oscillating water column 2

7/23/2019 Oscillating Water Column 2

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Model-prototype similarityof oscillating-water-column waveenergy converters

Antnio F.O. Falco , Joo C.C. Henriques

IDMEC, LAETA, Instituto Superior Tcnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

Article history:

Received 4 April 2014

Revised 8 May 2014

Accepted 8 May 2014

Available online 27 May 2014

Keywords:

Wave energy

Oscillating-water-column

Dimensional analysis

Model testing

Air turbines

a b s t r a c t

Model testing in wave tanks or under sheltered sea conditions is an

essential step in the development of wave energy converters. The

paper focuses on the rules for geometric, hydrodynamic, thermody-

namic and aerodynamic similarity in model testing of wave energy

converters of oscillating-water-column (OWC) type, with emphasis

on air compressibility effects in the air chamber and on air turbine

aerodynamics. It is shown that the correct volume scale ratio for the

air chamber is far from identical to the volume scale ratio for the

submerged part of the converter, and should take into account

the thermodynamics of the compressible flow through the air tur-

bine or through the turbine simulator (orifice or other). For those

cases when the model is large enough to be fitted with a scaled

air turbine, dimensional analysis is applied to obtain ratios for tur-

bine size and rotational speed, and also to establish relationships

between rotational speed control algorithms. A numerical example

is presented to illustrate the importance of appropriately simulat-

ing the air compressibility effects when testing at model scale.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

The theoretical modelling based on linear water wave theory is an essential step in the develop-

ment of wave energy converters. It provides insights and important information at relatively low

http://dx.doi.org/10.1016/j.ijome.2014.05.002

2214-1669/2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +351 919190017; fax: +351 218417398.

E-mail address:[email protected](A.F.O. Falco).

International Journal of Marine Energy 6 (2014) 1834

Contents lists available at ScienceDirect

International Journal of Marine Energy

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j o m e
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Nomenclature

Latin lettersAw incident wave amplitude

B radiation susceptanceC proportionality constantD diameter of turbine rotorf frequencyF function (see Eq.(1))Fr Froude numberg acceleration of gravityG radiation conductanceHs significant wave heightI rotational inertiak polytropic exponentK wq1

at

q1 flow-rate/pressure-head ratioL characteristic length of devicem mass of air in chamberN scale ratio (with subscript)p pressure oscillation in chamberp0 complex amplitude ofpP powerq qe qr volume flow rate displaced by OWCqe excitation flow rateqr radiation flow rateQ,Qe,Qr complex amplitude ofq ,qe,qrr p=pat pressure ratio

Re Reynolds numbers specific entropySf variance density spectrum of wavesTe energy period of wavesU function (see Eq.(2))V air chamber volumeV0 value ofVwithout wavesw turbine mass flow rateYe electromagnetic torque on generator rotorYt aerodynamic torque on turbine rotorGreek letters

c cp=cv specific heat ratio of air

C see Eq.(9)d qm=qF water density ratio Lm=LF length scale of deviceg turbine efficiencym kinematic viscosityN dimensionless aerodynamic torqueP dimensionless quantityq densityr variance (with subscript)U dimensionless flow rate of turbinev proportionality constant (orifice, porous plug)W dimensionless pressure head of turbine

X rotational speed of turbine

A.F.O. Falco, J.C.C. Henriques / International Journal of Marine Energy 6 (2014) 1834 19
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costs, in general in a relatively fast way. However, there are important non-linear effects that are not

accounted for by this kind of modelling, namely those associated with large amplitude waves, largeamplitude motions of the wave energy converter or oscillating-water-column, and real fluid effects

due to viscosity, turbulence, and vortex shedding. Commercially available computational-fluid-

dynamics (CFD) codes, usually based on the numerical integration of the Reynolds-averaged Navier-

Stokes (RANS) equations, may be used to account for such effects. However, even such codes, apart

from being computationally demanding, require some experimental validation. Therefore physical

model testing in a wave flume or wave tank is normally the next step. The scales range between about

1:100th in small wave flumes to about 1:10th in the largest wave tanks. Tests at larger scales (typi-

cally 1:4th scale) sometimes take place in sheltered sea locations. We may apply dimensional analysis

techniques to relate the conditions in model testing to those of the full-sized prototype in real sea

conditions.

