Efforts Towards a Validated Time-Domain Model of an Oscillating Water Column with Control Components

Transcription

1 Proceedings of the 11th Euroean Wave and Tidal Energy Conference 6-11th Set 215, Nantes, France Efforts Towards a Validated Tie-Doain Model of an Oscillating Water Colun with Control Coonents Thoas Kelly, Thoas Dooley, John Cabell and John Ringwood Deartent of Electronic Engineering Maynooth University, Co. Kildare, Ireland E-ail: thoas.e.kelly.212@uail.ie E-ail: john.ringwood@eeng.nui.ie Centre for Renewable Energy at Dundalk IT Dundalk Institute of Technology, Dublin Rd., Dundalk, Ireland E-ail: thoas.dooley@dkit.ie Wave Energy Ireland Ltd., Unit F1 Nutgrove Office Park, Rathfarnha, Dublin 14, Ireland E-ail: info@waveenergyireland.ie Abstract As art of the develoent rocess of a roosed offshore floating wind/wave latfor, a scale-odel testing rograe has been carried out at a narrow tank facility on a single oscillating water colun (OWC) device. Previous roof-ofconcet testing was erfored on a scale odel of the roosed latfor corising thirty-two OWCs with control coonents. That testing rograe, ileented at the Hydraulic and Maritie Research Centre, deonstrated the feasibility of the concet. However, the large nuber of interacting coonents in this colex syste results in difficulties when atteting to study and odel nuerically the hydrodynaic and therodynaic rocesses at work within the OWCs during oeration. In order to better study these rocesses, the subsequent hase of narrow tank testing was lanned based on a single-chaber OWC odel as described herein. This odel was constructed with siilar diensions and cross-sectional rofile to that of one OWC chaber in the larger odel. Various control coonents can be added to and reoved fro the new odel to investigate in a systeatic fashion the effect of each coonent. Theory to nuerically odel the OWC in various configurations has been develoed. A coarison between the tank test results and those obtained for a secific setu is resented. Index Ters tank testing, nuerical odelling, oscillating water colun, control valves, lenu. I. I NTRODUCTION Of the nuerous wave energy conversion technologies currently under investigation worldwide, erhas the ost established is that of the oscillating water colun (OWC). A nuber of develoers have considered the benefit of cobining wave energy conversion with offshore wind energy conversion. One develoer currently working on a cobined offshore wind/wave energy conversion lant is Wave Energy Ireland Ltd. (WEI). The concet being develoed envisages a floating, V-shaed latfor corising ultile oscillating water coluns (MOWC) anifolded with airflow rectification for use with a unidirectional turbine. One or ore offshore wind turbines would be installed on this latfor. Proof-ofconcet testing on the wave energy conversion eleents of ISSN this latfor was erfored on a stainless steel, 1:5 scale odel in the ocean basin facility at the Hydraulic and Maritie Research Centre (HMRC) located in Co. Cork, Ireland. During this testing, air adittance valves were used to rectify the air flow, and venturis were used to siulate ower take-off daing, as described in [1] and [2]. The riary focus of the testing at the HMRC was on the energy catured by the device in various configurations when subject to regular waves. Soe work on characterising the hydrodynaic and therodynaic behaviour of the device was erfored, but the large nuber of interacting coonents resulted in difficulty in studying the behaviour of individual rocesses in the syste. To further investigate these rocesses, and also characterise the individual coonents and rovide data for the validation of nuerical odels, a single-chaber, fixed OWC odel of siilar diensions to one chaber of the MOWC odel was constructed in arine lywood for test in a narrow tank facility located at the Dundalk Institute of Technology (DkIT). As noted by [3], there is a lack of available exeriental data for the validation of nuerical odels of OWCs. The work outlined in [3], which has arallels with the work described herein, details efforts towards a floating, MOWC using selfrectifiying turbines. The ain difference to the work in [3], is the inclusion here of control coonents and additional air volues which can be introduced to the single-chaber OWC odel to investigate the effect of, and characterise, each coonent in turn. This aer discusses the design and construction of this odel and the various configurations in which the odel was tested. Theory used for the generation of a nuerical odel is resented, and soe results fro the nuerical odel are coared to those obtained fro the narrow tank testing. Further, a ethodology for the derivation and verification of the frequency deendent added ass coefficients for an OWC based on results obtained fro a coercial hydrodynaic solver is resented. Coyright Euroean Wave and Tidal Energy Conference 215 8A1-4-1

