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1 Title Author(s) Primary wave energy onversions of osillating water olumns Sheng, Wanan; Alorn, Raymond; Lewis, Anthony Publiation date Original itation Tye of ubliation Link to ublisher's version Rights Sheng, W., Alorn, R. and Lewis, A. (213) 'Primary wave energy onversions of osillating water olumns', Proeedings of 1th Euroean wave and tidal energy onferene, 2-5 Setember, Aalborg University, Denmark. Conferene item htt:// Aess to the full text of the ublished version may require a subsrition. 213 Euroean Wave and Tidal Energy Conferene Item downloaded htt://hdl.handle.net/1468/2678 from Downloaded on T2:4:49Z

2 Primary Wave Energy Conversions of Osillating Water Columns Wanan Sheng #1, Raymond Alorn #2, Anthony Lewis #3 # Beaufort Researh-HMRC, University College Cork Youngline Industrial Estate, Pouladuff Road, Cork, Ireland 1 w.sheng@u.ie 2 r.alorn@u.ie 3 t.lewis@u.ie Abstrat This aer resents a study on the numerial simulation of the rimary wave energy onversion in the osillating water olumn (OWC) wave energy onverters (WECs). The new roosed numerial aroah onsists of three major omonents: otential flow analysis for the onventional hydrodynami arameters, suh as added mass, daming oeffiients, restoring fore oeffiients and wave exitations; the thermodynami analysis of the air in the air hamber, whih is under the assumtions of the given ower take-off harateristis and an isentroi roess of air flow. In the formulation, the air omressibility and its effets have been inluded; and a timedomain analysis by ombining the linear otential flow and the thermodynamis of the air flow in the hamber, in whih the hydrodynamis and thermodynamis/aerodynamis have been ouled together by the fore generated by the ressurised and de-ressurised air in the air hamber, whih in turn has effets on the motions of the struture and the internal water surfae. As an examle, the new develoed aroah has been alied to a fixed OWC devie. The omarisons of the measured data and the simulation results show the new method is very aable of rediting the erformane of the OWC devies. Keywords Wave energy onverter, osillating water olumn, air omressibility, ower take-off I. INTRODUCTION OWC wave energy onverters have been, often referred as the first generation of wave energy onverters, advaned a great deal sine the earliest OWC devies have been studied and widely imlemented for owering the navigation buoys sine 194s (see Falao [1]), and now some ratial OWC lants have been built and generated eletriity to the grid. It is reorted that the LIMPET OWC lant has generated eletriity to the grid for more than 6, hours in a eriod of more than 1 years (Heath [2]). A reent develoment is the Mutriku OWC wave energy lant in Sain [3], a multi- OWC wave energy onverters, inluding 16 sets of Wells turbines + eletrial generator (18.5 kw eah). It is estimated eletriity generation of 6 MWh so far [4]. To imrove wave energy onversion by the OWC devies, numerial analyses and hysial model tests are often used. Earlier theoretial work on the hydrodynami erformane of OWCs has shown that OWC devies ould have a high rimary wave energy onversion effiieny if the otimized daming an be attained (Sarmento et al [5], Evans [6], Evans and Porter [7]). However, the exerimental studies on the bottom-fixed or floating OWCs have shown that the wave energy onversion effiieny of an OWC devie very muh deends on the daming oeffiients of the flow assing through the ower take-off system, as well as the size of the water olumn (water olumn setional area and length). Toyota et al. [8] have shown that both the size of the air hamber and the length of the horizontal dut length of a BBDB devie have signifiant effets on the rimary ower onversion of the OWC wave energy onverters. Imai et al.[9] have also studied the influene of the horizontal dut length to the wave energy ature aaity in a BBDB devie, and shown that a longer horizontal dut has inreased the maximum internal-water-surfae (IWS) resonse to a longer resonane eriod. As a result of this, a longer horizontal dut may be desirable for tuning the BBDB to the wave states of longer wave eriods. Morris-Thomas et al. [1] have exerimentally studied the hydrodynami effiieny on fixed OWCs with different front shaes. From the omarison, it an be seen that the front shaes have some but limited effets on the wave energy onversion effiienies. For the tested four different front shaes, the wave energy ature

