Proceedings of the ASME 213 32nd International Conference on Ocean, Offshore and Arctic Engineering OMAE213 June 9-14, 213, Nantes, France OMAE213-11213 AIR TURBINE AND PRIMARY CONVERTER MATCHING IN SPAR-BUOY OSCILLATING WATER COLUMN WAVE ENERGY DEVICE J. C. C. Henriques A. F. O. Falcão R. P. F. Gomes L. M. C. Gato IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 149-1 Lisboa, Portugal ABSTRACT The oscillating water column (OWC) equipped with an air turbine is possibly the most reliable type of wave energy converter. The OWC spar-buoy is a simple concept for a floating OWC. It is an axisymmetric device (and so insensitive to wave direction) consisting basically of a (relatively long) submerged vertical tail tube open at both ends and fixed to a floater that moves essentially in heave. The air flow displaced by the motion of the OWC inner freesurface, relative to the buoy, drives an air turbine. The choice of air turbine type and size, the regulation of the turbine rotational speed and the rated power of the electrical equipment strongly affect the power performance of the device and also the equipment s capital cost. Here, numerical procedures and results are presented for the power output from turbines of different sizes equipping a given OWC sparbuoy in a given offshore wave climate, the rotational speed being optimized for each of the sea states that, together with their frequency of occurrence, characterize the wave climate. The new biradial self-rectifying air turbine was chosen as appropriate to the relatively large amplitude of the pressure oscillations in the OWC air chamber. Since the turbine is strongly non-linear and a fully-nonlinear model of air compressibility was adopted, a time domain analysis was required. The boundary-element numerical code WAMIT was used to obtain the hydrodynamic coefficients of the buoy and OWC, whereas the non-dimensional performance curves of the turbine were obtained from model testing. NOMENCLATURE Lowercase Latin c sound speed d buoy reference diameter g gravity acceleration h equivalent OWC chamber height, V /S 2 m i mass of body i n sea state number ṁ t turbine mass flow rate p pressure t time t x, t p see Eq. (23) x i vertical displacement of body i Uppercase Latin A ω amplitude of a wave with frequency ω A i j added mass on body i due to body j A i j lim ω A i j (ω) D turbine rotor diameter F di excitation force on body i K i j impulse response function on body i due to body j Lc dimensionless capture width, Eq. (37) H s significant wave height N turbine rotational speed P shaft power R i j radiation force on body i due to body j S i water plane area of body i T e energy period V volume Address all correspondence to this author. 1 Copyright 213 by ASME
Lowercase Greek η ε k γ ρ ρ w τ ω turbine aerodynamic efficiency residual of equation k specific heat ratio, c p /c v air density water density relative time starting at the beginning of the current absolute time step angular frequency Uppercase Greeks Γ excitation force amplitude coefficient Φ turbine flow rate coefficient, Eq. (19) Ψ turbine pressure coefficient, Eq. (18) Subscripts and superscripts initial conditions x average value of variable x ẋ derivative of x with respect to time ẍ second derivative of x with respect to time x dimensionless variable, x/x ref at atmospheric value n time step number ref reference value 1 INTRODUCTION The oscillating water column (OWC) equipped with an air turbine is possibly the most reliable type of wave energy converter. It has been extensively studied, and several prototypes have been deployed into the ocean [1, 2]. The OWC device comprises a partly submerged (floating or seabottom-fixed) structure, open below the water surface, inside which