2 II. S CALE M ODEL The single-chaber OWC odel was ade riarily fro slices of arine lywood and sans the width of the DkIT narrow tank. The odel reresents a single chaber of the roosed MOWC device at a scale of 1:5. The geoetry of the wetted surface of this odel is siilar in for to that of the wetted surface of the individual chabers described in [1]. Figure 1 is an exloded diagra illustrating the assebly of the odel. The chaber corises a rounded front li 1, a curved back wall 2, two side walls 3 and 4 and to ieces 5, 6 and 7. To investigate the effect of control coonents used during the tests described in [1], the odel ay be reconfigured as described in Section IV below. Two 42- holes are cut in 5 which snugly receive a standard Wavin ie of external diaeter One of these holes is used to instal an adjustable orifice in the for of an iris valve onto the odel, which ay be used to introduce varying aounts of quadratic daing to the water colun (see Phase 2 of the testing rograe). These holes also allow air adittance valves 8, which allow air to flow in one direction only, to be installed on the odel during soe hases of the testing. Around these holes, rectangular tracks are routed in 5. The tracks allow for the installation of additional air volues or lenus 9 as required in certain test hases. Finally, two lenus ay be connected using standard Wavin fittings with the iris valve 1 located between the. used in the testing described herein were set with no odel installed using the sae water deth as resent during the test described in Section IV. Reflections fro the tank wall were also easured and found to be within accetable liits. A range of sensors were fitted to the tank and the OWC odel. In order to record the free surface elevation within the tank, resistive wave robes were used. Two such robes were located at the transverse centre of the tank between the waveaker and the odel to onitor the generated wave height and frequency during the testing. A third wave robe easured the free surface elevation behind the odel. The otions of the water colun within the odel were onitored by a fourth wave robe located at the centre of the water colun at still water level conditions. Air ressure within the various air chabers (described in Section II above) were easured using 176PC14HD2 differential ressure transducers. Data was acquired and recorded using a USB-based CoactDAQ syste sulied by National Instruents Cor. Figure 2 illustrates a scheatic of the test facility and the odel. Fig. 2: Scheatic of test facility with odel. IV. T ESTING P ROGRAMME There were six distinct hases to the DkIT narrow tanking rograe. Fig. 1: Exloded single-chaber OWC odel diagra. III. T EST FACILITY The testing rograe was conducted in a narrow tank facility located at DkIT. The tank is 18 long, 35 wide and is filled to a deth of 1. The waveaker corises a hinged, wedge-shaed fla driven by a servo-otor. The setu incororates a force feedback control syste to reduce the effect of reflected waves. An absorbing arabolic beach is located at the oosite end of the tank fro the waveaker. The tank was calibrated during the suer of 214. During this calibration rocess, all wave frequencies and alitudes Phase 1: This hase served to characterise the exciting force acting on the water colun. The air chaber above the water colun was first sealed, and the odel was fixed in the tank with the li suberged to a deth of The odel was then subjected to regular waves of alitudes ranging fro 5 to 3, in increents of 5, for frequencies ranging fro.4 Hz to 1.4 Hz, in increents of.1 Hz. These wave conditions were chosen to reresent the tyical range of waves found in the North Atlantic at a scale of 1:5, and are equivalent to wave eriods of 5 s to aroxiately 17.7 s and wave heights of.5 to 3 at full scale. During this testing, the differential air ressure within the chaber of air above the water colun, and the otion of the water colun, were recorded. As air is effectively incoressible at this scale, the otion of the water colun is negligible, as confired during this hase. Hence, the water colun is effectively held fixed during this hase. Assuing the ressure of the air contained within the sealed chaber is unifor throughout, the force acting on the water colun 8A1-4-2

3 can be found by ultilying the gauge ressure within the chaber by the surface area of the water colun itself at any instant in tie. This hase is odelled nuerically using Equations 1 and 2 with i =, (see Section V). Phase 2: The urose of this hase was to characterise the orifice (by deterining the coefficient of discharge) and investigate non-linear effects within the syste. An adjustable orifice was fitted to the base odel using one of the 42- holes in 5 to allow air flow between the air chaber above the water colun and atoshere. The second 42- hole was sealed. The daing alied to the water colun through the adjustable orifice was varied by changing the oening diaeter of the orifice fro 5 to 3 in stes of 5. The odel was subjected to the sae incident waves of varying alitudes and frequencies as in Phase 1 for each orifice diaeter. As in Phase 1, the otions of the water colun and the gauge ressure within the chaber were recorded. For Phases 1 and 2, data was saled at 32 Hz. Phase 2 is odelled nuerically using Equations 1, 2 and 3. the ressure within the OWC chaber siked as it overcae the stickage in the valve. To fully cature this effect, the saling rate was increased to 1 khz, and this rate was used for the reaining hases of testing. Phase 3 is odelled nuerically using Equations 1, 2 and 4. Phase 4: Phase 4 was intended to investigate the effect of a single lenu installed so that air flowing fro the OWC chaber