Rotor Sail Thrust and Route Fuel Saving

True wind is specified as a FROM bearing $\theta$ off the bow. In the ship frame, $x=V_s+U_t\cos\theta$, $y=U_t\sin\theta$, $U_a=\sqrt{x^2+y^2}$ and $\beta=\operatorname{atan2}(y,x)$. The other side of the ship gives the same forward force when rotation is reversed.

Height $H=6D$, top-disc diameter $2D$, projected area $A=HD$. Lift, drag and electrical power coefficients use the fifth-order fits in Tillig & Ringsberg (2020), attached PDF page 5, Eqs. 7–9. These fits represent the specified geometry and were adjusted against full-scale data. De Marco et al. (2016), sections 6–7, provides the earlier geometry-dependent study and force interpretation. Its rounded multivariable coefficient table is not used here because cancellation makes the reproduced fit sensitive to coefficient precision.

For all rotors, $L=\tfrac12\rho N A U_a^2 C_L$, $D_f=\tfrac12\rho N A U_a^2 C_D$, $T=L\sin\beta-D_f\cos\beta$. Electrical demand is $P_r=\tfrac12\rho N A U_a^3 C_P$ plus the entered running allowance. The power coefficient already represents electrical consumption; motor efficiency is not subtracted a second time. A stopped rotor uses $C_{D0}=0.8$, the approximation in Vigna & Figari (2023), attached PDF page 6, Eq. 6.

RPM limiting reduces the selected spin ratio; if the resulting ratio is below 1, the rotor stops. Exceeding the electrical or apparent-wind operating limit also stops it. Otherwise the controller compares the fuel saving of spinning and stopped states. It does not continuously optimize the spin ratio.

Main-engine unloading is $\Delta P=\min(TV_s/\eta_D,\ P_0(1-f_{min}))$; negative values add propulsion demand. Bin fuel saving is $\Delta\dot m=\Delta P\,s_{ME}-P_r s_{AE}$. Normalize time weights, multiply by bin hours, and sum. Baseline fuel includes propulsion and hotel demand. The $TV_s/\eta_D-P_r$ balance is consistent with MEPC.1/Circ.896, attached PDF page 13, but the route average uses all entered weather bins, not the circular's strongest-half global probability calculation.

Diameter-based Reynolds number is $Re=U_aD/\nu$, using the entered air kinematic viscosity. Tillig & Ringsberg §3.5, attached PDF page 9, compares the coefficients against a 4 m × 24 m full-scale rotor and reports substantial Reynolds-number scale effects. A warning flags a changed diameter, and another flags nonzero-flow route bins below $Re=10^6$. This threshold is an author-selected screening trigger, not a validated cutoff from the paper. The fits themselves contain no Reynolds correction.

• Three piecewise steady weather bins; constant ship speed, baseline propulsion power, hotel load, efficiencies and specific fuel consumptions throughout the voyage.
• Winds are representative at rotor height in the water reference frame. No automatic height correction or current correction is applied.
• Identical rotors experience the same wind without aerodynamic interaction. They reverse spin for the opposite wind side.
• Supplier rpm, power and wind limits are entered by the user. Defaults are screening assumptions, not ratings for a named product.
• The running electrical allowance is additional to the published power coefficient. Stopped electrical demand is zero.
• Unloading above the retained-engine cap receives no additional credit, while full candidate rotor demand is charged. This is conservative accounting, not a solved constant-speed force equilibrium under excess thrust.
• The schematic and animation are original qualitative illustrations. Particle paths and relative speeds do not determine the force coefficients.
• Air kinematic viscosity defaults to 15 mm²/s as a nominal screening assumption; change it for the local air conditions. Reynolds number changes the applicability warnings, not the fitted coefficients.

Preliminary estimates only. Coefficient geometry is fixed to aspect ratio 6 and top-disc ratio 2. The selectable spin range is restricted to 1–3. Below the minimum achievable spin ratio, only stopped drag is used. This avoids extrapolating the fits in spin ratio; it does not establish Reynolds-number applicability.

Full-scale applicability and scale extrapolation. The coefficients were adjusted against measurements on a 4 m diameter, 24 m high full-scale rotor (Tillig & Ringsberg §3.5, attached PDF page 9). The paper reports substantial scale effects; the same coefficients are not established for every diameter and wind speed. A different diameter explicitly extrapolates that calibration. Route bins below Re = 1 million receive a separate warning. That is a conservative screening trigger selected here, not a universal physical transition or a validity limit stated by the paper. Even above it, the source does not establish a complete Reynolds-independent operating envelope. Confirm rotor-specific force and electrical-power data at the intended Reynolds numbers before relying on the estimate.

No heel, leeway, rudder loss, hull–propeller interaction, wind gradient, rotor wake, stability, foundation strength, fatigue, start-up energy or equipment availability model is included. A transverse force is reported to reveal the load that the simplified fuel balance omits. Large arrays especially need an interaction model. The default route is hypothetical; the result is not an annual saving claim.

Two source checks verify a published rotational-speed example and a published coefficient value. They do not validate a complete voyage or establish an uncertainty percentage. A vessel-specific performance model and measured route weather are required for procurement or guarantees.

IMO basis: MEPC.1/Circ.896, 14 December 2021, used for energy-balance context only. This tool does not calculate regulatory EEDI/EEXI credit. Check against the guidance in force for regulatory use.