Ocean Engineering 26 (1999) 1275–1295 www.elsevier.com/locate/oceaneng OWC wave energy devices with air ﬂow control A.F. de O. Falca˜oa,*, P.A.P. Justinob aDepartment of Mechanical Engineering, Instituto Superior Te´cnico, 1096 Lisbon Codex, Portugal bDepartment of Renewable Energies, Instituto Nacional de Engenharia e Tecnologia Industrial, Estrada do Pac¸o do Lumiar, 1699 Lisbon Codex, Portugal Received 9 July 1998; accepted 9 September 1998 Abstract A theoretical model is developed to simulate the energy conversion, from wave to turbine shaft, of an oscillating-water-column (OWC) plant equipped with a Wells air-turbine and with a valve (in series or in parallel with the turbine) for air-ﬂow control. Numerical simulations show that the use of a control valve, by preventing or reducing the aerodynamic stall losses at the turbine rotor blades, may provide a way of substantially increasing the amount of energy produced by the plant, particularly at the higher incident wave power levels. From the hydrodynamic point of view, a by-pass valve or a throttle valve should be used depending on whether the wave energy absorbing system is over-damped or under-damped by the turbine. © 1999 Elsevier Science Ltd. All rights reserved. Keywords: Wave energy; Oscillating water column; Equipment; Wells turbines; Valves; Flow control 1. Introduction The oscillating-water-column (OWC) wave energy device is probably the most extensively studied type of wave power plant and one of the few to have reached the stage of full-sized prototype. The device consists essentially of a ﬂoating or (more usually) bottom-ﬁxed structure, whose upper part forms an air chamber and whose immersed part is open to the action of the sea. The reciprocating ﬂow of air * Corresponding author. 0029-8018/98/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 9 8 ) 0 0 0 7 5 - 4 1276 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 displaced by the inside free surface motion drives an air turbine mounted on the top of the structure. A system of non-return valves may be used to rectify the ﬂow in order to allow a conventional turbine to be used, as proposed by Moody and Elliot (1982). Alternatively the plant may be equipped with a self-rectifying turbine, which eliminates the need for the rectifying valve system. Several versions of self-rectifying air turbines for wave energy conversion have been proposed and studied, namely McCormick’s counter-rotating turbine (Richards and Weiskopf, 1986; McCormick and Surko, 1989), the impulse turbine with self-pitch-controlled guide vanes (FSaelctoa˜gou, c1h9i8e8t; al., 1991) and Raghunathan, the Wells turbine (Raghunathan 1995). The latter was adopted in et al., 1985; Gato and most OWC wave pilot plants built so India (Koola et far, al., namely in 1993), UK Norway (Falnes, (Whittaker et al., 11999932)),anJadpPanort(uOghanl o(Featlcaa˜lo.,, 1993), 1998). A general feature of turbomachines lies in their efﬁciency being strongly depend- ant on ﬂow rate. Large drops in efﬁciency at far-from-design conditions are known to arise from ﬂow separation when the angle of incidence of the ﬂow approaching the rotating and/or stationary blades or vanes becomes excessive. One way of reduc- ing the sensitivity of the efﬁciency to ﬂow changes (at the expense of higher mechan- ical complexity and cost) is to employ variable geometry machines, as is the case of Kaplan water turbines and variable pitch propellers and wind turbines. Wind tur- bines, as well as turbomachines equipping wave energy devices, are especially sub- ject to large variations in ﬂow conditions. Variations associated with changes in wind or wave average conditions with time scales much larger than the response time of the machine itself can be matched by adjusting adequately the rotational speed as far as this is allowed for by the coupling to the electric generator. The inertia of the turbo-generator rotating elements makes it unfeasible to adjust the machine speed to faster oscillations of ﬂow (those whose time scales are less than, say, 1 min); especially for turbomachines of ﬁxed geometry, such ﬂow oscillations may severely reduce their efﬁciency. The efﬁciency of oscillating water column (OWC) wave energy devices equipped with Wells turbines is particularly affected by ﬂow oscillations basically for two reasons. First, because of the intrinsically unsteady (reciprocating) ﬂow of air dis- placed by the oscillating water free surface. Second, because increasing the air ﬂow rate, above a limit depending on, and approximately proportional to, the rotational speed of the turbine, is known to give rise to a rapid drop in the aerodynamic efﬁciency and in the power output of the turbine. A method which has been proposed to partially circumvent this problem consists in controlling the pitch of the turbine rotor blades in order to prevent the instantaneous angle of incidence of the relative ﬂow from exceeding blades (see