through the exhalation valve would discharge into this lenu before discharging to atoshere. The ai was relicate the effect of one of the lenus which were used in [1] to anifold airflow fro ultile chabers as art of the airflow rectification rocess and to sooth ower delivery to the ower take-off. The variable orifice used in Phase 1 was installed in the oening of the lenu. As before, various wave frequencies, alitudes and orifice oening-diaeters were tested. In addition to recording the otion of the water colun and the differential ressure within the chaber, the differential air ressure in the lenu was also recorded during this hase. The otion of the water colun during Phase 4 is odelled nuerically using Equations 1 and 6 as described in Section V. The ressure within the chaber is odelled using Equation 2. The ressure within the lenu is odelled using Equation 5 and the ass flow equations are described in Table I. Fig. 3: Scheatic of the OWC with an orifice (Phase 2). Phase 3: This hase was designed to characterise the air adittance valves. The variable orifice was reoved and two air adittance valves were installed on the odel. If the air ressure within the OWC was greater than atosheric ressure, air could flow through one valve fro the OWC chaber to atoshere: this valve is tered the exhalation valve. Should the air ressure within the OWC be less than atosheric ressure, air could flow fro atoshere through the second valve into the OWC chaber: this valve is tered the inhalation valve. It was found that for wave alitudes below 15, the air ressure within the chaber did not increase or decrease sufficiently to overcoe the valve inertia and stickage of either valve. As a result, data was only recorded for wave alitudes varying fro 15 to 3, in increents of 5, for the sae range of wave frequencies as before. It was found that the iniu wave alitude required to generate sufficient ressure to overcoe valve inertia and stickage varied: the abient teerature and tie since the valve last oened aear to be two of the factors that influence this effect. As before, the water colun otion and the differential ressure within the air chaber were recorded. It was found during this hase that Fig. 4: Scheatic of an OWC with air adittance valves and a single lenu (Phase 4). Phase 5: Phase 5 was intended to investigate the effect of installing a second lenu so that air flowing through the inhalation valve into the OWC chaber would be drawn fro this second lenu. The variable orifice was reoved fro the setu so that both lenus were in counication with atoshere through a single, identical 42- hole in the lid of each lenu. In addition to the values recorded in Phase 4, the differential ressure in the second lenu was also recorded during this hase. The otion of the water colun during Phase 5 is odelled nuerically using Equations 6 and 8 as aroriate. The ressure within the chaber is odelled using Equation 2. The ressures within the lenus are odelled using Equations 5 and 7, and the ass flow equations are described in Table II. 8A1-4-3

4 V. T HEORY Fig. 5: Scheatic of one OWC chaber with lenus installed on the inhalation and exhalation valves (Phase 5). Phase 6: The urose of the final hase of testing was to exaine the effect of air coression within the syste. In this hase, both lenus were connected via Wavin fittings with the variable orifice installed between the. This setu ost closely resebled that of the stainless steel MOWC odel tested in the HMRC. However, the air flow between lenus in the testing on the MOWC odel deends on the hase difference between the otions of the water coluns in that syste. In contrast, in the single OWC narrow tank testing, any air flow between lenus which occurs can only arise due to air coressibility, and will thus be inial. The differential ressure within the OWC chaber would thus be exected not to differ significantly fro that in Phase 1. The sae araeters were recorded during this hase as in Phase 5. The otion of the water colun during Phase 5 is odelled nuerically using Equations 6 and 8 as aroriate. The ressure within the chaber is odelled using Equation 2. The ressures within the lenus are odelled using Equations 5 and 7, and the ass flow equations are described in Table III. Fig. 6: Scheatic of one OWC chaber, the high- and low-ressure lenu and conduit including orifice (Phase 6). The governing systes of couled ordinary differential equations (ODEs) used to reresent nuerically the odel are derived in [4]. The equations for each hase of testing are outlined below. They are derived by considering the ass flow rates and ressure changes throughout the syste, as well as the otion of the water colun. The rocess by which the ass flows are treated is siilar to that described in [5]. In Phase 1, where no ass flow takes lace, the otion of the water colun is described by the odified Cuins Equation [6] resented in Equation 1. For this hase, and all subsequent hases, it is assued that the water colun acts in a uing ode only, which is denoted by the subscrit 7. Air ressures are denoted by P, and the subscrit i denotes a roerty relating to the air chaber above the water colun. Z Fe = (7 + a ) U7 + k7 (t τ )U 7 (τ )dt (1) + c77 U7 + (Pi Patos ) Aowc In Equation 1 above, Fe is the exciting force due to the incident wave, 7 is the