The power of the motive fluid through the turbine relates to the kinetic energy by replacing the mass variable, m in the above equation, with the Mass Flow Rate variable (in kg/s):
Where one defines the Mass Flow Rate variable, as a function of the motive fluid density, ρ, the area of the gate of the turbine, A_{T}, and the fluid velocity v:
Given the above definition one may substitute the motive fluid density, ρ, the area of the gate of the turbine, A_{T}, and the fluid velocity v for the Mass Flow Rate variable, back into the Mass Flow Rate Dependent Power Equation. From here we can express the power extracted by the turbine as the change in power, ΔP_{T}, from that which drives the Turbofoil® equipped vessel when its turbine gate is closed. The power extracted, ΔP_{T}, manifests in a change in velocity of the Turbofoil® equipped vessel when its turbine gate opens, here expressed in terms of the difference between v_{0}, the vessel velocity with the turbine gates closed, and the resultant v_{1}, the vessel velocity with the turbine gates open:
Rearranging terms:
One finally arrives at an expression determining the final velocity, as analyzed in the Turbofoil® Hydrodynamic Analysis Spreadsheet:
The Turbofoil® Hydrodynamic Analysis Spreadsheet also makes use of a Froude Number, Hull Speed, and Speed:Length (kts/ft^{½}) Ratio in evaluating a Turbofoil® operating around The Big Island, Hawaii. A Turbofoil® equipped vessel at the Big Island of Hawaii with its turbine gate opened is probably still planing if: the vessel velocity is greater than Hull Speed; Speed:Length greater than an empirical value ranging from 1.34 to 1.51; and the Turbofoil® Froude Number =~ l'Hydroptère Froude Number and Turbofoil Speed greater than 12kts, the purported speed at which l'Hydroptère, the worlds fastest hydrofoil catamaran, or "HydroCat", begins to plane. Note that the Hull Speed in cell B15 of the Turbofoil® Hydrodynamic Analysis Spreadsheet is still less than the Open Turbine Gate Operational Speed derived in cell B16 even when the empirical number in cell B15 is changed from 1.34 to 1.51.
The Hawaii regime was chosen because its hindcast wind data indicates it has the least strong winds of the regimes proposed in Integrated Power Technology Corporation™'s Configuration Geographic database.
The greater the Froude Number, the greater the "resistance" due to bow and stern wave interference. This method is a generally accepted estimation in lieu of complete CFD analysis determining drag coeficients, etc. The Froude Number is given as:
Where L_{WL} is the hull length at the waterline in meters, g is the acceleration due to gravity, 9.8m/s. Hull Speed is given as:
Hull Speed in knots and L_{WL} is the hull length at the waterline in feet for this approximation.
The above graph illustrates results from an Interactive Boundary Layer analysis determining parameters such as Drag, Lift-to-Drag Ratio, Propulsive Power, Turbine Output Power, Sail Thrust Required, Levelized Cost of Energy (LCOE), 10 year Return-on-Investment (ROI), and Investment Rate of Return (IRR), amongst many others, thus creating a model from which to run Monte Carlo simulations and perform Trade Space Exploration to enable optimization of design and operational trade-offs. Integrated Power Technology Corporation™ has initiated this Lift-to-Drag model to further a Trade Space Exploration analysis for optimal Turbofoil® design, and the above graph is just one of many the workbook can generate. The model has 32 input and 88 output variables. Variables output from the model include Power Ratio, e_{p}, and Hydrodynamic Performance Rating, HPR, to compare candidate Turbofoil® designs to other existing hydrofoil supported vessels of similar displacement and performance as described below.