air is trapped above the water free surface. The oscillating motion of the internal free surface produced by the incident waves makes the air flow through a turbine that drives an electrical generator. In most cases, a self-rectifying air turbine (i.e., a turbine that does not need rectifying air valves) is used. Floating-structure OWCs deployed offshore are more adequate that bottom-standing OWCs for the extensive exploitation of the wave energy resource. An OWC spar-buoy is a simple concept for a floating OWC. It is an axisymmetric device (and so insensitive to wave direction) consisting basically of a (relatively long) submerged vertical tail tube open at both ends and fixed to a floater that moves essentially in heave. The length of the tube determines the resonance frequency of the inner water column. The air flow displaced by the motion of the OWC inner freesurface, relative to the buoy, drives an air turbine. Several types of wave-powered navigation buoys have been based on this concept [3, 4], which has also been considered for larger scale energy production [5]. The OWC spar buoy with a tube of uniform cross section was the object of two of the earliest published theoretical studies on wave energy converters [6,7]. The advantages of using a tube of non-uniform inner cross section were discussed in detail in [8, 9], where a procedure was presented for the optimization of the tube and buoy geometry and the damping coefficient of the assumedly linear air turbine for maximum wave energy absorption. Phase-control by latching of a spar-buoy in regular waves was numerically studied in [1]. The choice of air turbine type and size, the regulation of the turbine rotational speed and the rated power of the electrical equipment strongly affect the power performance of the device and also the equipment s capital cost. This matching was studied in [11, 12, 13] based on the numerical modelling of a bottom-fixed OWC equipped with a Wells turbine. Since the Wells turbine may be assumed with good approximation as exhibiting a proportionality relationship between pressure oscillation and air flow rate, the numerical modelling in [11, 12, 13] could be performed based in the frequency domain analysis, more precisely on a frequencydomain-based stochastic model for irregular waves. Here, numerical procedures and results are presented for the power output from turbines of different sizes meant for a given OWC spar-buoy in a given offshore wave climate, the rotational speed being optimized for each of the sea states that, together with their frequency of occurrence, characterize the wave climate. The new biradial self-rectifying air turbine [14, 15] was chosen as appropriate to the relatively large amplitude of the pressure oscillations in the OWC air chamber resulting from the energetic wave climate and from the large inertia of the water contained in the long tail tube. The biradial air turbine, like other impulse air turbines, is strongly non-linear, and so a time-domain, rather than a frequency-domain, analysis is required. A fully non-linear model of the air compressibility was adopted. The boundary-element numerical code WAMIT was used to obtain the hydrodynamic coefficients of the buoy and OWC, whereas the non-dimensional performance curves of the turbine were obtained from model testing as reported in [15]. 2 DEVICE MODELLING 2.1 Spar-buoy description The spar-buoy is represented in Fig. 1. Its geometry resulted from an optimization procedure as described in [14]. 2 Copyright 213 by ASME
