## Abstract

The energy available for a sailing ship is a combination of wind energy and the energy in the water. The wind energy propels the ship blowing into the sail and in this way generates the thrust. The energy in the water can be transformed into electrical energy by means of a hydrokinetic turbine. The electrical energy can be stored in a battery or it can be converted into hydrogen by means of an electrolyzer. In such a combination the ship is called an energy ship.

In this work a new extended design method for the hydrokinetic turbine of the energy ship is presented and three different design variations based on Betz theory have been developed and verified with computational fluid dynamics CFD. The first Betz Standard (BS) design is based on the optimum turbine design according to Betz which is based on linear momentum theory. The second Betz Extended (BE) and third Propeller Like (PL) designs are also based on the theory of Betz but with an optimized extended airfoil length.

The theory and the design methods for each turbine are presented. The setup and the results of the numerical simulations are shown in detail and the advantages and disadvantages of each design method are discussed. Especially the different turbine characteristics, i.e. the axial force acting on the turbine, the torque and power including their dimensionless coefficients are analyzed and compared.

As an example, in a first analytical ideal design calculation according to the Betz theory, assuming a diameter of 890 mm and a ship velocity of 5.2 m/s, a power output of 25.8 kW was predicted for the BS design. With tip and profile losses the expected output is 21.9 kW. The results of the numerical calculation of the hydrokinetic turbine characteristics show that it has a typical behaviour as also found in wind turbines. The BS and BE design have its maximum power output near the design point at the design tip speed ratio λDBS = λDBE = 7. For the PL design λDPL is not known a priori but by means of the CFD results it is shown to be in the range of 4 < λDPL < 5. The BS design shows a maximum power output of about 17 kW with a power coefficient of cp = 0.4 at λOBS = 6.5. The BE and the PL designs show approximately the same maximum power output of about 21 kW with a power coefficient of cp ≈ 0.5 and hence are close to the predicted design output with losses. The BE and PL turbines show their maximum power output at λOBE = 5 respectively at λOPL = 4.2. However, the BE design has a much flatter power characteristics delivering the 21 kW over a much larger range of tip speed ratio, showing the advantage of this new design method. The extended airfoil surface of the BE design and the BS turbine design leads to a higher hydrodynamic resistance but also to a higher torque and power output. With those two designs, a power coefficient of almost 50% was achieved, quite close to the maximum theoretical possible power coefficient of Betz cp,Betz = 16/27 = 59.3%. Hence this is in the range achieved by wind turbines.