The books on dimensional analysis by Bridgman[1], Langhaar[2]and Sedov[3]may be regarded asclassics. Most textbooks on fluid mechanics include a chapter on dimensional analysis. Applications to

coastal engineering and to naval architecture can be found in[4]and[5], respectively. Dimensional

analysis in model testing of wave energy converters is addressed in the pioneer book by McCormick

[6]and more recently in[7,8]. Air turbines of oscillating-water-column converters are subject to the

same similarity laws as other types of compressible flow turbomachines, as analysed in [911].

Oscillating-water-column (OWC) devices, of fixed structure or floating (Fig. 1), are an important

class of wave energy converters. A large part of wave energy converter prototypes deployed so far into

the sea are of OWC type [12,13].

In an OWC, there is a fixed or oscillating hollow structure, open to the sea below the water surface,

that traps air above the inner free-surface. Wave action alternately compresses and decompresses the

trapped air which forces air to flow through a turbine coupled to a generator. Unless rectifying valvesare used, which is not practical except possibly in small devices like navigation buoys, the turbines are

self-rectifying, i.e. their rotational direction remains unchanged regardless of the direction of the air

flow. Several types of such special turbines have been developed. The axial-flow Wells turbine,

invented in the mid-1970s, is the most popular self-rectifying turbine, but other types, namely axial-

and radial-flow self-rectifying impulse turbines, have also been proposed, studied and used[14].

In OWC wave energy converters, the air in the chamber above the water column is subject to

oscillating pressure, and therefore its density is also time-varying according to some pressure-density

relationship. This spring-like effect, that affects the device performance, was theoretically modelled for

the first time in [15]and shortly afterwards in [16]. In both papers, an isentropic relationship was

assumed. A more realistic model, in which variations in air entropy are related to viscous losses in

the air turbine can be found in [17]. Model testing of an OWC where for the first time air compressibilityeffects in the chamber are accounted for is reported in [18]. The representation of aero-thermodynamic

effects in small scale physical modelling of OWCs was addressed more generally in[19].

The present paper focuses on model testing of wave energy converters of OWC type. Dimensional

analysis techniques are employed to relate the results from the tests to the performance of the

Subscriptsavai available to turbine (power)at atmospheric conditionsch air chamber conditions

F full-sized prototypeimp impulse turbinem modelt turbineWells Wells turbineSuperscripts dimensionless value time-average value

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full-sized prototype. Froude similarity is considered in Section 2. Particular attention is devoted, in

Section 3, to compressibility effects in the air chamber, known to be important at full size but fre-

quently ignored in model testing. Thermodynamics and turbine aerodynamic efficiency are shown

to play an important role here. Rules are derived for turbine size and rotational speed scale ratios

in Section 4. Dimensional analysis is extended to turbine rotational speed control in Section 5. A

numerical example is presented in Section 6 to illustrate the importance of appropriately simulating

the air compressibility effects when testing at model scale. Conclusions are presented in Section 7. The

results presented in Section 2 on Froude similarity are well known. They are presented here for com-

pleteness and because they are needed for the derivations in the more innovative sections 3 to 5where new results are obtained.

2. Dimensional analysis in hydrodynamics of model testing

When performing model testing in waves, it is assumed that the wetted part of the model is an

exact geometric representation of the full-scale prototype. This geometric similarity is supposed to

apply also to the bottom and surrounding walls. This condition is assumed here to be fulfilled. It

should be noted that, in most cases in practice, there are limitations in the test facilities that prevent

perfect geometric similarity to be attained. This is the case of the presence of walls in tanks and

flumes, or the impossibility of reproducing the sea bottom bathymetry.

We assume the incident waves to be represented by a given variance density spectrum(Pierson-Moskowitz or other, see e.g. [20]) SfHs; Te;f, where Hs 4 ffiffiffiffiffiffiffim0p is the significant waveheight, Te m1=m0 is the energy period, and mn is the n-th moment ofSf with respect to the fre-quency f. Directional wave spread is ignored here for reasons of simplicity. In the case of regular

waves, Hs andTe are simply the wave height and period.