-1-1 -2-2 -3-3 FIGURE 2. Perspective representation of biradial turbine. FIGURE 1. -4-5 View of the axisymmetic OWC spar-buoy. The floater diameter is 16 m and the draught is 48 m. The tail tube is tapered at the bottom, where the wall thickness is enlarged, as required by the geometry optimization. The buoy is equipped with a vertical-axis biradial selfrectifying air turbine, appropriate to the relatively large amplitude of the air pressure oscillations of a converter of this type in an energetic wave climate. The turbine is symmetrical with respect to a plane perpendicular to its axis of rotation, and is a radial-flow machine at both inlet and outlet, Fig. 2. The rotor is surrounded by a pair of axisymmetric ducts, each duct being formed by a pair of parallel discs. There are two rows of guide vanes rigidly connected to each other, as shown in Fig. 1. These guide vane rows may be removed from, or inserted into, the flow space by axially displacing the whole guide vane set, so that the downstream guide vanes are prevented from obstructing the flow coming out of the rotor (see Fig. 2). The radial distance between the rotor and the guide vanes is small, as shown in the figures. The turbine rotational speed is to be numerically optimized for each of the sea spectra that make up the local wave climate and for a set of turbine diameters. 2.2 Governing equations of the spar-buoy and OWC The spar-buoy (floater and tail tube) is named here as body 1. We neglect the deformation of the inner water free surface and define body 2 as an imaginary piston floating on the OWC surface, whose length is assumed small and -4-5 density equal to water density. Let x i be the coordinates of body i for the heaving motion, with x i = at equilibrium position and the x i -axes pointing upwards, as seen in Fig. 3. The motion equations for the two-body system are (m 1 + A 11) ẍ 1 + ρ w gs 1 x 1 + A 12 ẍ 2 p ref S 2 p = F d1 R 11 R 12, (m 2 + A 22) ẍ 2 + ρ w gs 2 x 2 + A 21 ẍ 1 + p ref S 2 p = F d2 R 22 R 21. (1) Here ρ w is the water density, g is the acceleration of gravity and m i is the mass of body i, S 1 is the annular cross sectional area of body 1 defined by the undisturbed free-surface plane, S 2 is the area of the piston flat top, F di is the hydrodynamic excitation force on body i, and R i j is the hydrodynamic radiation force on body i produced by the motion of body j. The air pressure inside the chamber is p + p at, where p at is the atmospheric pressure. The dimensionless relative pressure oscillation inside the chamber is defined as p = p/p ref with p ref = ρ w gh and H is the incident wave height. The radiation force is computed in the time domain as R i j = t K i j (t s)ẋ j (s)ds, (2) where K i j is the impulse response function.the added masses are given by A i j = lim ω A i j (ω), where A i j (ω) are the frequency-dependent added mass coefficients for the set of bodies spar-buoy/piston. Similar relations are applied to radiation forces on the piston. Since we are assuming linear water wave theory, the resulting diffraction force is obtained as a superposition of the frequency components F di (t) = n j=1 Γ i (ω j )A ω j cos(ω j t + 2πα j ), (3) where the phase of each component α j is a random variable governed by a uniform distribution over (, 1). The excitation force coefficients Γ i (ω j ) are computed using WAMIT. 3 Copyright 213 by ASME