3 effiienies are overall similar, and the maximum wave energy onversion effiieny is about 7%. Another imortant aset for the OWC wave energy devies is the air omressibility in the air hamber due to the large sae and large ressure in the air hamber. Sarmento et al [11] have roosed a linearized formula for the flowrate through the ower take-off system, based on an assumtion of an isentroi flow. Sheng et al. [12] have reently formulated a full thermodynami equation for the air flow in the hamber and through the ower take-off system for studying the air omressibility. To validate the air omressibility redited by the numerial method, a iston onneted to a linear test rig, whih models an OWC devie, an be driven in a muh owerful manner so that an obvious air omressibility an be reated in the hamber. It is shown that the numerial method has well redited the omressibility when omared to the exerimental data. In the aer, a new numerial analysis method in the rimary wave energy onversion of the OWC devies has been develoed and desribed. The numerial analysis rinially onsists of following three omonents: Frequeny domain analysis: Conventional otential analysis to the hydrodynamis of the OWC devie, inluding the erformane of the water olumn (iston) in frequeny domain, and from whih the relevant arameters for time-domain are derived; in the frequeny domain analysis, a two-body system has been adoted. Thermodynami analysis of the air flow in hamber: The air flow in the hamber is studied under the given harateristis of the PTO system, i.e., the relation between the ressure dro aross the PTO and the flow rate through the PTO. The PTO devie an be linear, suh as the Wells turbine, or nonlinear (e.g., imulse turbine) or a more generi relation using a seond-order olynomial funtion. Time domain analysis: Time-domain (TD) equation is established for the dynamis of the water olumn (iston) for an OWC devie by ombining the struture of the devie (Rigid Body 1) and the water olumn (the iston, Rigid Body 2). As an examle, the OWC devie onsidered here is a fixed OWC whih has only one motion mode: heave. In the time domain analysis, the fore due to the hamber ressure is inluded for the iston motion (internal water surfae motion). A. Numerial aroahes II. RESEARCH ON OWC WECS Potential theory has been widely used to study the OWC devies. For some seifi OWC devies, suh as twodimensional OWC devies, or some three-dimensional OWCs with very simle strutures, analytial solutions are ossible (Evans and Porter [7], Martins-rivas et al [13], Mavrakos et al. [14]), but more oular aroahes are the numerial analysis using the well-develoed boundary element method, suh as WAMIT [15], ANSYS AQWA [16]et al. These ommerial odes are readily available for any geometry of interest. Based on the assumtion of the otential flow, the veloity otential of the flow around the floating struture satisfies the Lalae equation: 2 φ where φ is the frequeny-domain veloity otential of the flow around the floating struture (the orresonding timedeendent veloity otential should read Φ φe iω ). t An earth-fixed oordinate system is defined for the otential flow roblem. The oordinate is fixed in suh a way that the x-y lane is on the alm water surfae and z-axis ositive u vertially. Its origin oinides with the entre of gravity of the devie in alm water. In the oordinate, the free surfae onditions an be exressed in frequeny domain (see Lee et al. [17]), as 2 φ ω z g, φ iω ρg, ( on S f ( on S where ω is the wave frequeny, ρ the density of water, g the aeleration of gravity, the ressure amlitude ating on the interior water surfae, S i the interior water surfae in the water olumn, and S f the free surfae but exludes the interior free surfae. To solve the hydrodynami roblem, two different aroahes an be used: massless iston model [17, 18]and ressure distribution model (Evans et al. [19]). In the former aroah, the internal free surfae is assumed to behave as a massless rigid iston. In the latter aroah, the internal freesurfae ondition and the satial variation of the internal free surfae is reresented in terms of the dynami air ressure in the hamber. In this researh, the first aroah is alied. To inlude the interior water surfae as a boundary in the onventional BEM ode WAMIT [15], Lee et al.[2] introdued the soalled generalized modes for the interior water surfae motions, suh as the iston-tye and the first sloshing modes of the interior water motions so that the resonse of the interior water surfae an be obtained in WAMIT without signifiant modifiations to the ode. Following this rinile, the numerial simulations in this researh are all onduted using WAMIT. B. Exerimental investigations For OWC devies, the air assing through the PTO system is a reiroating roess during a wave eriod. To onvert ) i ) (1) (2)