Gato and tFhaelcca˜roit,ic1a9l9v1a)l.uAe lathboouvgehwchoinchsidseevreedretesctahlnliincgalloyccfeuarssiabtleth(eSaroltteorr, 1993) this has never been implemented at full scale owing to mechanical difﬁculties. Alternately, the ﬂow rate through the turbine can be prevented from becoming excessive by equipping the device with air valves. Two different schemes can be envisaged. In the ﬁrst one the valves are mounted between the chamber and the atmosphere in parallel with the turbine (by-pass or relief valves, on or near the roof of the air chamber structure) and are made to open (by active or passive control) in A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1277 order to prevent the overpressure (or the underpressure) in the chamber to exceed a limit which is deﬁned by the aerodynamic characteristics of the turbine at its instantaneous speed. In the second scheme a valve is mounted in series with the turbine in the duct connecting the chamber and the atmosphere. Excessive ﬂow rate is prevented by partially closing the valve. In both schemes, the air ﬂow through the turbine is controlled at the expense of energy dissipation at the valves. Theoretically the two methods, if properly implemented, are equivalent from the point of view of limiting the ﬂow rate through the turbine. However, the resulting pressure changes in the chamber are different (reduction and increase in pressure oscillations in the ﬁrst and second cases, respectively). Consequently the hydrodynamic process of energy extraction from the waves is differently modiﬁed by valve operation in the two control methods. The main purpose of this work is to analyze theoretically the performance of an OWC wave energy device when valves are used to limit the ﬂow through the turbine. Both schemes are considered and compared: a valve (or a set of valves) mounted in parallel with the turbine (by-pass or relief valve) or a valve mounted in the turbine duct. The hydrodynamic analysis is done in the time domain for regular as well as for irregular waves. The spring-like effect due to the compressibility of the air is taken into account and is discussed in some detail. Realistic characteristics are assumed for the turbine. Numerical results are presented for a simple two-dimensional chamber geometry for whose hydrodynamic coefﬁcients analytical expressions are known as functions of wave frequency. 2. Governing equations We consider an OWC wave energy device, ﬁxed with respect to the sea bottom (Fig. 1). The geometry of the device and of the surrounding submerged solid bound- aries is arbitrary, and the waves incident upon the device are supposed in general to be irregular. Let pa ϩ p(t) (pa ϭ atmospheric pressure) be the pressure (assumed uniform) of the air inside the chamber, q(t) the volume ﬂow rate displaced by the motion of the inside free-surface of water, and m(t) the mass of air contained inside the chamber. If we denote by w the mass ﬂow rate through the turbine and by wv the mass ﬂow rate through the by-pass valve (positive for outward ﬂow), it is w ϩ wv ϭ Ϫdm/dt. We may write m ϭ V, where (t) and V(t) are respectively the density and the volume of the air inside the chamber. Then, taking into account that dV/dt ϭϪ q, we obtain w ϩ wv ϭ Ϫ V d dt ϩ q (1) 2.1. Hydrodynamics Assuming linear water wave theory to apply, we may write q(t) ϭ qr(t) ϩ qi(t) (2) 1278 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 Fig. 1. Sketch of the OWC chamber. where qr(t) is the radiation ﬂow rate due to the air pressure oscillation p(t) inside the chamber in the absence of incident waves, and qi(t) is the diffraction ﬂow rate due to the incident waves if the air inside the chamber were kept at atmospheric pressure (i.e. p ϭ 0). In the following analysis, it is assumed that qi(t) is a known function of time. Within the framework of linear water wave theory, we may write ͵t qr(t) ϭ gr(t Ϫ )p()d (3) Ϫϱ where gr(t) is a function depending on the geometry of the system. This function can easily be related to the system’s radiation coefﬁcient by setting p(t) ϭ Peit, qr(t) ϭ Qreit, where P() and Qr() are in general complex amplitudes. The complex coefﬁcient of radiation is by deﬁnition (Evans, 1982) B() ϩ iC() ϭ Qr()/P() (B, C real), and so, assuming gr(t) to be an even function, we ﬁnd B as the result of a Fourier transform ͵ϱ B() ϭ 1 2 gr()eid Ϫϱ from which, by inversion, we obtain A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1279 ͵ϱ gr(t) ϭ 2 B()cos t d (4) 0 The function B() depends on the geometry of the chamber and surrounding submerged walls, and is assumed to be known either from measurements (Sarmento, 1991) or theoretically. The formulation of the problem will be closed by introducing the thermodynamic relations governing the compressibility of the air in the chamber (which will allow d/dt to be related to dp/dt) and the aerodynamic characteristics of the turbine and valves. 