ass of the water colun, a is the infinite frequency added ass of the water colun and U7 is the vertical dislaceent of the water colun fro the at rest osition. Patos is atosheric air ressure, c77 is the hydrostaticr stiffness of the water colun, while the convolution ter k7 (t τ )U 7 (τ )dt reresents the radiation forces, or the so-called eory effect of the water colun otions. The kernel of the convolution is reresented by k7, that is the iulse resonse function of the water colun in the uing ode. Aowc is the cross-sectional area of the water colun. As the hydrodynaic araeters of the water colun are odelled using linear, otential codes, non-linear forces, which are ainly due to viscous daing and vortex foration, are neglected in the following analysis. Furtherore, for the wave conditions used, non-linear hydrodynaic effects are inial. The gauge ressure in the chaber, under the assution of adiabatic conditions, is given by: 2 Pi = cs (2) i γ V i P Vi Vi where cs is the seed of sound in air, Vi is the volue of air above the water colun within the OWC and γ is the heat caacity ratio of air. In the above equation, P = Pi Patos. This equation alies to all hases and has been derived fro consideration of the first law of therodynaics. The sae result was obtained in [7] for a recirocating coressor. For Phase 1, the first ter on the right-hand side of Equation 2 is zero as the ass of air in the OWC chaber, i, is constant: the flow of ass into or out of the chaber, i, is zero. In Phase 1, the otion of the water colun is due to the coression of air within the OWC chaber. In Phase 2 of the testing, the OWC is in counication with atoshere through an orifice. In this figure, ho denotes the original height of the air chaber. The otion of the water colun ay be described by Equation 1 as before. Mass flow of air into the chaber is considered ositive, and ass 8A1-4-4

5 flow out of the chaber is considered negative. Under the assution of incoressible flow, orifice theory [8] gives the ass flow rate into or out of the chaber as: i = Cdo Ao 2ρair P sign P (3) where Cdo is the coefficient of discharge of the orifice and Ao is the cross-sectional area of the orifice. The variation in chaber ressure is given by Equation 2. The volue of air in the chaber is roortional to the dislaceent of the water colun, and the rate of change of that volue is roortional to the velocity of the water colun. The above equations are ileented to siulate nuerically the OWC odel to generate the results resented in Section VII. In Phase 3 the adjustable orifice was reoved and two air adittance valves were installed. A difference in ressure across a valve causes a rubber diahrag to lift and allows airflow through the valve in one direction only. Equations 1 and 2 still aly. However, the ass flow rate through a valve is given as [9]: i = Cdv Lg hax 2ρair P sign P (4) where Cdv is the coefficient of discharge of the orifice, Lg is the distance between the rubber ring and the inside wall of the valve and hax is the height the rubber ring lifts. It is assued that the valves oen and close instantaneously in resonse to a ressure difference. The fors of Equations 3 and 4 are identical, i.e., a constant is ultilied by the sae function of a ressure difference. In Equation 3, this constant is given by Cdo Ao, and in Equation 4, this constant is given by Cdv Lg hax, assuing the two air adittance valves are identical. During Phase 4 an air lenu was installed on the odel so that the exhalation valve is in counication with the lenu, and the lenu is in counication with atoshere through an orifice. In addition to the three equations required to odel Phase 2 nuerically, a further two equations are needed to odel the syste setu used in Phase 4 in order to describe the rate of change of ressure, and the rate of change of ass, within the lenu. As derived in [2], the rate of change of ressure within the lenu, P d, with resect to the seed of sound, cs, the ass flow rate into or out of the lenu, d, and the volue of the lenu, Vd, is given by: P d = c2s Vd d and the inhalation valve is closed, Equation 1 is odified to: Z 7 + Fe = (7 + a ) U k7 (t τ )U 7 (τ )dt (6) + c77 U7 + (Pd Patos ) Aowc Further, the rates of change of the asses of air in the chaber and in the lenu are governed by different equations deending on the current state of the valves. These ass flow equations are resented in Table I. The subscrit o refers to a roerty of the orifice between the air lenu and atoshere, and the subscrit v refers to a roerty of an air adittance valve. The sign of the ressure difference is oitted but ilied as shown in Equation 3. TABLE I: Mass flow equations - Phase 4 Case Pi > P d & Pi > Patos Pi < P d & Pi > Patos Pi < P d & Pi < Patos The equations describing the ass flow rates and the otion of the water colun at any instant deend on the state of the syste. The ter (Pi Patos ) Aowc in Equation 1 reflects the force acting on the water colun due to the relative change of air ressure in the chaber with resect to atoshere. In Phase 4 this force deends on the state of the air adittance valves. If the inhalation valve is oen and the exhalation valve is closed, Equation 1 alies. If the exhalation valve is oen i= 2ρair (Pd Patos ) i = Cdv Lg hax 2ρair (Patos Pi ) d = Cdo Ao 2ρair (Pd Patos ) d = Cdo Ao