The Hydrodynamic Performance Rating workbook assembles hydrofoil performance data, and using Professor Gunter Hoppe's suggested equations, (2) (for Power Ratio, e_{p}, cells I19,20,22,23 in the HydroPerfRating spreadsheet tab; "the lower the better") and (7) (for Hydrodynamic Performance Rating, HPR, cells K19,20,22,23 in the HydroPerfRating spreadsheet tab; "the higher the better") from his paper Performance Evaluation of High Speed Surface Craft with Reference to the Hysucat Development. Here we compare two different Turbofoil® regimes, Hawaii and Chile to the other existing hydrofoils:
In the Hydrodynamic Performance Rating workbook, the Hawaii_Froude and Chile_Froude spreadsheet tabs use the power numbers from the original First Principles, Conservation of Energy analysis spreadsheet tabs, Hawaii_Turbo and Chile_Turbo, respectively, based on Turbofoil® Configuration Geographic Data, Hawaii and Chile sourced from AWStruewind and 10 year NASA hindcast wind data.
The Hawaii_Froude and Chile_Froude spreadsheet tabs give the Turbofoil® v_{1} speed (cell B16 in both sheets) after the turbine gates open, a change from the gate-closed speed v_{0} (Hawaii_Turbo tab and Chile_Turbo tab, cell B7 in both tabs), due to the power extracted by the turbine as the change in power, ΔP_{T} as described in the Hawaii_Froude tab and Chile_Froude tab, (cell B29 in both tabs).
Thus the Hydrodynamic Performance Rating workbook allows one to iteratively make estimates on the remaining Prime Mover power (cells G19,20,22,23 in the HydroPerfRating spreadsheet tab), needed to keep the vessel at the v_{1} speed after the Turbofoil® turbine gates open to extract the ΔP_{T}, ultimately an estimation of the Prime Mover Power based on estimates guided by conservative Power Ratio, e_{p}, (cells I15,16,18, 19; higher means more conservative) and Hydrodynamic Performance Rating, HPR (cells K15,16,18,29; lower means more conservative) in comparison to the other hydrofoils.
For the Turbofoil® regime in Hawaii, the Prime Mover (High Altitude Sail) power needs to be 4MW, (cell G19,20 in the HydroPerfRating spreadsheet tab), plus 4.8MW, (cell B29 in the Hawaii_Froude spreadsheet tab) or a total of about 9MW.
For the Turbofoil® regime in Chile, the Prime Mover (High Altitude Sail) power needs to be 10MW, (cell G22,23 in the HydroPerfRating spreadsheet tab), plus 10.8MW, (cell B29 in the Chile_Froude spreadsheet tab) or a total of about 21MW.
Several other Seaworthy vessels in operation today share similar concepts with a Turbofoil®, either in the form of a hydrofoil sailboat, a large vessel pulled by High Altitude Sails, or a vessel powered solely by Renewable Energy, three examples of which are documented here whose specifications are referenced elsewhere such as the Turbofoil® Hydrodynamic Analysis Spreadsheet.
L'Hydroptère, the worlds fastest hydrofoil catamaran sailboat, or "HydroCat" holds the record for over 57 knots in a short burst of time. A 60ft trimaran which, on reaching a speed of about 12 knots, is designed to lift all its hulls out of the water simultaneously to reduce drag; it planes along 5m above the surface on two hydrofoils that extend at an angle down into the water from the outer stabilisers. Five tons of titanium and carbon fiber materials comprise the vessel's superstructure, and can reach speeds of 50 knots over the water in winds of 25 to 30 knots.
Here are basic specifications for the l’Hydroptère:
Length | 18 meters |
Beam | 24.5 meters |
Height of Lateral Foils | 6.5 meters |
Weight at Takeoff with 5 crew | 6.5 tons |
Weight in Flight with Wind Ballast | 7 tons |
Main Sail Area | 165 square meters |
Sailplan | Fore-and-Aft Rigged |
The ship is 132m long with a deadweight 9,821t, a beam of 15.8m, a draft of 7.73m, an air draft of 37.50m to keel (29.77m +7.73m draft) and a cruising operational speed of 15.5kt.
The Beluga SkySails has been equipped with a 160m² sail similar to a paragliding sail. The area of the kite can be increased to 320m² if required for even more pulling power.