4 the neumati ower into mehanial ower, different design strategies for the ower take-off have been roosed, but the most oular tyes of the neumati ower onversion are the air turbines, suh as the Wells turbine [21-23], imulse turbine [24-26], and the self-ith ontrolled air turbine [27] et al. A review on the air turbines for wave energy onversion has been reently given by Takao et al. [28]. The Wells turbine is well known to have a linear daming relation between the ressure dro aross the PTO and the flowrate, and their relation an be exressed as ressure, the extrated ower an be alulated either by the hamber ressure only or by the motion of the interior water surfae (from whih the flowrate an be easily alulated). k1q where is the hamber ressure, Q the airflow through the ower take-off system, and k 1 the daming oeffiient. For imulse turbines, the nonlinear relation between the hamber ressure and the flowrate an be aroximated by the following exression as, (3) k 2 Q Q where k 2 is the nonlinear daming oeffiient. In small sale OWC model tests, the PTO systems (air turbines) have been suosed to be saled for testing aording to the relevant similitude laws, the size and the daming harateristis and so on. However, it is ratially diffiult to manufature a small saled air turbine whih maintains the orret harateristis of the air turbines, due to the large fritions in the saled ower take-off system (Payne [29]). Alternatively, it is more ratial to model the relation between the hamber ressure and flowrate, for examle, the linear relation of the PTO system an be modeled by orous membrane (see Lewis et al.[3] and Forestier et al.[31]), while the nonlinear relation has been widely modelled by orifie lates [8-1, 32, 33]. It is shown if the orifie ratio (defined as the orifie area divided by water olumn area) is between.5%-2.%, the orresonding daming levels will give the OWC devies an otimal wave ower onversion effiieny. III. PERFORMANCE OF OWC DEVICES As an examle, a generi ylinder OWC is studied. Figure 1 shows a generi bottom-fixed OWC whih is fixed on a fixed frame on the tank floor. This fixed OWC model is well isolated from the influene of the ambient strutures. The generi OWC has a water olumn of a diameter of.14m, a draught of.3m, and a wall of thikness of.16m. The details of the model an be found in Sheng et al. [34]. To measure the wave ower extrated by the OWC devie, a ressure transduer is mounted on the to of the water olumn. In addition, the motion of the interior water surfae is also measured by a float whih suorts a marker. In rinile, the ower an be alulated by the ressure dro aross the orifie and the flowrate assing the orifie. Due to the wellestablished relation between the flowrate and the hamber (4) Figure 1 Fixed ylindrial OWC tested in HMRC oean wave tank In the numerial study, the internal water surfae (IWS) resonse is redited as large as 26 (see Figure 2) if no visous daming oeffiient has been inluded. The added visous daming brings down the IWS resonse greatly to 7. It must be noted that the added visous daming is urely indued by the motions of the struture and the water olumn, where the daming from the PTO system is not inluded. For examle, in the exeriment, an orifie of 12mm diameter has been installed on the to of the OWC devie, the measured IWS resonse is given in Figure 3. Prinially, the measured IWS resonse has inluded all damings, suh as hydrodynami daming, visous daming and the daming from the PTO. The omarison in Figure 3 shows the differene between the IWS resonses with and without PTO daming, where the numerial redition is without PTO daming. The numerial redition with PTO will be dealt with later in the reort. IV. EFFECTS OF AIR COMPRESSIBILITY As ointed out by Sarmento et al. [5], the sring-like effet of the air omressibility in the air hamber an not be negligible for the full sale OWC devies in whih the air hamber may be large enough. In addition, for the full sale OWC devies, the hamber (gauge) ressure and its gradient with regard to time an be large enough to reate omressibility in the air hamber. Sarmento et al. [35] give a linearized formula for alulating the air flow through the ower take-off devie as, Q Q w V γ d (5)