The setu in Phase 5 of the testing requires the introduction of a further variable, Ps, which is the ressure within the lenu fro which the inhalation valve draws air. The ass flow rate into or out of this lenu is tered s. Table II resents the ass flow rate equations deending on the valve conditions. The variation in ressure within the lenu on the inhalation TABLE II: Mass flow equations - Phase 5 Case Pi > Pd & Pi > Patos Pi < Pd & Pi > Patos Pi < Pd & Pi < Patos (5) Mass flows i = Cdv Lg hax 2ρair (Pi Pd ) d = Cdv Lg hax 2ρair (Pi Pd ) Cdo Ao 2ρair (Pd Patos ) Mass flows i = Cdv Lg hax 2ρair (Pi Pd ) d = Cdv Lg hax 2ρair (Pi Pd ) Cdo Ao 2ρair (Pd Patos ) s = Cdo Ao 2ρair (Patos Ps ) i= d = Cdo Ao 2ρair (Pd Patos ) s = Cdo Ao 2ρair (Patos Ps ) i = Cdv Lg hax 2ρair (Ps Pi ) d = Cdo Ao 2ρair (Pd Patos ) s = Cdo Ao 2ρair (Patos Ps ) Cdv Lg hax 2ρair (Ps Pi ) side is given by an equation with is analagous to Equation 5, where Vs is the volue of the lenu on the inhalation side: c2 s P s = s Vs (7) As in Phase 4, the force acting on the surface of the water colun due to the ressure acting on it deends on the state of 8A1-4-5

6 the valves. If the exhalation valve is oen and the inhalation valve is closed, Equation 6 alies. If the inhalation valve is oen and the exhalation valve is closed, this equation is odified to: Z Fe = (7 + a ) U7 + k7 (t τ )U 7 (τ )dt (8) + c77 U7 + (Patos Ps ) Aowc in counication with atoshere (Phase 2). This syste is reresented by: No further variables are required in Phase 6. The ass flow rate equations for this hase are suarised in Table III. Again, the force acting on the water colun due to the ressure X(1) = U7 = X(2), X(t) = AX(t) + Bu(t) where X (1) = U7, X (2) = U 7 and X (3) = P = (Pi Patos ). Using Equations 1 and 2 and aking X(1), X(2) and X(3) the subjects yields: R k7 (t τ )X(2)(τ )dt F e 7 = X(2) =U [ + a ] [ + a ] c77 Aowc X(1) X(3), [ + a ] [ + a ] TABLE III: Mass flow equations - Phase 6 Case Pi > Pd & Pi > Patos Pi < Pd & Pi > Patos Pi < Pd & Pi > Patos Mass flows i = Cdv Lg hax 2ρair (Pi Pd ) d = Cdv Lg hax 2ρair (Pi Pd ) Cdo Ao 2ρair (Pd Ps ) s = Cdo Ao 2ρair (Pd Ps ) i= d = Cdo Ao 2ρair (Pd Ps ) s = Cdo Ao 2ρair (Pd Ps ) i = Cdv Lg hax 2ρair (Ps Pi ) d = Cdo Ao 2ρair (Pd Ps ) s = Cdo Ao 2ρair (Pd Ps ) Cdv Lg hax 2ρair (Ps Pi ) on the free surface deends on the state of the valves. If the exhalation valve is oen and the inhalation valve is closed, Equation 6 alies. If the inhalation valve is oen and the exhalation valve is closed, Equation 8 alies. The density of air within a chaber is one araeter uon which a ass flow rate deends. This density varies due to the coression of air. As shown in [1], for an adiabatic rocess, the linearised relationshi between air density and ressure is given by: P ρair = ρatos 1 + (9) γρatos VI. N UMERICAL M ODELLING Tyically, the develoent of any rotoye wave energy converter corises two colientary strands: hysical testing and nuerical odelling. Each strand infors the other, with the goal of develoing a reliable odel which can liit the need for further otentially exensive and tie-consuing hysical testing. A nuber of tools were eloyed to solve the couled systes of ODEs used to reresent the odel OWC atheatically. The equations were rewritten in atrix for as a syste of couled first order ODEs. Each hase of testing is odelled by a different syste of couled equations. By way of exale, Equation 1 illustrates the couled first order syste for the OWC when configured with a single orifice (1) and = X(3) = P γ (x(3) + Patos ) cs2 i + X(2) Aowc [ho X(1)] [ho X(1)] The atrices A, B and u of Equation 1 are now written as: 1 c77 owc [A7 +a A = [7 +a ] ] γ(x(3)+patos ) [ho x(1)] B = (F e R k7 (t τ )U 7 (τ )dt) 2 cs Aowc [ho x(1)] and u= 1 [7 +a ] i This syste of ODEs can be exanded to odel the reaining test hases. Once oulated, the syste is solved using Runge-Kutta ethods ileented in the rograing environent MATLAB. In order to oulate the values of these equations, AutoCAD is used to deterine the volue and hence the ass of the water colun, as well as the free surface area of the water colun (to which the coefficient of buoyancy is directly related). As the water colun ass and the coefficient of buoyancy vary with the height of the water colun, AutoCAD was used to deterine these values for water colun heights varying in increents of 1 around the still water level. By fitting quadratic curves, the relationshi between the dislaceent and the ass of the water colun, the dislaceent and the volue of air and the dislaceent and the area of the water colun can be included in the nuerical odel. These values are recalculated at each tie ste based on the instantaneous water colun dislaceent. The hydrodynaic araeters, such as the frequency deendent exciting force, added ass and radiated daing, are deterined using the boundary roble solver WAMIT V7 [11]. WAMIT is a 3-diensional solver, whereas the testing described herein took lace in a 2-diensional tank. Thus, the results ust be considered caution. Future work will coare 8A1-4-6