2.2. Thermodynamics The spring-like effect due to air compressibility is known to increase with the average chamber height above the water free surface (i.e. the air volume divided by the inside free-surface area) and should not be neglected in full-scale devices in wFahlicca˜ho,th1e98ch5a).mIbtewr ihlleibgehtaisssutympeicdaltlhyatofthtehethoerrdmerodoyfnsaemveicrapl rmoceetesrss (Sarmento and taking place in the chamber, turbine, valves and connecting ducts is adiabatic, since the amount of heat exchanged in the relatively small period of time of a wave cycle (of the order of 10 s) is likely to be only a small fraction of what would be required to keep the air at constant temperature. In addition we assume that the process of ﬁlling and discharging is slow enough for the thermodynamic state of the air in the chamber to be approximately uniform, and denote by pa ϩ p, ϭ m/V, T, u, s and h its pressure, density, absolute temperature, speciﬁc internal energy, speciﬁc entropy and speciﬁc enthalpy, respectively. We have an open system made up of air bounded by the chamber walls, by the inside water free-surface, and by surfaces 1 and 2 (conventionally separating the chamber from the turbine duct and from the by-pass valve, respectively) where the air kinetic energy is assumed small. The thermodyn- amic properties of air in the atmosphere are denoted by the subscript a. The air is considered a perfect gas, with (pa ϩ p)/ ϭ RT, du ϭ cvdT, dh ϭ cpdT, ␥ ϭ cp/cv, R ϭ cp Ϫ cv (cp, cv constant). We consider ﬁrst the discharge process when air is ﬂowing out of the chamber to the atmosphere (p > 0, w > 0, wv Ն 0). As in the case when air is slowly discharged from a rigid vessel through a throttling valve to an atmosphere at lower pressure (the difference being that in our case the water free-surface is a moving boundary) it can easily be found that the discharge is reversible as far as the change in the thermodynamic state of the air particles remaining in the chamber is concerned, and therefore is an isentropic process (see e.g. Kestin, 1966). Consequently we may write (pa ϩ p)−1/␥ ϭ constant during each discharging process and d dt ϭ 1 ␥RT dp dt (5) 1280 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 The ﬁlling process (p Ͻ 0, w Ͻ 0, wv Յ 0), during which air enters the chamber from the atmosphere, is more difﬁcult to model, since the air speciﬁc entropy changes (from its atmospheric value) due to viscous losses in the turbine, valves and connecting ducts, and its speciﬁc enthalpy also changes due to work performed in the turbine. The process is idealized by considering that the air ﬂowing in is instantly mixed inside the chamber, where the thermodynamic state of the air is assumed uniform at all times. Considering as before an adiabatic process, the ﬁrst law of thermodynamics applied to our open system yields (see Kestin, 1966) du (h1 Ϫ u)w ϩ (h2 Ϫ u)wv ϩ m dt Ϫ (pa ϩ p)q ϭ 0 where subscripts 1 and 2 denote conditions respectively at the open boundaries S1 (duct exit) and S2 (by-pass valve exit). Since no work is done at the valves, it is h1 ϭ ht (where the subscript t denotes conditions at the turbine exit) and h2 ϭ ha. As we are dealing with a perfect gas, the equation above can be written as dT (cpTt Ϫ cvT)w ϩ (cpTa Ϫ cvT)wv ϩ mcv dt Ϫ (pa ϩ p)q ϭ 0 (6) where T ϭ (pa ϩ p)V/(mR). From turbomachinery theory (Dixon, 1978) it is known that the viscous loss in the turbine (which here is assumed to include the losses in the connecting ducts but not in the throttle-valve) per unit mass of air is ht Ϫ h1s ϭ h1 Ϫ h1s ϭ cp(T1 Ϫ T1s), where ͫ ͬ T1s ϭ Ta pa ϩ pt pa (␥ Ϫ 1)/␥ is the temperature of the thermodynamic state whose pressure is equal to the pressure pa ϩ pt at the turbine exit and whose speciﬁc entropy is the same as that in the atmosphere outside. Denoting by L ϭ Ϫ (h1 Ϫ h1s)w the power lost due to viscous effects in the turbine and its connecting ducts (excluding the throttle-valve), we may write ͫ ͬ Ϫ cpTtw ϭ Ϫ cpTaw pa ϩ pt pa (␥ Ϫ 1)/␥ ϩL (7) Introducing this result into Eq. (6) and eliminating q between the resulting equation and Eq. (1), we ﬁnd after some rearrangement ͫ ͬ ͫ ͬ ͫ ͬ cp 1 ␥RT dp dt Ϫ d dt 1L w ϭ T m Ϫ cp m T1s Ϫ 1 T Ϫ cp wv m Ta Ϫ 1 T (8) This is the equation which relates density variation to pressure variation during the ﬁlling process. The power loss L is assumed to be a known function of the pressure difference pt for given rotational speed of the turbine. The instantaneous value of the mass m is related to w and wv by A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1281 ͵t m(t) ϭ Ϫ (w ϩ wv)dt ϩ m(ti) ti where ti is some initial time. From well known thermodynamic relations for a perfect gas, it turns out that the left-hand side of Eq. (8) is equal to the derivative ds/dt of the speciﬁc entropy of air in the chamber during the ﬁlling process. Ultimately, the variation in s is due to the entrance of air (through surfaces S1 and S2) whose pressure is assumed equal to the chamber