5 with Q and Q w being the flowrate through PTO and driven by water free surfae, V the undisturbed volume of the air, and the atmosheri ressure and the hamber ressure, γ the seial heat ratio of air (γ=1.4 for air). indiated that the air omressibility may redue the effiieny of the air hamber by about 8%. This following analysis will address the omressibility roblem, and hek the ossible redution of the effiieny of the air hamber due to omressibility, via both exerimental data and numerial redition. Figure 2 Numerial reditions of the IWS motion resonses of a fixed OWC (without and with added visous daming) Figure 3 IWS resonses of a fixed OWC: numerial resonse against the exerimental data with an orifie φ=12mm Based on the flowrate formula, the ower available to the PTO, P, and the inut ower rovided by the water free surfae, P w, are alulated as, P P w Q Q Due to omressibility of the air, the instantaneous flowrates driven by the interior water surfae and through the PTO may be different. Hene, the instantaneous inut ower P w and the ower through the PTO, P, may be different. A numerial study onduted by Thakker et al. [36] has w (6) A. Thermodynamis of air flow For studying the air omressibility of the air, Sheng et al. [12] have roosed a method to redit the air omressibility in the air hamber based on the known PTO harateristis and an isentroi roess of the air in the hamber. For omleteness, the method is outlined here. The mass of air enlosed in the air hamber an be exressed as m ρ V where m is the time deendent air mass in the air hamber, V the air volume of the air hamber, ρ the air density in the hamber. Differentiating eq. (7) yields, dm ρ dv dρ V It is noted that a ositive value of the mass rate with regard to time means some air is inhaled in through the PTO system (mass inrease), and a negative value of the mass rate means some air is driven out of the air hamber (mass redution). Due to the air omressibility, the air density hanges in the air hamber in exhalation and inhalation. For ressurized hamber, the air is ressured to have a higher density (than that of atmoshere) and is driven out of the air hamber through the ower take-off system. In inhalation, the hamber is de-ressurized, and atmoshere is inhaled into the hamber through the ower take-off system. Obviously, the air flow through the PTO system with different density must be onsidered for exhalation and inhalation searately. Exhalation: Inhalation: Q 1 ρ 1 Q ρ dm dm (7) (8) (9) (1) where ρ is the atmosheri density, and a ositive flowrate Q through the PTO means the flow exhaled from the air hamber. Following many other researhers for this toi, the analysis is simlified: the air in the hamber is assumed as isentroi roess, and under suh an assumtion, a state equation for the oen system is,

6 = onstant γ ρ (11) whih an lead to a linearized form for the density due to the fat of the hamber (gauge) ressure is normally muh smaller than the atmosheri ressure, as so Thus for exhalation, Q Similarly, for inhalation, Q γ ρ ρ 1 dρ Q w ρ γ V γ d 1 Qw γ B. Orifie ower take-off system d V γ d (12) (13) (14) (15) In this reort, an orifie is the nonlinear PTO for this study. The general relation of the hamber ressure and the flowrate through the PTO is given by Eq. (4). Seifially, for exhalation roess, it is k 2 2Q where k 2 is the daming oeffiient of the PTO. It orresonds to (16) (19) The flowrate driven by the interior water surfae an be exressed as Q w dv (2) Substituting (17) and (2) into (14) yields a dynami equation for exhalation dv V γ d k 2 (21) Substituting (19) and (2) into (15) yields a dynami equation for inhalation dv V d 1 γ γ k2 (22) V. EXPERIMENTAL VALIDATION By solving equations (21) and (22), we an obtain the air ressure in the hamber based on the known air flow driven by the interior water surfae, or we an obtain the hamber air volume based on the known hamber ressure. To validate the method, an exeriment is onduted to simulate an osillating water olumn and the ower take-off system. In the exeriment setu, a well-ontrolled iston is used to simulate the air hamber and the interior water surfae, and on to of the iston, an orifie is installed for simulating a non-linear ower take-off (for instane, an imulse turbine), see Figure 4. In this study, the diameter of the iston is.3m, and the orifie.19m (the ratio of orifie is about.4, or.4%). Q k 2 (17) And for inhalation whih orresonds to k 2 2Q (18) Q k 2 Figure 4 Piston on a test rig (left) and orifie on the to of the iston (right)

7 In the test, the iston motion, the hamber ressure and the temerature of the air in the hamber are measured. Figure 5 shows the measured data (triangles) for a ase of iston amlitude of.45m and frequeny of 1. Hz, and the orifie has a diameter of.19m. Under suh ondition, the ower generated in the hamber is u to 8W, whih is muh more owerful than that we an get in the wave tank test for a same size devie. As a result of this, a hysteresis loo between the hamber ressure and the flowrate an be obviously seen (Figure 5), whih is a basi feature of the omressible air. To indiate the differene between the inomressible and omressible air, the relation of inomressible air is also lotted (thik solid line in