7 the 3-diensional nuerical results with soe techniques to generate 2-diensional results fro WAMIT and also with the results of tank testing. However, the issues described in this section reain valid. Only the uing ode, assigned the subscrit 7, is considered. The geoetry of the ean wetted surface of the odel OWC tested here, which is required as an inut to WAMIT, is generated using the CAD ackage, MultiSurf [12]. Figure 7 illustrates this geoetry. Note that, for clarity, the free surface of the water colun is shown in blue in this figure. Fig. 7: MultiSurf odel of the ean wetted surface of the OWC. Initially, when solving the couled syste of ODEs described in Equation 1, the iulse resonse function was constructed fro the frequency doain data obtained fro WAMIT, and Equation 1 was solved while erforing the convolution integral between the kernel, k7, and the velocity, U 7, at each tie ste. However, this is coutationally exensive see, for exale, [13] and the convolution integral was relaced with a state-sace reresentation as described in [14]. This rocess aroxiates the convolution integral with a statesace syste with zero initial conditions as follows: X P (t) = A X(t) + B u (t) y (t) = C X (t) Z k7 (t τ )U 7 (τ )dt (11) The order of the couled ODEs in Equation 1 increases by the order of the syste in Equation 11, and the atrices A and B are re-written as: A=... A... C 77 (t) (7c(t)+a (7 (t)+a ) )... B 1 γ(x(7)+patos ) (ho x(5))... owc (7A(t)+a ) B=... Fe... cs2 Aowc [ho x(1)] The vector u reains unchanged. In order to oulate and deterine the order of the atrices A, B and C above, the Marine Systes Siulator (MSS) frequency doain identification tool [15] develoed at the Norwegian University of Science and Technology (NTNU), was used. This toolbox requires vectors corising the frequency deendent radiated daing, added ass and corresonding frequencies, and, otionally the infinite frequency added ass, for the ode of otion in question. The outut is a state-sace reresentation of the radiation forces for that ode. However, it was found that when the araeters deterined by WAMIT for the uing ode of the OWC were assed to this tool, the resultant statesace syste (irresective of what order syste was generated by the MSS toolbox) was unstable. Thus, the iulse resonse of the state-sace syste did not aroxiate the iulse resonse function constructed fro the frequency deendent radiated daing as found by truncating the integral given by: Z 2 B(ω) cos(ωt)dω (12) K(t) = π This roble would aear to relate to the values calculated by WAMIT for the added ass at higher frequencies and for the infinite frequency added ass. A lot of the diensionalised added ass versus frequency as calculated by WAMIT can be seen in Figure 8 labelled Unreconstructed. The diensionalised infinite frequency added ass returned by WAMIT V7 is given as 356 kg. The added ass curve should be asytotic to the infinite frequency added ass. However, not only is the curve not asytotic to the value calculated by WAMIT, this value is also an order of agnitude greater than all other values of added ass deterined by WAMIT. Further, the for of the added ass curve is unusual when coared to the equivalent curve for one of the six rigid body odes of regular shaes (see, for exale [14]) in so far as, at high frequencies, the values of the added ass for the water colun tend towards zero rather than towards the calculated (but anoalous) value for the infinite frequency added ass. While the cause of this issue is unknown at this tie, it ay relate to how WAMIT solves for the radiation otential, and the tendency for Green s integral equations to becoe singular as the thickness of the body (or art of the body) decreases [11]. Siilar results were yielded when the rocess was reeated for the redefined unodified cylindrical OWC in WAMIT, suggesting the issue does not lie with the geoetry of the odel. Finally, it is worth noting that the odel geoetry had reviously been analysed using WAMIT V6, and while the added ass versus frequency curve was of siilar for, the earlier version of WAMIT calculated an infinite frequency added ass of 46.8 kg, which, while of the sae order as the added ass values calculated at non-infinite frequencies, is still greater than all 8A1-4-7

8 Reconstructed Unreconstructed Frequency (rad/s) 2 Fig. 8: Diensionalised frequency deendent added ass fro WAMIT V7 and reconstructed fro radiated daing for the OWC shown in Figure 7. deendent iulse resonse are related through the KraerKronig relation as follows: Z 1 K(t) sin(ωt)dt (13) A(ω) = A ω Thus, it is ossible to use the iulse resonse as constructed fro the radiated daing using Equation 12 to calculate the added ass using Equation 13 once the correct value of A is known. While the radiated daing calculated by WAMIT for the OWC (shown in Figure 9) would aear to be of the correct for, in order to establish confidence in these values, a thinwalled narrow ie of draft.18 and radius.225 was odelled in WAMIT as a single diole atch with a second atch reresenting the water colun surface. The radiated daing of the water colun in this ie was deterined in WAMIT, and the results coared to those calculated using the analytical solution given in [16] as: 2 π b Kae 2Ka (14) B(ω) = M ω 2 a where M is the ass of the water colun, a is the draft 2 of the ie, b is the radius of the ie and K = ωg. The coarison between these results is shown in Figure 1 where the closed for results were obtained fro the solution of Equation 14. It can be seen there is good agreeent between these two data