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The Power of Many? ..... Coupled Wave Energy Point Absorbers Paul Young MSc candidate, University of Otago Supervised by Craig Stevens (NIWA), Pat Langhorne & Vernon Squire (Otago) Talk outline Motivation WECs… WTF? The big idea The physics Results Where to next? World resource Wave energy flux magnitude (kW per metre of wavefront) Source: Pelamis Wave Power website Source: Smith et al (NIWA), Analysis for Marine Renewable Energy: Wave Energy, 2008 Source: Smith et al (NIWA), Analysis for Marine Renewable Energy: Wave Energy, 2008 • Practical worldwide resource ~ 2000-4000 TWh/year1 • (Current global demand ~ 17000 TWh/year) Advantages: High energy density Low social & environmental impact (?) Reliability & predictability (c.f. wind) Low EROEI (?) Direct desalination AND... 1. Estimate by UK Carbon Trust Talk outline Motivation The big idea The physics Results Where to next? Point absorbers Pros: Cons: Suitable for community scale Less disruption in event of device failure Non-resonant in typical sea conditions Lower efficiency Cheaper per kW/h? Maybe a linked chain of point absorbers will 'see' long wavelengths better than a lone device? Key questions Is it possible to obtain better power output (per unit) with a linked chain? (Can we improve peak efficiency and/or widen bandwidth?) How are the mooring forces affected? (Survivability) What is the interplay between the device spacing and the wavelength? My scheme: model device 1-D (surge only) idealisation Talk outline Motivation The big idea The physics Results Where to next? Further assumptions/simplifications Small-body approximation Linear, small amplitude waves Neglect hydrodynamic interaction between devices Forces Mooring forces Hydrodynamic forces: excitation, drag and radiation Master equation: maK FM K FE K FD K FR K (not including power take-off) Technical issues… Importance of memory effects Talk outline Motivation The big idea The physics Results Where to next? Validating numerical code For lone device with zero drag, easy to solve equation of motion analytically. Discrepancy between models with and without memory effects noticeable when nonlinear drag introduced, but small. HOT OFF THE PRESS: Things get interesting with multiple devices. Some good agreement... ...some poor agreement... Talk outline Motivation The big idea The physics Results Where to next? Mooring and linkage forces Chacterise as tension-only spring Spring stiffness FM J ,K = (Linkage force on device J from device K) { Device spacing − S x J − x K , ∣x J − x K∣ d 0, ∣x J − x K∣ d Position of device K Hydrodynamic forces (The tricky part...) Inline force on small(ish) bodies in oscillatory flow often described by Morison equation: Submerged volume Drag coefficient Fluid velocity 1 F = V s u˙ ma u− C d A∣x˙ − u∣ x˙ − u ˙ x¨ − 2 Area 'seen' Fluid density by fluid Added mass BUT added mass depends on the oscillation frequency... For device with a ≈ 2m, energy-bearing wavelengths in typical sea state are 0.056 ka 0.126 Semi-submerged sphere moving in surge But under nonlinear conditions, device response may be over much broader range of frequencies... How big is the effect? Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982. Falnes' formulation Wave forces are decomposed in frequency domain into excitation and radiation forces. For surge, under small-body approximation, these are1: F E ≈ [ V s ma F R≈ − ma i ] u˙ x¨ ( + damping term) 1 C d A∣x˙ − u∣ x˙ − u ˙ x¨ − (c.f. F = V s u˙ ma u− 2 1. Falnes, J.: Ocean Waves and Oscillating Systems: linear interactions including wave-energy extraction. 2002. ) Radiation force in time domain t ∞ x¨ − ∫ K F R = − ma x˙ t− d 0 Added mass at infinite frequency Impulse response function Added damping ∞ K t =2 ∫ cos t d 0 This expression is exact, but added mass and damping depend on body geometry. Thankfully... ...can fit an analytic function that isn't horrible Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982. ∞ ∫ Evaluate integrals 0 with MATLAB symbolic math toolbox to get: K t =2 cos t d Master equation m x¨J t = F M F M J= FM FM { J , K= J ,J − 1 FD J FM FE J FR J J ,J 1 − S x J − x K , ∣x J − x K∣ d 0, ∣x J − x K∣ d F D= − F E J= J V s ma i 1 C d A∣x˙ − u∣ x˙ − u 2 n.b. u= u x J , t u˙ x J , t t F R= − m a ∞ x¨ − ∫ K 0 x˙ t− d Solution method x1 v1 x2 Cast as 1st order vector equation for y= v 2 ⋮ xn (n.b. will be 4n entries with internal mass included) vn [] Solve numerically with 4th order Runge-Kutta procedure on MATLAB Memory integral giving good agreement for linear motion over wavelength range