Figure 5). The numerial redition of the ressure based on the above theory is also lotted (dashed line in Figure 5). From the omarison, it an be seen that the hamber ressure has been well redited. It must be noted that in the figure, some oints are aart from the hysteresis loo for both measured data and the redited data. The reason for this is atually aused by the measurement errors in the measurement of the osition of the iston (in the numerial simulation, same iston osition is used for both exeriment and redition. If we aly a lowerass filter to the iston motion (-12.5Hz frequeny band), then the measured and redited data an be seen in Figure 6, where all the measured and redited data are very lose to the loo. And it is also found that there are some small differenes between the measured and redited data when the ressure returns from its eak. Figure 7 shows the omarison of measured hamber ressure and the redited hamber ressure for the ase of A=.45m, f=1.hz (ϕ=.19m). It an be seen that the redited ressure is in a very good agreement with the measured hamber ressure. Figure 8 shows a omarison of the measured hamber ressure and the redited ressure in a slower iston motion in whih the frequeny of the iston motion is.5hz. It an be seen that the ase with a smaller frequeny has a muh smaller air omressibility. Again, the redited ressure is very lose to the measured ressure. Figure 6 Filter has been alied (omonents of high frequeny are filtered out) After the solution of the ressure in the air hamber, the inut ower and the ower through the orifie an be alulated by Eq. (6), with the flowrate Q w being driven by the iston, and the flowrate through the orifie Q being redited via the numerial method outlined above. Figure 9 shows the omarison of the alulate owers (available to PTO) and the inut ower by the internal water surfae. Then the ower loss an be alulated as Pw P R P where the overbar means the time average value. w (23) Figure 7 Measured and redited hamber ressures Figure 5 Comarison of measured and redited ressureflowrate

8 Figure 8 Comarison of the theoretial result and the measured data (A=.45m, f=.5hz, ϕ=.19m). It is found that a small ower loss is still seen from the alulation. Figure 9 shows a omarison of the time series of the inut ower given by the iston and the PTO ower extrated by an orifie PTO. From the time series, it an be seen that there are some differenes between the inut ower (rovided by the iston) and the PTO ower (whih is available to the PTO system), eseially at the negative eak of the ressure. The maximum differene of the owers an be about 1% in this ase. However, detailed analysis has shown that the differene between the average inut ower and PTO ower is muh smaller than the aearane, whih is only.52%. The reason for this is that when the air in the hamber is ressurized or de-ressurised, art of the inut ower is stored in the omressible air. When the hamber ressure returns from its ositive or negative eaks, the stored ower may be released. This is atually onsistent with the assumtion of isentroi roess in the hamber. Generally the inut ower and the PTO ower are similar, and a small differene between the inut and the PTO owers an be seen, whih an be regarded the ower loss through the oen boundary (the PTO system). It an be exlained as when the air is ressurized and driven out of the hamber, its temerature is higher (thus internal energy is higher) than the atmoshere whih is suked into the hamber in the inhalation roess. The differene between the internal energy of the exhaled and inhaled air is the ower loss due to air omressibility in the air hamber. Obviously, the ower loss will very muh deend on the maximum hamber ressure and the flowrate. Figure 9 Power redition for an orifie (A=.45m, f=1.hz, ϕ=.19m) VI. TIME-DOMAIN ANALYSIS For an OWC wave energy onverter, the aliation of the ower take-off (PTO) system will dam the flow through the PTO and thus reate ressurized and de-ressurised air in the hamber. For full sale OWC wave energy onverter, the air omressibility may not be ignored due to the large air volume and hamber ressure (ressure gradient with regard to time). For suh a system, regardless of the linear or nonlinear ower take-off alied, the whole dynami system is generally nonlinear, and thus a linear analysis is not enough. In this study, a hybrid frequeny-time domain aroah has been adoted by ombining the onventional otential flow analysis and the thermodynamis of the air flow outlined above. In the hydrodynami analysis, the struture of the devie is fixed as shown in Figure 1, and is identified as a rigid body (Body 1), whilst the water olumn (the so alled iston) is the seond rigid