LetLbe a characteristic length (this could be a diameter in the case of an axisymmetric device). We

denote bypat the atmospheric pressure and by pat pt the pressure in the air chamber of the OWCconverter; the pressure oscillation p(t) is related to the action of the power take-off system (PTO)

which is essentially an air turbine driving an electrical generator.

Let us consider in general a dimensional quantity a as a function ofn dimensional independent

quantities a Fa1; a2; :::;an; where the function F represents a definite physical law independent

of the choice of the system of units. In the case of a wave energy converter, this may have the formP FL; Hs; Te;p;g;q; m; 1

wherePis power (possibly power absorbed from the waves),gis the acceleration of gravity (which is

assumed as a physical constant), and q and m are the density and kinematic viscosity of water,

Fig. 1. Different types of OWCs: (a) bottom-standing structure (Pico plant); (b) Backward Bent Duct Buoy (BBDB); (c) spar-buoy

OWC.

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respectively. Here we ignored the surface tension of water, which is an acceptable assumption pro-

vided that the wavelength is larger than about 0.1 m[21]. Buckinghams theorem of dimensional anal-

ysis (see e.g.[2]) allows us to replace Eq. (1) by

P1 UHs=L; Fr; Re; Pp: 2

Here, U is a function, Fr L1=2g1=2T1e is a Froude number, Re L2m1T1e is a Reynolds number,Pp pL1q1g1 is dimensionless pressure and P1 PL7=2q1g3=2 is dimensionless power. (Notethat the reciprocal of the Froude number,Te Fr1 g1=2L1=2Te, may be regarded as a dimensionlesswave period.) Identical relationships could be established by replacing P1 by other dimensionless

quantities. For example, this could be dimensionless volumetric flow rate P2 qL5=2g1=2; where qis flow rate displaced by the motion of the OWC free surface. If the four dimensionless variables Fr,

Re, Hs=L and Pp take equal values in the model and the full-sized prototype, the same will be true

for P1; P2;:::

Note that, in the case of floating devices, mooring forces may be significant, and, if so, should be

appropriately represented in model testing [22]. However, if the mooring system is adequately

designed, the effect of mooring forces on wave energy absorption is relatively small, as found in[23]. Such effects will be ignored here.

Let the subscripts m and Fdenote the model and the full-sized prototype. The length scale is

defined as e Lm=LF. The constancy of the Froude number Fr and the Reynolds number Re cannotbe satisfied simultaneously, since this would require mm=mF e3=2, a condition that is obviouslyunachievable in practice ifeis not close to unity. In model testing, the effects due to variations in Fro-ude number are almost always much more important than those associated with changes in Reynolds

number. Following general practice, we will keep here the constancy of Froude number, and ignore

Reynolds number as a modelling rule.

The expression of dimensionless power P1 PL7=2q1g3=2 shows that the scale ratio for power ise7=2 (if variations in water densityq are neglected). In model testing of wave energy converters in thelargest wave tanks, the length scale e in general does not exceed about 1:10th. This implies a maxi-

mum power ratio of about 1:3200. In the case of an OWC wave energy converter, this scale is too small

for the turbine to be simulated adequately by a mini-turbine. The usual procedure is to simulate the

turbine by an orifice, if the turbine is of impulse type ( Fig. 2), or by a layer of porous material, where

the flow is approximately laminar and simulates a linear turbine like the Wells turbine; the approx-

imation provided by this testing procedure will be analysed in Sections 3, 4 and 5. Only in tests

Fig. 2. Model at 1:16th scale of a spar-buoy OWC developed at Instituto Superior Tcnico being tested at the large wave tank of

the National Renewable Energy Centre (NAREC), Blyth, England, in 2012. The turbine was simulated by an orifice. The tank was

filled with sea water.



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performed under real sea conditions at scales not less than about 1:4th, is a real turbine fitted to the

model. This was done in the sheltered waters of Galway Bay, Ireland, at scale 1:4th, with a Backward

Bent Duct Buoy (BBDB) fitted first with a Wells turbine and later with an impulse turbine[24],Fig. 3.