The MAK 8M32 diesel engine from Caterpillar has an output of 3,840kW with a volcano clutch, a transmission and a high-screw four-blade variable pitch propeller (Wärtsilä GMCP 800). There is one Wärtsilä electric bowthruster of 500kVA and the rudder is a free-hanging balance type from Rolls Royce.
There is a shaft generator (Stamford HCM 534 F2) of 813kVA, two diesel generators (Scania DI 12-62M) of 335kW, and one diesel emergency generator of 85kW (Stamford UCM 224 G SISU 420 DSRG). There are tanks for 3,540m³ of ballast water, 73.1m³ of fresh water and 621.7m³ of heavy fuel oil.
On route, the efficiency of the SkySails system was tested for up to eight hours a day were applicable in winds of up to force five (17–21 knots). The system was hailed as a success, with calculated savings of up to 2.5t of fuel/$1,000 a day. With larger sails of up to 600m², fuel savings of between 10% and 35% are possible. The Beluga SkySails will have a new 320m² sail installed as a continuation of its pilot testing.
Depending on the prevailing wind conditions, the latest SkySails product generation has a maximal propulsion power of more than 2 MW (approx. 2,700 hp; equivalent ship engine).
The worldwide patented SkySails System generates tractive force using large, dynamically flying towing kites, which in terms of physics is the most effective form of utilizing wind energy. With a good wind the SkySails SKS C 320 can produce a pulling force in the towing rope of more than 320 kilonewton (kN), a force greater than the thrust of both engines on an Airbus A321. The 32 meter width of the towing kite is just about as broad as the total wingspan of the A321.
SkySails towing kites for ships operate at altitudes between 100 and 500 m where stronger and more stable winds prevail. By means of dynamic flight maneuvers, e.g. "Figure 8's", SkySails easily generate five to 25 times more power per square meter sail area than conventional sails. The propulsion system requires a towing rope that is light weight and incredibly strong, since the weight of the rope reduces the power generated by the kite. Dyneema® ropes withstand the high tractive forces of more than 32 tonnes, generated in air speeds of up to 180 kilometres per hour(50m/s). Dyneema® is by far the world’s strongest fibre, it is 15 times stronger than steel, light weight and extremely resistant to corrosive chemicals, abrasion and friction. Gleistein Ropes developed the Dyneema® towing ropes to meet the specific needs of SkySails kites.
Specification | Data |
Length (with flaps) | 31 m (35 m) |
Beam (with flaps) | 15 m (23m) |
Height above waterline | 6.10m |
Draft | 1.55m |
Deadweight est. | 85,000 kg |
Crew planned for circumnavigation | 4 persons |
Working hours for completion | 68,000 |
Cruising Speed | 7.5 kts (14 km/h) |
Max. Speed | 14 kts (26 km/h) |
Solar Generator | |
Surface Area | 537 m² |
Power (STC) | 93.5 kW peak |
Efficiency | 18.8% |
Battery | |
Chemistry | Lithium Ion |
Battery Voltage | 388V |
Capacity | 2910 Ah (485Ah / cell) |
Total Cells | 648 |
Weight per cell | 13kg |
Total Weight Battery with chassis | about 11 tons |
Efficiency | > 95% |
Motor | (2 Motors each side) |
Type | Permanent Magnet Synchronous Motor |
Nominal Power | 2 x 10kW @ 1000 rpm (only 1 Motor each side) |
Maximal power | 2 x 60kW @ 1600 rpm (2 Motors each side) |
Transmission Ratio | 1:10 |
Efficiency @ nominal load | 92% |
Propeller and Steering System | Principle Vector Prop, Rudderless Steering System Pitch Control for optimization efficiency |
Propeller | 5 blades |
Diameter | 2 meters |
Nominal Speed /Maximal Speed | 100 rpm / 160 rpm |
Materials used | Carbon Fibre 20.6 tons Sandwich Core 11.5 tons Epoxy Resin + Hardener 23 tons |
Construction Cost | EUR 15 Million |
Maintenance (docking incl.) | Antifouling and Germanische Lloyd certification renewal: about EUR 120,000 |