body (Body 2). In the WAMIT analysis, two-body system is alied, and the first body is fixed and the seond body has only heave motion (Mode 9 in WAMIT analysis). The frequeny domain analysis results are onverted into the orresonding time-domain arameters, whih in here are used for establishing a time-domain equation for the dynami system. In the formulation, due to the ressurised and deressurised air, in the time-domain equation for the iston motion, an additional fore due to the hamber ressure is alied (see Eq. (24)), whilst the hamber ressure an be solved via the eqs. (21) and (22), with the hamber air volume formulation given by Eq. (25). F M 99 A99 ( ) x9 C99x9 b99x 9 K99 ( t τ) x (τ) dτ 9 A t 9 (24)

9 where x 9 is the heave motion of the iston/water olumn (reresenting the internal water surfae motion); M 99 the mass of the iston/water olumn; A 99( ) the orresonding added mass at the infinite frequeny; b 1 the added visous daming; C 99 the restoring oeffiient; K 99 the retard funtion; F 9 the exitation fore on the iston; the hamber ressure; A the setional area of water olumn. The hamber air volume an be alulated as V V A x9 (25) with V and V being the air volumes of time-deendent and in alm water. VII. RESULTS AND ANALYSIS Solving the time-domain equation outlined above, we an obtain the hamber ressure resonses (defined as the differene of the eak and the trough ressures divided by wave height H w) and the IWS resonses (defined as the differene of the IWS eak and trough motions divided by H w) in regular waves under different orifies (therefore different daming from the PTO). Figure 1 and Figure 11 are the omarisons of the measured and redited hamber ressure resonses and the IWS resonses for an orifie of 12mm in a wave height of about 4mm. Figure 12 and Figure 13 are the omarisons of the measured and redited hamber ressure resonses and the IWS resonses for an orifie of 14mm in a wave height of about 4mm. From these omarisons, it an be seen that the numerial reditions roosed in the researh have well redited the hamber ressure and the internal water surfae. It must be emhasized that the dynami system of the OWC devie with nonlinear orifie PTOs is nonlinear, hene the resonses resented for the nonlinear system are wave height deendent. In the numerial simulation, the wave heights are those measured from the tank tests. Though in the tank test, the wave heights are suosed to be a onstant in a series regular wave tests, but the measured wave heights are normally different from those. For examle, in the tank tests, the wave heights are suosed to be 4mm, but in reality the measured wave height are atually ranged from 38.4mm to 51.7mm, deending on the wave frequenies. This may be the reason why the numerial reditions are not very smooth. Figure 1 Resonses of hamber ressure in regular waves (φ =12mm, H w=4mm) Figure 11 Resonses of internal water surfae in regular waves (φ =12mm, H w=4mm) Figure 12 Resonses of hamber ressure in regular waves (φ=14mm, H w=4mm)

10 REFERENCES Figure 13 Resonses of internal water surfae in regular waves (φ =14mm, H w=4mm) VIII. CONCLUSIONS In numerial simulations of OWC wave energy onverters, the nonlinear effets, eseially the nonlinear effets due to the nonlinear PTO and the relevant air omressibility, must be inluded, hene a hybrid time-domain analysis is often a onvenient aroah to solve the nonlinear dynami roblem and to assess the erformane of the OWC devies in waves aroriately. By ombining the otential flow analysis and the thermodynamis of the air flow in the air hamber, the time-domain equation for OWC wave energy onverter has been formulated in this researh. From the researh and the examles given in this reort, following onlusions an be drawn: 1) The omarisons between the exerimental data and numerial simulation have shown the new method is very aable of assessing the OWC erformane, inluding the hamber ressure resonses and the internal water surfae motion. 2) Though the examle given in this researh is for a fixed OWC wave energy onverter, its rinile an be used for floating OWC wave energy onverter. 3) The internal water surfae motion in the OWC devie is reresented by a iston, whose length is same as that of the water olumn. 4) The otential theory has well redited the flow around the OWC devie. ACKNOWLEDGMENTS This material is based uon works suorted by the Siene Foundation Ireland (SFI) under the Charles Parsons Award at Hydraulis and Maritime Researh Centre (HMRC). Statistis and data were orret at the time of writing the artile; however the authors wish to dislaim any resonsibility for any inauraies that may arise. 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