where ρ is the air density and V the volume of air inside the chamber. The time-dependent volume of air is with V = (h + x 1 x 2 ) S 2, (9) h = V S 2, (1) where V is the volume of air inside the chamber in calm water. It should be noted that it must be h x 2 x 1 since the volume of air cannot be negative. Considering air a perfect gas and the air compression/expansion inside the chamber as an isentropic process, as proposed in [18], we have FIGURE 3. OWC. Cross section view of the axisymmetic floating In the current work, the wave amplitudes, A ω j, are related to the Pierson-Moskowitz spectrum [16] ( S ζ (ω) = 262.9 H2 s ω 5 Te 4 exp 154 ) ω 4 Te 4, (4) through A ω j = 2 ω j S ζ (ω j ). (5) The Pierson-Moskowitz spectrum is characterized by its significant wave height H s and energy period T e. The wave motion to be modelled is random and so non-periodic. To avoid periodicity in long time-series, we use non-equally spaced frequencies [17] ω j = (1 + δ rand()) ω, (6) where ω j = ω j 1 + 1 ( ) ω j + ω j 1, j = 2,...,n, (7) 2 with ω 1 =.1 rad/s, ω = 3./n rad/s, δ =.2 and rand() is a random number generator between and 1. All the spectra S ζ (ω) calculations were performed assuming n = 3. 2.3 Governing equations for the turbine The mass flow rate of air, ṁ t, through the turbine (positive for outward flow) is ρ V + ρv = ṁ t, (8) p + p at ρ γ = p at ρ γ, (11) at where γ = c p /c v is the specific heat ratio, approximately equal to 1.4 for air. The mass flow rate of air passing through the turbine is γ (p + p at)(ẋ 1 ẋ 2 ) + (h + x 1 x 2 ) ṗ = γ (p + p at )ṁ t, ρs 2 (12) where use was made of the relation dp p + p = γ dρ at ρ. (13) The differential equation (12) can thus be written as where q 1 ẋ 1 q 1 ẋ 2 + q 2 ṗ = q 3, (14) q 1 = γ (p + p at), (15) q 2 = h + x 1 x 2, (16) q 3 = ṁt c 2 ( )γ 1 at p γ p ref S 2 p + 1, (17) at and c at = (γ p at /ρ at ) 1 2 is the speed of sound under atmospheric conditions. The system of differential equations (1) and (14) is subject to the initial conditions x 1 () = x 1,, ẋ 1 () = ẋ 1,, x 2 () = x 2,, ẋ 2 () = ẋ 2, and p () = p. The performance characteristics of a turbine are defined in terms of the mass flow rate, the pressure head and the efficiency. They can be presented in dimensionless form, neglecting the effects of the variations in Reynolds number (since the Reynolds number is in general large enough for that) and Mach number (see [19]), as Φ = f m (Ψ) and η = f P (Ψ), where Ψ = p ref p ρn 2 D 2, (18) 4 Copyright 213 by ASME
Φ = ṁt ρnd 3, (19) where ρ is a reference density, usually defined in stagnation conditions at turbine entrance. Here we will assume simply ρ = ρ at. In (18) and (19), N is the turbine rotational speed in radians per unit time and D is the turbine rotor diameter. The time averaged turbine power output will be given by p t1 ref P = η (Ψ(p )) ṁ(ψ(p )) p dt. (2) ρ at (t 1 t ) t 2.4 Numerical solution of the equations of motion Usually the equations of motion are discretized with a four step Runge-Kutta or a similar ODE integrator that gives us the value of the function at the end of the time step with a specified order of accuracy. With this type of methods, the convolution integral can only be discretized with a trapezoidal rule thus reducing the order of accuracy of any high-order method to O(h 2 ). So, to achieve highorder accuracy we need a high-order representation of the velocity in the integration interval. From this point onwards we will refer to a relative time τ starting at the beginning of the current absolute time step, t n, such that τ = t t n. (21) The time step will be denoted by τ. Let us approximate x i and p by n th order polynomials x i = x i, + ẋ i, τ + t x a i, p = p + t p p, (22) where a i = (a 2,a 3,...,a n ) T i and p = (p 1, p 2,..., p n) T are column vectors of the unknown coefficients and t x = (τ 2,τ 3,...,τ n ), t p = (τ,τ 2,...,τ n ) are line vectors with the powers of τ. Consequently, (23) ẋ i = ẋ i, + ṫ x a i, (24) ẍ i = ṫ x a i, (25) and ṗ = ṫ p p. (26) The use of the relative time allows us to impose the initial conditions directly in the polynomial approximations (22). Introducing this approximating polynomials in Eqs. (1) and (14), we can write the resulting system of equations in a block matrix form Ay = f, (27) [ ].8.7.6.5.4.3.2.1. η Φ.1 