Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2018

ADDITIONAL EXERCISES

Additional Exercise 1 Consider a floating oscillating-body wave energy converter, with a vertical axis of symmetry, that can oscillate in heave (vertical oscillations). The converter reacts against the sea bottom through a rigid rod and a linear damper, whose damping coefficient is C, as represented in Fig. 1A. The incident waves are regular (sinusoidal) with radian frequency w and period T. You may assume deep water. The body is of cylindrical shape with a conical bottom. The radius of the cylinder is a = 7 m. The volume of water displaced by the floater in the absence of waves is V = 3.3 a3. The sea water density is r = 1025 kg m-3 . The added mass A and the radiation damping coefficient B of the floating body are given in dimensionless form in Figs 3 and 4. It is Figure 1B represents the same floater, with a linear spring of stiffness K > 0 mounted in parallel with the same linear damper (damping coefficient C). Figure 1C represents the same floater and the same linear damper (damping coefficient C) as in Fig. 1A. There is an additional body, deeply submerged, rigidly attached to the floater. The mean density of the fully submerged body is equal to the sea water density. Figure 2 represents, for each of the three devices A, B and C, of Figs 1A, 1B and 1C, the dimensionless averaged power output For deep water and axisymmetric body, it is

Deep water. Regular waves Fig. 1

T (s) Fig. 2

Indicate the correspondence between the three curves (curve 1, curve 2 and curve 3) of Fig.2, and the three devices (A, B and C) of Fig. 1. Justify. Determine, at least approximately, the value of the PTO damping coefficient C of devices A, B and C (the PTO damping coefficient C has the same value for all three devices). Determine, at least approximately, the stiffness K of the linear spring of device B. Determine, at least approximately, the mass plus added mass, m2+A2, of the fully submerged body of device C. (Note that, since the body is deeply submerged, its added mass is independent of frequency.) Assuming wave amplitude Aw independent of wave period T, which of the three devices, A, B and C, has the greatest peak power output ? Justify. Assuming wave amplitude Aw independent of wave period T, which of the three devices, A, B and C, has the greatest oscillation amplitude |X| at peak power output? Justify. Determine, at least approximately, that value for one of the three devices (at your choice), for Aw = 1 m. Which of the three devices, A, B and C, has the greatest absorption width at peak power? Justify.

Fig. 3

Fig. 4

Additional Exercise No. 2

Additional Exercise No. 3

Fig. 1

Fig. 2

Fig. 3

Frequency domain X X X

X How to represent the (linear) air turbine and the air chamber more accurately?

X

X

Additional Exercise 4

Additional Exercise 5

Additional Exercise 6

Additional Exercise 7

Additional Exercise 8 Fig. 1. Schematic representation of the wave energy converter.