sets. However, before the frequency deendent added ass values ay be reconstructed using Equation 13 with the iulse resonse function constructed fro radiated daing values using Equation 12, it is necessary to deterine the aroriate value of A. It can be seen fro Equation 13 that the shae of a curve lotting the frequency deendent added ass deterined fro this equation against frequency is not deendent on A. However, the y-intercet of this curve is deendent on A. The curve labelled Reconstructed Frequency (rad/s) 2 Fig. 9: Radiated daing calculated by WAMIT V7 for the OWC shown in Figure Radiated daing (Ns/) Added ass (Kg) 25 Radiated daing Radiated daing (Ns/) the non-infinite frequency added ass values deterined by WAMIT V6. However, the frequency deendent added ass and the tie WAMIT Closed for Frequency (rad/s) 2 Fig. 1: Coarison of radiated daing calculated fro WAMIT V7 and closed for solution for a thin-walled ie. in Figure 8 illustrates the frequency deendent added ass deterined using Equation 13 with the radiated daing values fro WAMIT V7 and A set equal to kg. The effect of varying A would be to cause the y-intercet of this curve to increase or decrease, and the Reconstructed curve would follow, while retaining the shae shown in Figure 8. The value of A used was chosen so that the values of the reconstructed added ass would be coincident with the unreconstructed values calculated by WAMIT V7 fro to 8 rad/s. Note further that the Reconstructed added ass curve is of siilar for to those in the literature for rigid body odes, for exale [14]. The reconstructed added ass values found using Equation 13, the infinite frequency added ass value of kg and the radiated daing values calculated by WAMIT V7 were used as inuts to the MSS toolbox. The resulting state-sace syste is stable, and Figure 11 illustrates the coarison between the iulse resonse for this statesace syste and the iulse resonse deterined using 8A1-4-8

9 1 Iulse resonse fro Equation 12 Iulse of state-sace syste Iulse 5 6 Exciting force - WAMIT 2-D Haskind fro daing 3-D Haskind fro daing Exciting force (N) Equation 12. It can be seen that good agreeent exists between the two suggesting that the aroach used to deterine a value for the frequency deendent added ass values and for A is valid in this case Frequency (rad/s) Tie (sec) Fig. 11: Coarison of the iulse resonse of the uing ode of the OWC as constructed fro Equation 12 and the iulse resonse of the state-sace aroxiation. Consider now the exciting force. For the two-diensional case, the Haskind relation between the agnitude exciting force and the radiated daing is given by [17]: r ρg 2 B(ω) (15) Fe (ω) = A ω where A is the wave alitude. The relationshi between the agnitude of the exciting force and radiated daing for the heave ode of a body in the three-diensional case is given by: r 4ρgVg Fe (ω) = A B(ω) (16) K where ω2 g,k = Vg = 2ω g Figure 12 illustrates the agnitude of the exciting force for ode 7 of the OWC as calculated by WAMIT V7, and the two(equation 15) and three-diensional (Equation 16) Haskind forces based on the radiated daing calculated in WAMIT V7 for the OWC shown in Figure 7. If it is assued that the radiated daing is correct, it can be seen that the agnitude of the exciting force calculated by WAMIT aears to be a cobination of the two- and three-diensional Haskind forces. Future work will coare the results shown in this figure with soe techniques to generate 2-diensional results fro WAMIT and also results fro tank testing. 25 Fig. 12: Exciting force acting on the uing ode on the water colun shown in Figure 7 calculated by WAMIT V7, and showing the 2-D and 3-D Haskind forces calculated fro the radiated daing. the testing are coared to the rediction fro the current nuerical odel. This nuerical odel uses a value of kg for the infinite frequency added ass and a state-sace aroxiation for the convolution integral as deterined fro the reconstructed added ass values derived as above in Equation 1. A value of.6 for the coefficient of discharge is used in Equation 3 as recoended in [8]. Figure 13 illustrates the redicted gauge ressure in the OWC chaber lotted against that obtained during the test for a wave alitude of 2, with a frequency of.4 Hz and an orifice oening diaeter of 2, equivalent to a full-scale wave of alitude 1 and eriod 17.7 s. At this frequency, the otion of the water colun is doinated by the uing ode. Figure 14 shows Predicted Measured 1 Pressure (N/2 ) Tie (sec) VII. S AMPLE R ESULTS Fig. 13: Gauge ressure in OWC chaber as redicted by nuerical odel and easured during tank test. Work on calibrating and validating the nuerical odel based on the results of the narrow tank testing is ongoing. By way of exale of the efforts towards roducing a validated odel, soe results fro a single test run during Phase 2 of the redicted uing otion of the water colun coared to that observed during the test. It can be seen that good agreeent has been obtained for the water colun otions 8A1-4-9