1 2 3 4 5 6 7 8 Ψ [ ] FIGURE 4. Efficiency, η, and dimensionless mass flow rate, Φ as function of the dimensionless pressure head, Ψ, of the biradial turbine used in the numerical simulations. where with and h 1 A 12 ẗx p ref S 2 t p A = A 21 ẗx h 2 p ref S 2 t p, (28) q 1 ṫ x q 1 ṫ x q 2 ṫ p y = a 1 a 2, (29) p f = l + l, (3) h k = (m k + A kk )ẗ x + ρ w gs k t x, (31) F d1 R 11 R 12 l = F d2 R 22 R 21, (32) q 3 ρ w gs 1 [x 1, + ẋ 1, τ ] + p ref S 2 p l = ρ w gs 2 [x 2, + ẋ 2, τ ] p ref S 2 p. (33) q 1 ẋ 1, + q 1 ẋ 2, The vector of the polynomial coefficients y is computed using the normal system of equations associated with the continuous least squares problem A T Ay = A T f, (34) after integrating the entries of A T A and A T f (since they depend on τ) between and τ. This continuous least squares 5 Copyright 213 by ASME
TABLE 1. Characteristics of the 12 sea states used in the calculation of the average annual power absorption. n H s T e φ P wave [s] [m] [%] [kw/m] 1 1.1 5.49 7.14 3.2 2 1.18 6.5 12.53 4.4 3 1.23 7.75 8.29 5.7 4 1.88 6.33 11.74 1.9 5 1.96 7.97 2.96 14.9 6 2.7 9.75 8.73 2.4 7 2.14 11.58.6 25.9 8 3.6 8.3 9.55 36.7 9 3.18 9.93 1.22 49.1 1 3.29 11.8 2.6 62.4 11 4.75 9.84 4.79 18.5 12 4.91 12.3 2.85 141.8 formulation is a simplification of the one presented in [17]. For any column vector v, we have v T A T Av = (Av) T (Av) = (Av) (Av), therefore A T A is positive semi-definite. As a result, the matrix A T A is factorized using the Cholesky decomposition LDL T [2]. To simplify the integration of the A T f, we approximate the entries of l, Eq. (32), by interpolating polynomials of degree n at n + 1 equally spaced points in τ. Since Eq. (14) is non-linear and we need also to know the velocity in the current time step to compute the convolution integral, the system of Eqs. (34) must be solved iteratively until convergence. In this iterative process we have used the following stopping criterion where ε k = 1 τ [max(ε 1,ε 2 )] 1 2 < 1 9, (35) τ ( x (n) k ) x (n 1) 2 k dτ, (36) and (n) denotes the n th iteration. The pressure reference p ref was chosen to assure that x i and p have comparable orders of magnitude. 3 RESULTS A spar-buoy of 16 m diameter and a draught of 48 m, as shown in Fig. 3, were chosen for the numerical simulations. The hydrodynamic radiation damping and added mass coefficients of the spar-buoy were computed with a boundary element code [21], using 165 equally spaced frequencies between.1 and 2.476 rad/s and also the extrema and. The biradial turbine characteristic curves were taken from L c [ ].24.235.23.225.22.215.21.25 h =1. m h =12.5 m h =15. m h =2. m h =3. m h =5. m.2 1. 1.25 1.5 1.75 2. D [m] FIGURE 5. Dimensionless annual-averaged capture length Lc (based on turbine power output P) as function of the turbine diameter, D, for the wave climate of table 1. The rotational speed was optimized for each diameter and wave spectrum. model testing as reported in [15] and are shown in dimensionless form in Fig. 4. It should be noted that the relationship between pressure head and flow rate is non-linear. The study considered a set of turbine rotor diameters: 1., 1.25, 1.5, 1.75 and 2. m. A fourth degree polynomial was used to compute the spar-buoy and the OWC body positions as well as the air chamber pressure at each time step. The simulation time used was 14 s with a constant time step τ =.1 s. The time-averaged turbine power output was computed for the last 12 s. Figure 5 shows the dimensionless annual-averaged capture length, based on annual-averaged turbine power output P, Lc P = (37) P wave d as a function the turbine rotor diameter D, for the wave climate of Table 1 (for details see [8]) and for several values of the chamber height h. The rotational speed