pressure pa ϩ p but whose entrance temperature (T1 ϭ Tt, T2 ϭ Ta respectively at S1 and S2) and hence entropy (s1 > sa, s2 > sa) are in general different from the corresponding values (T, s) in the chamber. Assuming constant atmospheric conditions, it is s > sa (as can be concluded from the second law of thermodynamics), but s must be bounded and so the time-average of the quantity on the right-hand side of Eq. (8) over a sufﬁciently long period of time (several ﬁlling periods) must be zero. Let us examine the meaning of the three terms on the right-hand side of Eq. (8). The ﬁrst one is intrinsically positive and is directly related to the entropy increase due to viscous losses in the turbine. The sign of the other two terms depends on the outer and inner air temperatures. The second term represents the contribution from air entering the chamber from the duct if the expansion in the turbine were isentropic (ideal turbine). Finally, the third term represents the contribution of bypassing air. Since the three terms on the right-hand side of Eq. (8) depend differently on time, their sum (equal to ds/dt) will oscillate about a zero time-average. It follows that the isentropic relation (5) is expected to provide a poorer approximation to the ﬁlling process than it does for the exhausting process. 2.3. Turbine The turbine is subject to a pressure difference pt ϭ p Ϫ pv, where ͉pv͉ is the pressure loss at the throttle-valve (pv ϭ 0 if the valve is fully open or if there is no throttle-valve). Neglecting the effect of the variations in Reynolds number and Mach number, the performance characteristics of the turbine can be written in dimen- sionless form as (see e.g. Dixon, 1978) ⌽ ϭ fw(⌿), ⌸ ϭ fP(⌿) where (9a,b) ⌽ ϭ w *ND3 , ⌿ ϭ pt *N2D2 and ⌸ ϭ P *N3D5 are dimensionless coefﬁcients of ﬂow, pressure and power respectively, and D is the outer diameter of the turbine rotor, N its rotational speed (in radians per unit time) and P the power output. The variable * is a reference density, the usual choice being the stagnation density at the turbine entrance. The functions fw and fP depend on the geometry of the turbine, but not on its size and rotational speed. Since we 1282 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 are assuming the turbine to be self-rectifying, w and pt take positive as well as negative values, and so do the corresponding dimensionless coefﬁcients ⌽ and ⌿. In addition, we assume that the turbine and its connecting ducts are symmetrical with respect to a plane normal to the rotational axis, and consequently fw is an odd function and fP is an even function. For some types of turbines, including especially the Wells turbine, the curve of ⌸ ϭ fP(͉⌿͉) exhibits a maximum, for a critical value ͉⌿͉ ϭ ⌿cr. It will be assumed that the device is equipped either with a by-pass valve, or with a throttle-valve in the turbine duct. In either case, the purpose of the valve is to prevent the pressure difference across the turbine from exceeding (in absolute value) its critical value, i.e. to keep ͉⌿͉ Յ ⌿cr. This will be done by opening the by-pass valve or alternately by throttling the valve in the turbine duct. Moreover it will be assumed that the valve (either the by-pass valve or the throttle-valve) will not overperform its function, i.e. ͉⌿͉ ϭ ⌿cr whenever it is limiting the pressure drop pt through the turbine. We note that the critical value pcr of pt depends on the rotational speed of the turbine (it is proportional to N2). 2.4. Valves It may be useful to introduce dimensionless variables to represent the aperture position of the valves. Assuming fully turbulent ﬂow through the by-pass valve, we may write ␣A ϭ ͉wv͉ √2͉p͉ where ␣A is the effective area of passage of the valve, A is a reference area (we set A ϭ D2/4) and wv ϭ Ϫ dm/dt Ϫ w is the ﬂow rate through the valve. The dimensionless variable ␣ should take the values ␣ ϭ 0 for ͉p͉ Ͻ pcr, | |dm ␣ϭ dt Ϫ wcr A√2pcr for ͉p͉ ϭ pcr (10) Here, wcr is the (positive) critical value of w corresponding to the critical pressure pcr (Eq. (9a). Likewise, for the throttle-valve we deﬁne the dimensionless variable A  ϭ ͉w͉ √2͉p Ϫ pt͉ where A is the same reference area as above. (It is  ϭ 0 if the valve is fully open or if there is no throttle-valve.) The variable  should take the values A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1283 ·  ϭ 0 for ͉p͉ Յ pcr, A  ϭ wcr √2(͉p͉ Ϫ pcr) for ͉p͉ > pcr (11) We note that in practice the valves are not in general symmetrical with respect to the ﬂow direction, and so a given value of ␣ does not represent necessarily the same position of aperture of the by-pass valve for outward ﬂow and for inward ﬂow. The same can be said about  and the throttle-valve. 