10 odel successfully redicted the otions of the water colun. However, while the nuerical odel redicted the gauge ressure within the chaber during the inhalation hase of the otion of the water colun with a degree of accuracy, it was less accurate during the exhalation hase for this case. This is ossibly due to the assution of isentroic conditions, and the nuerical odel will be refined in further work to investigate this henoenon. in this instance. Predicted Measured Dislaceent ().2 ACKNOWLEDGEMENTS Tie (sec) Fig. 14: Water colun dislaceent as redicted by nuerical odel and easured during tank test. The authors wish to acknowledge the Carentry and Joinery Deartent of DkIT for their assistance in the construction of the arine lywood odel used during the narrow tank testing described herein. The first author further wishes to acknowledge the anageent of WEI Ltd. for roviding access to aterials and intellectual roerty within the fraework of research being undertaken ursuant to his PhD, and the Irish Research Council for roviding funding for this research. R EFERENCES Good agreeent has also been obtained for the inhalation ressures for the secific test conditions listed. However, the exhalation ressures are over-redicted by the nuerical odel. The gauge ressures in the OWC chaber obtained during the narrow tank testing exhibit asyetry about zero. This attern is aarent for all tests during Phase 2, and the asyetry increases as the alitude of the water colun otion increases. It was considered that this effect ay be due to the nature of the iris valve eloyed, which corises a nuber of oving, overlaing segents. However, the effect is also resent in the results obtained during Phase 3 of the testing when the iris valve was not eloyed. As noted in various texts including [1], the rocesses at work within the chaber are not identical for the inhalation and exhalation stages. As observed in [1], the air ixing rocess takes lace outside the OWC chaber during exhalation, but inside the chaber during inhalation. The results here suggest that the assution of isentroic conditions ay be valid during the inhalation hase, but ay not be valid during the exhalation hase. Soe success has been had in odelling the exhalation hase as an isotheral rocess, work which it is hoed will be the subject of future ublications. VIII. C ONCLUSIONS A rograe of narrow tank testing on a single, fixed OWC has been coleted. Data has been gathered which ay be used to characterise various control coonents and exlore non-linear effects within the syste. This work continues and will serve to infor future nuerical odels of a floating, MOWC device. A rocess to overcoe an issue with the values calculated by WAMIT for the frequency deendent added ass and infinite frequency added ass for use with the frequency doain identification toolbox roduced by NTNU has been resented. Theory to odel the behaviour of an OWC with an orifice in which the convolution ter in the Cuins Equation is relaced by a state-sace syste has been introduced. For a resented instance, the nuerical [1] T. Kelly, T. Dooley, J. Cabell, and J. V. Ringwood, Coarison of the exeriental and nuerical results of odelling a 32-oscillating water colun (OWC), V-shaed floating wave energy converter, Energies, vol. 6, no. 8, , 213. [Online]. Available: htt:// [2], Modelling and results for an array of 32 oscillating water coluns. Tenth Euroean Wave and Tidal Energy Conference, 213. [3] A. Iturrioz, R. Guanche, M. Alves, C. Vidal, and I. Losada, Tiedoain odeling of a fixed detached oscillating water colun towards a floating ulti-chaber device, Ocean Engineering, vol. 76, , 214. [4] Wave Energy Ireland Ltd, Analysis of wave flue testing of WEI odels at the HMRC, Cork, CREDIT DkIT, Tech. Re., Noveber 21. [5] D. P. Georgiou, K. F. Milidonis, and E. N. Georgiou, Sensitivity analysis for the encaged turbine concet in oscillating water colun lants, ISRN Renewable Energy, vol. 212, no. 1,. 1 8, 212. [6] W. Cuins, The Iulse Resonse Function and Shi Motions, ser. Technical Reort Deartent of the Navy, David Taylor Model Basin, [7] A. Haze, S. Noor, S. Bashi, and H. Marahban, Matheatical and intelligent odeling of electroneuatic servo actuator systes, Aust. J. Basic Al. Sci., vol. 3, no. 4, , 29. [8] ISO, Measureent of fluid flow by eans of ressure differential devices inserted in circular cross-section conduits running full Part 2: Orifice lates, International Organization for Standardization, Geneva, Switzerland, ISO :23, 23. [9] R. A. Habing, Flow and late otions in coressor valves, Ph.D. dissertation, Twente University Holland, 25. [1] W. Sheng, R. Alcorn, and T. Lewis, On therodynaics in the riary ower conversion of oscillating water colun wave energy converters, Journal of Renewable and Sustainable Energy, vol. 5, no. 2315, 213. [11] WAMIT Inc., WAMIT User Manual Version 7.. MA USA: WAMIT Inc, 212, vol. 1. [12] AeroHydro Inc., MultiSurf 8., Maine USA, 211. [13] R. Taghiour, T. Perez, and T. Moan, Hybrid frequency-tie doain odels for dynaic resonse analysis of arine structures, Ocean Engineering, vol. 35, no. 1, , 28. [14] Z. Yu and J. Falnes, State-sace odelling of a vertical cylinder in heave, Alied Ocean Research, vol. 17, no. 5, , [15] T. Perez and T. I. Fossen, A Matlab Toolbox for Paraetric Identification of Radiation-Force Models of Shis and Offshore Structures, Modeling, Identification and Control, vol. 3, no. 1,. 1 15, 29. [16] D. Evans, The oscillating water colun wave-energy device, J. Inst. Maths. Alied, vol. 22, , [17] N. Newan, The exciting forces on fixed bodies in waves, Journal of Shi Research, vol. 6, no. 3,. 1 17, A1-4-1