3. Aero-thermodynamic modelling of air chamber

The volume of the air chamber of the OWC converter should be large enough to avoid ingestion of

water by the air turbine under rough sea conditions. Typical design values of the air chamber volume

divided by the area of the OWC free surface range between 3 and 8 m. An increase in this ratio is not

necessarily detrimental to the efficiency of the energy conversion. Obviously, if the volume increases

to very large values, the amplitude of the air pressure oscillations becomes very small, and the capa-

bility of the device to absorb wave energy vanishes. The spring-like effect of air compressibility in the

chamber increases with chamber volume, and is important in a full-sized OWC converter.

The aerodynamic and thermodynamic processes that take place in the air chamber and turbine of

an OWC converter are quite complex. It seems reasonable to assume that they are approximately adi-

abatic. Indeed, the temperature oscillations in the air chamber are relatively small and their time

scales (a few seconds) are too short for significant heat exchanges to occur across the chamber walls

and across the air-water interface, in comparison with the energy flux in the turbine [17]. It should berecalled that the compressible flow through turbomachines is usually modelled as adiabatic, even in

gas turbines where large temperature differences take place (see e.g. [11]).

However, even if the process is assumed as adiabatic, significant changes in specific entropy occur

in the flow through the turbine, due to viscous losses. Such changes can be related to turbine effi-

ciency. We recall that the pressure in the chamber is pat p, where pat is the atmospheric pressure.During inhalation, it isp < 0, and air with specific entropy s > sat is admitted to the chamber, where

a highly turbulent mixing process takes place. During exhalation, p> 0, air leaves the chamber

through the turbine in a process at approximately constant specific entropy for the air remaining in

the chamber. The inhalation and exhalation processes in an OWC were studied in some detail in[17].

We assume air as a perfect fluid, with specific heat ratio c

cp=cv

ffi1:4. We may write, for the

pressure-density relationship of the compressible flow of a perfect gas through a turbine [11],p2qk2

p1qk1

; 3

Fig. 3. Backward Bent Duct Buoy (1:4th of full scale) equipped with a Wells turbine being tested in Galway Bay, Ireland, about

2008 (courtesy of OceanEnergy).

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where subscripts 1 and 2 refer to conditions at turbine entrance and exit, respectively. The polytropic

exponentk is related to the turbine polytropic efficiency (or small-stage efficiency)g by

k 1

1 c1c g: 4

In turn, the turbine polytropic efficiency gis related to the turbine (total-to-total) efficiencygby (see[11])

g1 p2=p1

gc1=c

1 p2=p1c1=c

: 5

Here, efficiency is defined as for gas turbines: g Dh=Dhs;where Dhis drop in specific enthalpy underreal conditions and Dhs is drop in specific enthalpy under isentropic conditions, for the same inlet and

outlet pressures. In the case of inhalation we are analysing here, we replace p1bypatand p2 by pat p,withp < 0. Eq.(3) becomes

patp

qkch patqkat ; 6

whereqch is air density in the chamber and qat is density in the atmosphere.We writer p=pat. From Eqs(4) and (5),we can express the polytropic exponent k as a function

of the turbine efficiency g and the pressure ratio r. This is represented in Fig. 4forc 1:4. It can beseen that the polytropic exponentkis almost invariant with the pressure ratior p=pat, even for val-ues ofras large as 0.6 (pressure in the chamber equal to 40% of atmospheric pressure). For g 1 (per-fectly efficient turbine), the flow is isentropic and k c 1:4, as expected. Forg= 0, it is k= 1. In thiscase, no work is done and therefore there is no change in specific enthalpyh. Since for a perfect gas it is

dh cpdT(hereTis absolute temperature), the process is isothermal (note that this is not a result fromheat exchange). The caseg 0; k 1 represents a throttling process and occurs if the pressure dropthrough the turbine is simulated by an orifice (which is frequently used to simulate a self-rectifyingimpulse turbine) or, alternately, represents the flow through a plug made of porous material (where

the flow is approximately laminar and simulates a linear turbine like the Wells turbine).