3. Theoretical simulation The beneﬁts from using by-pass or throttle valves to limit the air ﬂow rate through the turbine in an OWC wave power device will be illustrated by numerically simulat- ing the performance of a power plant. A simple, two-dimensional geometry will be adopted for the submerged solid boundaries of the system, in order to be able to use known analytical results for the hydrodynamic coefﬁcient of radiation B(). We assume unidirectional waves to propagate in water of constant depth h towards the OWC structure, which spans a (theoretically semi-inﬁnite) channel formed by two parallel vertical walls apart from each other by a distance b equal to the inner width of the chamber. This can be regarded as simulating an OWC device built in a gully, as the pilot plants erected and on the island of Pico, on the Azores i(sFlaanlcda˜oo,f Islay, Scotland (Whittaker 1998). The fully reﬂecting et al., 1993) back wall of the chamber extends vertically to the bottom, whereas the submerged part of the front wall is assumed to be of negligible draft and of small thickness so that the diffraction it produces may be ignored. We denote by a the distance between the back and front walls at the water free-surface level (chamber length). For this (Sarmento agnedomFaeltcrya˜o, ,th1e98h5y)drodynamic coefﬁcient of radiation B() is given by 2b B ϭ Ϫ wgk sin2ka (12) where g is the acceleration of gravity, w is the density of water, k is the wave number related to by the dispersion relationship k tanh kh ϭ , ϭ 2/g, and ϭ (1 ϩ h csch2kh)−1. Then, introducing the dimensionless variables Ί Ί gˆr ϭ w 2b gr, ˆ k ϭ kh, ˆ ϭ h g, ˆt ϭ t g h Eq. (4) may be written as gˆr(ˆt) ϭ Ci(⍀ˆt) Ϫ 2I1 ϩ I2 Ϫ 2I3 (13) where ⍀ is a positive number, Ci(z) is the cosine integral function (Abramowitz and Stegun, 1965), and 1284 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 ͵ ͵ I1 ϭ ⍀ ˆˆ k sin2 ˆ ka h cos ˆ ˆt dˆ , I2 ϭ ϱ 1 ˆ cos 2ˆ 2a h cos ˆ ˆt dˆ , 0 ⍀ ͵ϱ I3 ϭ (ˆ ) cos ˆ ˆt dˆ (14) ⍀ where (ˆ ) ϭ ˆˆ k sin2 ˆ ka h Ϫ 1 ˆ sin2 ˆ 2a h ͫ ͩ ͪ ͬ ෂ Ϫ 2e−2ˆ2 ˆ a sin2 2ˆ 2a ϩ h h ˆ Ϫ 1 ˆ ˆ 2a sin2 h as ˆ →ϱ It is known (Abramowitz and Stegun, 1965) that Ci(z) has a logarithmic singularity at z ϭ 0, and Ci(z) ෂ (1/z)sin z as follows that the integrals I1 and I3 integration by parts, that they are ztO→en(1ϱd/.ˆtt)oF. rBzoemyroftohalesloˆttw→heinϱogr;eimandpoerfeodcReiidetummreaanysni–bmLeilefaboreustnogdu,tehbaiytt used by Titchmarsh (1948, theorem 11) to prove a theorem on Fourier integrals of iPnscbnhooertoonnescwvg-aepersidaerdgluraiertIohenp3dsatliiotscfwtogˆOoiorrt(shf(ˆtcn[)agiuˆrlhhrelm(aas−ˆttp)e1sirenϩfiacgoctarl1foltuslo]gyeenav−ˆtcre2eti⍀avitrohna2a)nmdllusavia,incsatidiltnsOuigmmen(s1gaagˆ/uyyorˆt()lfˆtabb)araeeism/thynsˆtah.e→yaogtwlϱbeˆtn.ceϭtAetfhodd0audit,nfinatd⍀hngeidnuiisipnJstcuethOshgtoer(isnas1eelo/nˆtI)r(2el1aasi9srsug9lˆtuet3→sn),e.iϱnfiFoto.irmugTmg.halhy2ye. We consider now the incident wave and assume ﬁrst that it is regular with angular frequency and amplitude A. Its surface elevation may be written as (x,t) ϭ A sin(t Ϫ kx ϩ ) (15) where x is a horizontal coordinate whose positive direction is the direction of advance of the incident wave, with x ϭ 0 at the back wall. Since the diffraction due to the front wall is neglected, the so-called diffraction ﬂow rate qi is simply that displaced, between x ϭ Ϫ a and x ϭ 0, by the free surface of the stationary wave system consisting of the superposition of the incident wave and the wave reﬂected on the back wall. We easily ﬁnd qi(t) ϭ 2Ab k sin(2ka) cos(t ϩ ) (16) Wave motion and wave energy absorption are time-varying oscillatory phenomena, and it is difﬁcult to deﬁne the instantaneous wave power available to a device. Conversion efﬁciency is meaningful only in terms of time-averaged quantities. The energy ﬂux of the incident wave can be deﬁned as the rate of work done by the pressure forces at the vertical plane x ϭ constant. Its value averaged over an interval A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1285 Fig. 2. Curves of gˆr(ˆt) versus ˆt for a/h ϭ 0.5 (solid line), a/h ϭ 1 (broken line) and a/h ϭ 2 (chain line). t1 Յ t Յ t1 ϩ ⌬t is a function of x and t1, except if ⌬t is an integer multiple of the wave period 2/, in which case it is given by Ei ϭ A2bwg 4k (17) We will assume that real, irregular waves can be simulated by the superposition of a ﬁnite number N of regular waves, and write N (x,t) ϭ An sin(nt Ϫ knx ϩ n) (18) nϭ1 The diffraction ﬂow rate is now qi(t) ϭ 2b n N ϭ 1 Ann kn sin(2kna) cos(nt ϩ n) (19) We may write, to any desired degree of approximation, n ϭ 0/Nn, where Nn is an integer, and it follows that the wave represented by Eq. (18) is periodic with timeperiod T ϭ 2M/0, where M is the least common multiple of Nn (n ϭ 1, 2, …, N). Then it can be shown that the energy ﬂux of the incident wave averaged 1286 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 over a period of time t1 Յ t Յ t1 ϩ T is independent of x and t1 and is equal to the sum of the separate contributions of the components Ei ϭ bwg 4 n N ϭ 1 A2nn nkn (20) 4. Numerical results of We the chose the European geometry pilot plant of the chamber to built on the