was optimized for each diameter and wave spectrum. Here, P wave is wave energy flux per unit of wave-crest length and d is the buoy diameter. The figure shows a slight increase in dimensionless capture length Lc as h increases from h = 1 m, with Lc reaching a maximum value at about h = 15 m and becoming markedly smaller at h = 5 m (an unrealistic value). The maximum produced energy occurs for turbine rotor diameter D 1.31 m, but the decrease in produced energy is only about 4.4% if the D is reduced to 1. m. Table 2 shows that, as expected, the optimum rotational speed N, for each spectrum, is larger for smaller tur- 6 Copyright 213 by ASME
TABLE 2. Optimum turbine rotational speed in rad/s for each spectrum and rotor diameter combination for an equivalent OWC chamber height of h = 15 m. n D [m] 1. 1.25 1.5 1.75 2. 1 59.1 37.9 26.3 18.6 7. 2 16.4 74.7 54.9 43. 34.6 3 81.7 58.2 43.6 33.6 26.6 4 13.1 75.4 56.8 45.1 36.5 5 137.2 1.9 78.4 62.6 5.9 6 177.4 133. 13.5 83. 67.5 7 151.3 111.5 84.9 67.7 54.9 8 242.5 183. 142.9 114.3 93.4 9 192.3 143.4 111.2 88.8 72.3 1 145.2 16.3 81.2 64.3 51.8 11 183.2 136.7 15.4 83.7 67.5 12 227.9 171.9 133.3 15.8 86.1 TABLE 3. Turbine time-averaged power output in kw at optimum rotational speed, for each spectrum and rotor diameter combination, for an equivalent OWC chamber height of h = 15 m. n D [m] 1. 1.25 1.5 1.75 2. 1 2.6 1.9 1.4 1. 1.1 2 25.2 24.2 22.7 2.5 18. 3 11.6 11.1 1.2 9. 7.8 4 26.2 26.8 25.8 24.2 21.7 5 69. 73.4 73.3 7.2 65.2 6 163.6 178. 182.3 179.4 171.2 7 95.7 1.2 99.9 95.4 89. 8 46.1 514.2 532.2 528.4 511.4 9 215.3 232.8 236.4 23.7 219.6 1 84.1 87.4 86.7 82.7 77.4 11 184.5 198.4 199.7 194.3 184.8 12 378.3 418. 429.1 423.9 41.9 bine rotor diameter D, and, for fixed D, is, in general, larger in the more energetic sea states. Note that the decrease in N from sea-states 2 to 3, 6 to 7 and 8 to 9 are due to lower hydrodynamic efficiency of the OWC spar-buoy. Note that, in spite of relatively large air pressure oscillations, in no case the rotor tip speed exceeds 121 m/s, which seems acceptable in terms of Mach number and centrifugal stresses. Table 3 shows that, in general (but not in all cases), the power output from the smaller turbines is larger than from the larger turbines in the less energetic sea states. For the more energetic sea states, the best turbine diameter is D = 1.5 m, being P = 532.2 kw the maximum power output, L c [ ].32.28.24.2.16.12.8.4 N =9 rad/s N =11 rad/s N =13 rad/s N =15 rad/s N =17 rad/s N =19 rad/s N =21 rad/s. 1. 1.25 1.5 1.75 2. D [m] FIGURE 6. Dimensionless capture length Lc (based on turbine power output P) as function of the turbine diameter, D, for a seastate H s = 2.8 m, T e = 8.14 s and the Pierson-Moskowitz spectrum, with an average power equal to the annual-averaged power of the wave climate given in Table 1. The equivalent OWC chamber height used was h = 15 m. for sea state 8. This should be taken into account when choosing the rated power of the electrical equipment. Note that larger maximum values for the wave power and turbine power output would be expected if a much larger set of sea states were adopted to define the wave climate. Figure 6 gives the dimensionless capture length Lc (based on turbine power output P) as function of the turbine diameter, D, for the single representative Pierson- Moskowitz spectrum with H s = 2.8 m, T e = 8.14 s whose average power is equal to the annual-averaged power of the adopted multi-spectra wave climate. The figure shows that the time-averaged turbine power output is not very sensitive to turbine size, provided that the right rotational speed is adopted, which should be larger for the smaller turbines. The figure also indicates D = 1.42 m as the optimal diameter, which is larger than the value 1.31 m computed for the more realistic multi-spectra wave climate, and Lc =.315 rather than.239; this confirms that, in simulations, the wave climate representation by a single sea state is unreliable [17]. 4 CONCLUSIONS The performance of an OWC spar-buoy wave energy converter, with a 16-metre diameter floater, was analysed in the time domain for a wave climate characterized by a set of 12 sea states each represented by a Pierson-Moskowitz spectrum and its frequency of occurrence. The converter was 7 Copyright 213 by ASME
