can be achieved. Since there are many design alternatives, it is necessary to understand what kind of rotor sail design fits well and contributes to the reduction of CO

_{2}emissions at the maximum level.

_{2}emissions can be increased. Route optimization is one of the solutions to reducing energy consumption by traveling where strong winds exist in desirable directions. Reducing ship speed is another way of reducing energy consumption [7,8,9]. Additionally, changing the size and shape of the hull can contribute to a reduction in energy consumption.

## 5. Conclusions

_{2}emissions in the maritime industry. In the present study, it was aimed to develop a methodology to find the optimal rotor sail design that operates at optimal conditions depending on the environment. First, we have found that the design of rotor sails affects the optimal operating conditions. Then, a two-level optimization problem was proposed to find the optimal operating conditions to minimize average total power, considering different environmental conditions and the optimal rotor sail design by taking into account economic feasibility.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The distribution of the scenarios of (

**a**) environmental conditions given at Beaufort scale and (

**b**) true wind/wave directions.

**Figure 5.**(

**a**) The performance of the regression model, (

**b**) the effects of aspect ratio and (

**c**) diameter ratio on the lift, drag, and power coefficients.

**Figure 6.**System responses obtained for the ship without any wind-assistive system and with three different rotor sails.

**Figure 7.**Required capital, change in average power, and hourly operational cost of optimal rotor sail designs considering different investment periods.

**Figure 8.**(

**a**) Environmental characteristics and (

**b**) the contributions of optimal designs to the average power reduction under two different scenarios.

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

L (m) | 320 | ∇ (m^{3}) | 312,600 |

B (m) | 58 | x_{G} (m) | 11.2 |

d (m) | 20.8 | S_{rudder} (m^{2}) | 112.5 |

D (m) | 9.86 | C_{B} | 0.81 |

Lift (c_{L}) | Drag (c_{D}) | Power (c_{P}) | |||
---|---|---|---|---|---|

${a}_{0}$ | −0.0377 | ${b}_{0}$ | 0.1256 | ${c}_{0}$ | −9.5158 |

${a}_{1}$ | −0.0721 | ${b}_{1}$ | 0.2201 | ${c}_{1}$ | 2.2700 |

${a}_{2}$ | −0.0404 | ${b}_{2}$ | 0.1775 | ${c}_{2}$ | −0.1934 |

${a}_{3}$ | 2.9334 | ${b}_{3}$ | 0.1246 | ${c}_{3}$ | −0.0758 |

${a}_{4}$ | 0.0662 | ${b}_{4}$ | −0.0017 | ${c}_{4}$ | −0.0017 |

${a}_{5}$ | 0.0034 | ${b}_{5}$ | −0.0765 | ${c}_{5}$ | 2.4739 |

${a}_{6}$ | 0.0134 | ${b}_{6}$ | −0.0808 | ${c}_{6}$ | −0.2542 |

${a}_{7}$ | −0.0139 | ${b}_{7}$ | −0.0674 | ${c}_{7}$ | 0.0127 |

${a}_{8}$ | −0.0596 | ${b}_{8}$ | −0.0405 | ||

${a}_{9}$ | −0.8421 | ${b}_{9}$ | 0.5613 | ||

${a}_{10}$ | 0.0010 | ${b}_{10}$ | 0.0055 | ||

${a}_{11}$ | 0.0027 | ${b}_{11}$ | 0.0036 | ||

${a}_{12}$ | 0.0673 | ${b}_{12}$ | −0.0720 |

Beaufort Scale | Wind Speed (m/s) | Significant Wave Height (m) | Wave Period (s) |
---|---|---|---|

3 | 4.4 | 0.6 | 3 |

4 | 6.8 | 1 | 3.9 |

5 | 9.8 | 2 | 5.5 |

6 | 12.6 | 3 | 6.7 |

7 | 15.7 | 4 | 7.7 |

8 | 19 | 5 | 9.1 |

9 | 22.7 | 7 | 10.2 |

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**MDPI and ACS Style**

Guzelbulut, C.; Suzuki, K.
Optimal Design of Rotor Sails Based on Environmental Conditions and Cost. *J. Mar. Sci. Eng.* **2024**, *12*, 31.
https://doi.org/10.3390/jmse12010031

**AMA Style**

Guzelbulut C, Suzuki K.
Optimal Design of Rotor Sails Based on Environmental Conditions and Cost. *Journal of Marine Science and Engineering*. 2024; 12(1):31.
https://doi.org/10.3390/jmse12010031

**Chicago/Turabian Style**

Guzelbulut, Cem, and Katsuyuki Suzuki.
2024. "Optimal Design of Rotor Sails Based on Environmental Conditions and Cost" *Journal of Marine Science and Engineering* 12, no. 1: 31.
https://doi.org/10.3390/jmse12010031