island be of Paipcpor,oAxizmoaretesly(Ftahlacta˜oo,f the chamber 1998), which has a square planform of dimensions 12 ϫ 12 m2. The volume of the chamber above the still water surface is V0 ϭ 1050 m3, and the water depth is h ϭ 8 m. Atmospheric conditions are taken to be pa ϭ 1.013 ϫ 105 Pa, a ϭ 1.2 kg m−3, and the density of water is w ϭ 1.025 ϫ 103 kg m−3. The thermodynamic properties of air are ␥ ϭ 1.4, cp ϭ 1004.5 J kg K−1. Turbine performance curves were taken from laboratory test results of a turbine model described in detail by Gato et al. (1996). The model has a monoplane six- bladed rotor (outer diameter 590 mm, hub diameter 400 mm) placed between two rows of guide vanes. The rotor blades are of constant chord equal to 125 mm and their proﬁle is NACA 0015. The performance characteristics of the turbine are rep- resented in dimensionless form in Figs. 3 and 4 by the curves ⌽ ϭfw(⌿) and ⌸ ϭ fP(⌿). It should be mentioned that the mechanical losses were ignored (more precisely, they were determined separately and added to the measured shaft-power output). The numerical simulations were performed for a rotor outer diameter D ϭ Fig. 3. Curve of ﬂow rate coefﬁcient ⌽ versus pressure coefﬁcient ⌿ for the Wells turbine. A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1287 Fig. 4. Curve of power coefﬁcient ⌸ versus pressure coefﬁcient ⌿ for the Wells turbine. 2.3 m and rotational speed N ϭ 157.1 rad/s (1500 rpm), practically identical to the corresponding values for the turbine of the Pico pilot plant. For these values of D and N, it can be found, from Fig. 3, that the pressure pt may be assumed to be proportional to the mass ﬂow rate with a constant of proportionality equal to pt/w ϭ 99.4 Pa s kg−1. This value coincides practically with what resulted from the experimental optimization of the power take-off impedance carried out in irregular-wave basin with a model of the Pico OWC pilot plant (Brito-Melo et al., 1995). From Fig. 4 it may be found that the critical value of the pressure is pcr ϭ 10.5 kPa. The equation to be integrated is ͵t d VϪ dt gr(t Ϫ )p() d ϩ w ϩ wv Ϫ qi(t) ϭ 0 (21) Ϫϱ The derivative d/dt is related to dp/dt by Eq. (5) during the discharge period and by Eq. (8) during the ﬁlling period. In order to save computing time, most results presented here were computed using a linearized version of Eq. (21). In such cases, Eqs. (5) and (8) are replaced approximately by d/dt ϭ (␥RTa)−1dp/dt, and, instead of Eq. (21), we have ͵t V0 ␥pa dp dt ϩ 1 a (w ϩ wv) Ϫ gr(t Ϫ )p()d Ϫ qi(t) ϭ 0 (22) Ϫϱ where V0 is the volume of air in the chamber in undisturbed conditions. In any case, 1288 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 the variables w and wv are assumed to be known functions of p for given rotational speed of the turbine and given aperture position of the valves (see Sections 2.3 and 2.4). The integration of Eq. (21) or Eq. (22) was performed numerically for t Ն 0, it being assumed that p(t) ϭ 0 for t the equations, with time step equal to Յ 0. 0.04 sA. Stirnacpeegzˆor(iˆtd)ails rOu(l1e/ˆtw) aass ˆtu→seϱd, to integrate the convol- uvtailounesinˆtteϪgr1a4l 2in; Eq. tϭ (21) or present Eq. (22) time), in was truncated at t Ϫ 128 s (in order to save computing time. dimensionless Calculations were performed for regular and irregular waves. In the latter case, a spectrum was adopted (Fig. 5) which was regarded as representative of the wave climate at the site chosen for the erection of the pilot plant in the island of Pico, Azores, as measured by an ultrasonic probe suspended above the water (Pontes and Oliveira-Pires, 1992). In order to make comparisons easier, the same spectral distri- bution was kept, the energy level being changed by simply multiplying the ordinates in Fig. 5 by a factor independent of frequency. In the calculations the spectrum was actually replaced by a discrete one, consisting of the superposition of 100 compo- nents having 640−1 Hz as their least common multiple, which ensures periodicity with period ⌰ ϭ 640 s. The incident wave energy ﬂux was calculated by Eq. (20). In the case of regular waves, the frequency was taken equal to the peak frequency in Fig. 5 (0.086 Hz). In order to allow for transient effects related to the choice of initial conditions at t ϭ 0 to die out, results were taken only for t greater than six wave periods. A reference (design) value of the incident wave ﬂux (per unit wave Fig. 5. Representative spectrum of the wave climate at the site chosen for the erection of the power plant. Spectrum average power is 20.9 kW/m and the peak frequency is 0.086 Hz. A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1289 crest length) was deﬁned, for regular as well as for irregular waves, as the value Ei ϭ 20.9 kW/m ϭ Edes. A dimensionless incident wave power level was deﬁned as ⌳ ϭ Ei/Edes. Values for the time-averaged efﬁciency of the plant were calculated from ͵ ΄ ͵ ΅ t1 ϩ ⌰ t1 ϩ ⌰ −1 ϭ P(t)dt b Ei(t)dt (23) t1 t1 For regular waves, t1 was taken larger than six times the wave period (for the reasons stated above), whereas in the case of irregular waves t1 was set equal to zero. Figs. 6 and 7 show values of as a function of the dimensionless incident wave power ⌳, for regular and irregular waves, respectively. In any case, the results were computed from the linearized Eq. (22). The curves concern a plant equipped with a throttle valve, a by-pass valve or with no valve. In calm seas (low values of ⌳) the three curves coincide, since the valves are not required to limit the ﬂow through the turbine. The beneﬁts from using a ﬂow-limiting valve can be seen to become important for larger values of ⌳, especially in irregular waves (which, for equal ⌳, would produce higher turbine-ﬂow peaks than regular waves). The comparison of Figs. 6 and 7 shows that (except in calm seas), for the same incident wave power level, the plant efﬁciency is lower under irregular wave conditions than in regular waves; this is true whether valves to limit the turbine ﬂow are used or not. Fig. 6. Average efﬁciency for regular waves, with air ﬂow control by a throttle-valve (broken line), a by-pass valve (chain line), or no ﬂow control (solid line). 1290 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 Fig. 7. Average efﬁciency for irregular waves, with air ﬂow control by a throttle-valve (broken line), a by-pass valve (chain line), or no ﬂow control (solid line). As already mentioned in Section 1, if a by-pass valve is used to limit the ﬂow through the turbine, this is done by reducing the air pressure oscillations inside the chamber, whereas a throttle valve produces an increase in pressure oscillations. This means that the hydrodynamic process of energy extraction is modiﬁed in different ways by the operation of the two types of valves, and consequently different plant efﬁciencies should be expected (although the two methods are in principle equivalent from the point of view of controlling the turbine ﬂow rate). Figs. 6 and 7 show that it is so, and that, for the particular cases represented, better efﬁciencies are achieved with a by-pass valve as compared with a throttle valve. Let us analyze this matter in more detail. For given device geometry, it is known that the amount of energy absorbed (assumed from a linear) given power train of take-off regular waves depends on mechanism (Evans, 1982; the impedance Sarmento and Foaflcta˜hoe, 1985), and, in this context, we say that the system may be optimally damped, over- damped or under-damped. If the ﬂow through the turbine is reduced by partially closing a throttle valve in series, this is equivalent to increasing the impedance of the power take-off mechanism (which ceases to be linear). If a by-pass valve is used for the same purpose, this can be regarded (as far as the hydrodynamic wave-energy absorption process is concerned) as decreasing the impedance. It seems reasonable to conclude that a valve in series (rather than a by-pass valve) should be used if the system is under-damped by the turbine, since this would lead to more energy absorbed from the waves (it being assumed that both valves would perform equally well the task of limiting the turbine ﬂow to avoid aerodynamic blade-stalling). Con- A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1291 versely, a by-pass valve should be used to control the ﬂow through an over-damping turbine. These considerations can be extended from regular to irregular waves. Figs. 6 and 7 apparently indicate that the wave energy absorbing system is over-damped by the turbine (in regular as well as irregular waves), since more energy is produced by the turbine if a by-pass valve is used as compared with a throttle valve. For the present device’s geometry, the damping coefﬁcient that would allow maximum energy results absorption derived by Sfarormmen0t.o08a6ndHzFarlecga˜uola(1r 9w85av) easndcains be 95 calculated from Pa s kg−1. This is analytical about 5% less than the damping coefﬁcient (99.4 Pa s kg−1) of the turbine used in these simula- tions (see above), which conﬁrms that the system indeed is over-damped. A similar conﬁrmation for irregular waves was not attempted. All the numerical results presented here so far were computed from Eq. (22) and are based on a linearized isentropic pressure–density relationship for air. Results were also obtained based on the more realistic pressure–density relationship provided by the non-linear Eqs. (5) and (8), and are shown in Fig. 8 in terms of air entropy oscillations in the chamber for irregular incident waves of power level ⌳ ϭ 1; here no valve is assumed to operate for controlling the air turbine ﬂow rate. At time t ϭ 0 the inside air thermodynamic conditions were assumed equal to those of air outside. As should be expected, for t > 0 the entropy oscillates but is necessarily always larger that its initial value. The sharp jumps (the most remarkable of which occurs at t Х 590 s) correspond to large negative pressure peaks due to large wave troughs. A part of the plot of Fig. 8 is enlarged in Fig. 9, where air pressure oscillations are also represented. As should be expected, entropy changes can be seen to occur only Fig. 8. Numerical simulation for the speciﬁc entropy at the chamber. 