equipped with the new biradial self-rectifying air turbine whose rotational speed was optimized for each sea state and each turbine rotor diameter. The numerical results indicate that, as expected, the optimal rotational speed is higher for the smaller turbines, and that the annual-averaged power output is not very sensitive to variations in turbine size within a relatively wide range of rotor diameters D, with D 1.31 m yielding the maximum annual energy production, and a small penalty incurred if a smaller (say 1 m) rotor diameter is adopted instead. These results confirm the OWC spar-buoy as a wave energy converter characterized, under optimal hydrodynamic performance conditions, by large damping and relatively small air flow rate, which are convenient requirements for a relatively small biradial turbine at large rotational speeds (which however do not seem unacceptably high from the aerodynamic, mechanical and electrical viewpoints). ACKNOWLEDGEMENTS This work was funded by the Portuguese Foundation for Science and Technology through LAETA/IDMEC and contracts PTDC/EME-MFE/13524/28 and PTDC/EME- MFE/111763/29, and by Project Offshore Test Station, KIC InnoEnergy, European Institute of Technology. The first author was supported through Ciência 27 initiative. The third author was supported by Ph.D. grant SFRH/BD/35295/27 from the MIT-Portugal Programme. Thanks are due to Kymaner Ltda, Lisbon, for allowing some turbine results to be used. APPENDIX A Asymptotic extension of the radiation damping coefficient for high frequencies The radiation damping values were computed by the Boundary Element Method WAMIT [21] and the dimensionless impulse response function is evaluated as K i j ( t) = 2 π B i j ( ω) ω cos( ω t)d ω, (38) where t = t(g/d) 2 1, ω = ω(d/g) 2 1, B i j = B i j /(ρd 3 ω) and K i j = K i j /(ρgd 2 ). When the panels characteristic dimension used in the geometry discretization is comparable with the incident wavelength, the computation of the hydrodynamic coefficients become less accurate. Since the values of B i j are required also at those frequencies to compute the impulse response function, we adopt an asymptotic extension defined as b i j ( ω) = 4 q=1 α q exp( β q ω q ). (39) Dimensionless radiation damping, B 11 ( ω).45.4.35.3.25.2.15.1.5 Mode 11.8.7.6.5.4.3.2.1 Zoom WAMIT Exp. fitting. 2. 2.3 2.6 2.9 3.2 3.5...5 1. 1.5 2. 2.5 3. 3.5 4. Dimensionless frequency, ω FIGURE 7. Radiation damping function for the spar-buoy obtained with a BEM code (circles) and the non-linear least-squares exponential approximation (line). The constants α q and β q are computed using a nonlinear least-squares algorithm to fit the vector of computed points B i j ( ω) from the first entry m that satisfies the conditions: (i) d B i j ( ω m )/d ω < and (ii) B i j ( ω m ).2max( B i j ( ω)); to the last entry. The integral (38) is then divided in two intervals: (i) ω [, ω m ], where B i j is interpolated with a cubic spline; (ii) ω [ ω m, [, where the approximation Eq. (39) is used. Fig. 7 shows the radiation damping coefficient of the piston (OWC). The results form the asymptotic extension eliminates the spurious numerical oscillations when the radiation damping tends to zero. REFERENCES [1] A. F. de O. Falcão, Wave energy utilization: A review of the technologies, Renewable and Sustainable Energy Reviews 14 (21) 899 918. [2] T. V. Heath, A review of oscillating water columns, Philosophical Transactions of the Royal Society A- Mathematical, Physical & Engineering Sciences 37 (212) 235 245. [3] Y. Masuda, An experience with wave power generation through tests and improvement, in: D. V. Evans, A. F. O. Falcão (Eds.), IUTAM Symp. Hydrodynamics of Ocean-Wave Energy Utilisation, Springer-Verlag, 1986, pp. 445 452. [4] T. J. T. Whittaker, F. A. McPeake, Design optimization of axisymmetric tail tube buoys, in: D. V. Evans, A. F. O. Falcão (Eds.), IUTAM Symp. Hydrodynamics of Ocean-Wave Energy Utilisation, Springer-Verlag, 8 Copyright 213 by ASME
1986, pp. 13 111. [5] A. Tucker, J. M. Pemberton, D. T. Swift-Hook, J. Swift-Hook, et al., Laminated reinforced concrete technology for the SPERBOY wave energy converter, in: Proc. XI World Renewable Energy Congress, Abu Dhabi, 21, pp. 941 946. [6] M. E. McCormick, Analysis of a wave energy conversion buoy, Journal of Hydronautics 8 (1974) 77 82. [7] M. E. McCormick, A modified linear analysis of a waveenergy conversion buoy, Ocean Engineering 3 (1976) 133 144. [8] R. P. F. Gomes, J. C. C. Henriques, L. M. C. Gato, A. F. O. Falcão, Hydrodynamic optimization of an axisymmetric floating oscillating water column for wave energy conversion, Renewable Energy 44 (212) 328 339. [9] A. F. O. Falcão, J. C. C. Henriques, J. J. Cândido, Dynamics and optimization of the OWC spar buoy wave energy converter, Renewable Energy 48 (212) 369 381. [1] J. C. C. Henriques, A. F. O. Falcão, R. P. F. Gomes, L. M. C. Gato, Latching control of an OWC spar-buoy wave energy converter in regular waves, in: Proceedings of the 31 st International Conference on Offshore Mechanics and Arctic Engineering, Rio de Janeiro, Brazil, 212. [11] A. F. de O. Falcão, Control of an oscillating-watercolumn wave power plant for maximum energy production, Applied Ocean Research 24 (2) (22) 73 82. [12] A. F. de O. Falcão, Stochastic modelling in wave power-equipment optimization: maximum energy production versus maximum profit, Ocean Engineering 31 (11-12) (24) 147 1421. [13] L. M. C. Gato, P. A. P. Justino, A. F. de O. Falcão, Optimisation of power take-off equipment for an oscillating water column wave energy plant, in: Proceedings of the 6 th European Wave and Tidal Energy Conference, Glasgow, Scotland, 25. [14] A. F. O. Falcão, L. M. C. Gato, E. P. A. S. Nunes, A novel radial self-rectifying air turbine for use in wave energy converters, Renewable Energy 5 (213) 289 298. [15] A. F. O. Falcão, L. M. C. Gato, E. P. A. S. Nunes, A novel radial self-rectifying air turbine for use in wave energy converters. Part 2. Results from model testing, Renewable Energy 53 (213) 159 164. [16] Y. Goda, Random Seas and Design of Maritime Structures, 2nd Ed., World Scientific Publishing, Singapore, 2. [17] J. C. C. Henriques, M. F. P. Lopes, R. P. F. Gomes, L. M. C. Gato, A. F. O. Falcão, On the annual wave energy absorption by two-body heaving WECs with latching control, Renewable Energy 45 (212) 31 4. [18] A. F. de O. Falcão, P. A. P. Justino, OWC wave energy devices with air flow control, Ocean Engineering 26 (1999) 1275 1295. [19] S. L. Dixon, Fluid Mechanics and Thermodynamics of Turbomachinery, 4th Edition, Butterworth, London, 1998. [2] D. S. Watkins, Fundamentals of Matrix Computations, John Wiley & Sons, New York, 1991. [21] C. H. Lee, J. N. Newman, Computation of wave effects using the panel method, in: S. Chakrabarti (Ed.), Numerical Models in Fluid-Structure Interaction, WIT Press, Southampton, United Kingdom, 25. 9 Copyright 213 by ASME