1292 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 Fig. 9. Speciﬁc entropy (solid line) and pressure (broken line) variation for the air inside the chamber for a time interval of 100 s. during the ﬁlling period, i.e. while p Ͻ 0. The derivative ds/dt can be positive or negative depending on the sign of the quantity on the right-hand side of Eq. (8), as already commented in Section 2.2. It is known that, for a perfect gas, it is d ϭ cp ds ϩ 1 ␥RT dp (24) The two terms on the right-hand side of Eq. (24) represent the contributions to the variations in density due to entropy and pressure variations, respectively. Two types of errors are introduced when a linearized isentropic version is used (as in Eq. (22)): (i) by neglecting the ﬁrst term on the right-hand side of Eq. (24); (ii) by replacing the factor (␥RT)−1 by (␥RTa)−1. It may be of interest to know the order of magnitude of these errors. We write ͵s 1 cp ds Х a cp (s Ϫ sa) ϭ a⑀1 sa If the process is assumed isentropic, the second term on the right-hand side of Eq. (24) gives, upon integration, exactly ͵ ͫͩ ͪ ͬ p 1 ␥R dp T ϭ a 1ϩ p 1/␥ Ϫ1 pa ϭ a⑀2 0 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 1293 which, in linearized version, becomes more simply ap/(␥pa) ϭ a⑀3. We assume that the values shown in Fig. 9 are representative of the plant performing under design conditions (⌳ ϭ 1) and take, as orders of magnitude, p ϭ 5 ϫ 103 Pa, s Ϫ sa ϭ 0.4 J kg−1 K−1. We obtain ⑀1 Х 0.4 ϫ 10−3, ⑀2 Х 35 ϫ 10−3, ⑀2 Ϫ ⑀3 Х Ϫ 0.25 ϫ 10−3. These ﬁgures show that, under plant design conditions, the linearized isentropic relationship between density and pressure introduces an error not exceeding about 1%. The contributions to the error, arising from linearization (⑀2 Ϫ ⑀3) and from neglect of entropy changes (⑀1), are of the same order of magnitude. This agrees qualitatively with the results obtained for the device average efﬁciency in the presence of irregular waves of power level ⌳ ϭ 1: ϭ 0.3288 based on the linearized isentropic density–pressure relationship; ϭ 0.3273 based on the non-linear isentropic relationship; ϭ 0.3265 if a non-linear relationship is used allowing for entropy variations. The errors from imperfect modelling of air compressibility effects are expected to increase fast with wave power levels; however, the same can be said of the errors in the calculation of the energy absorbed from the waves if linear water wave theory is used. 5. Conclusions A theoretical model was developed to simulate the energy conversion, from wave to turbine shaft, of an OWC plant equipped with a Wells turbine. The model realistically allows for changes in entropy of air in the chamber, due to viscous losses, to be accounted for. The use of a valve or a set of valves to control the ﬂow through the turbine (and in this way prevent or reduce the aerodynamic stall losses at the turbine rotor blades) was found to provide a way of substantially increasing the amount of energy produced by the plant, particularly at the higher incident wave power levels. This increase is expected to be very important for turbines whose performance is drastically affected by rotor stalling, as is the case of the turbine used in the numerical simulations and is known to occur with most Wells turbines of ﬁxed-pitch type. In principle a by-pass valve should be used if the damping of the (assumedly linear) turbine is substantially larger than optimum (from the point of view of the hydrodynamic process of wave energy absorption); a throttle valve in series with the turbine is expected to lead to better results if the system is clearly under-damped. However the choice of one or the other type of valve will very likely depend basically on constructional and operational factors. The isentropic linearized relationship between density and pressure was found to provide a satisfactory approximation in the modelling of the spring-like effects due to air compressibility, except possibly under very rough sea conditions. The errors introduced by linearization and by the isentropic assumption were found to be of the same order of magnitude. 1294 A.F. de O. Falca˜o, P.A.P. Justino / Ocean Engineering 26 (1999) 1275–1295 Acknowledgements The work reported here was partly supported by the European Commission under contract nos JOU2-CT93-0314 and JOR3-CT95-0012, and by IDMEC, Lisbon. The second author is indebted to Program PRAXIS XXI for a research grant. References Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions. Dover, New York. Brito-Melo, A., Sarmento, A.J.N.A., Gato, L.M.C., 1995. 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