Although the dependence of the polytropic exponent k on pressure ratio r can reasonably be

neglected as shown inFig. 4, the same cannot be said of the dependence on turbine efficiencyg. Dur-ing the inhalation, the flow rate through the turbine increases from zero to a peak value, and then

decreases to zero; correspondingly, the turbine aerodynamic efficiency oscillates between a very small

value, at zero flow rate, and larger values at near design conditions. We now look for average values

for the efficiencyg and for corresponding average values for the polytropic exponent k.An OWC wave energy converter may be regarded as a dynamical system having as input the wave

elevation at a given point, and as output the pressure oscillation p(t) in the air chamber. In a given sea

Fig. 4. Polytropic exponent k versus pressure ratio r p=pat for different values of turbine efficiency g .



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state, the wave elevation may be regarded with good approximation as a stationary Gaussian process

with a variance density spectrumSff (see e.g.[20]). If the system is linear, then the air pressure oscil-lation is also a stationary Gaussian process whose variance density spectrumSpf can be obtained forthe pressure oscillation provided that the transfer function is known (see e.g. [25]). Then, the variance

(or mean-square value)rp, and the probability density function /p, of the pressure oscillation p, can

be obtained directly from Spf. From /p, by integration, we may find average values for turbinepower output and turbine aerodynamic efficiency, provided that the instantaneous performance

curves of the turbine are known. Such procedures are described in detail in [26].

The Wells turbine is known to be approximately linear in terms of flow rate versus pressure head,

at fixed rotational speed (see[14,27]). This means that an OWC wave energy converter equipped with

a Wells turbine may be regarded with fairly good approximation as a linear system. In self-rectifying

impulse turbines, the flow rate is approximately proportional to the square-root of pressure head [14],

which is a nonlinear relationship that makes the linear system assumption a rougher approximation;

as in[25], we accept this approximation at this point (and nowhere else in the paper) solely for the

purpose of obtaining typical values for the averaged efficiency of self-rectifying impulse turbines of

OWC plants subject to random waves (seeFig. 5). Dimensionless results from model testing are pre-

sented inFig. 5for a Wells turbine and for a biradial impulse turbine [28]. In both cases, the testedturbines may be regarded as state-of-the-art machines. The solid curves in Fig. 5give the instanta-

neous efficiencyg of the turbine versus dimensionless pressure head W. The dotted curves representthe average efficiency g versus the variance or root-mean-square rW of W. Here, it isW jpjq1airX2D2, whereqair is air density, X is rotational speed (in radians per unit time) and D isturbine rotor diameter. These results indicate that, if a good turbine is employed and its rotational

speed is adequately controlled to match the sea state, the average efficiency of the turbine g isexpected to be in the range 0.6 to 0.7.

Having defined an average efficiencyg for the turbine, it seems reasonable to define also an averagevalue for the polytropic exponentkof the density-pressure relationship in the air chamber. Fig. 4indi-

cates this value to be in the range 1.2 to 1.25 for state-of-the-art air turbines.

We consider now the spring-like effect due to air compressibility in the chamber. Let qch, Vandm qchVbe the instantaneous values of the density, volume and mass of air in the chamber. The massflow rate of air through the turbine (positive for inward flow) is

wdm

dt V

dqchdt

qchq; 7

whereq dV=dtis the volume flow rate displaced by the motion of the OWC free surface (in a struc-ture-fixed frame of reference for a floating device). The first term on the right-hand-side of Eq.(7), pro-

portional todqch=dt, represents the spring-like effect of air compressibility in the chamber. If dynamicsimilarity is to be respected, then the ratio between the two terms on the right-hand-side of Eq. (7),

Fig. 5. Aerodynamic efficiency curves for a Wells turbine (left) and a biradial impulse turbine (right). Solid curves:

instantaneous efficiencygversus dimensionless pressure head W. Dotted curves: average efficiency gversus variance (or rms)rW ofW. From[28].

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C V

qchqdqch

dt ; 8

should take equal values under corresponding conditions in model and at full-scale. We saw above

that pressure and density of air in the chamber,p

patand qch;may be related to pressure and air den-sity in the atmosphere,patand qat;by a polytropic relationship (6), where the polytropic exponentkisrelated to an averaged efficiency gof the turbine as shown inFig. 4. From Eqs.(6) and (8), we find

C V

kqp pat

dp

dt: 9

For perfect Froude similarity, the scale ratios aree5=2 forq,edfor p, (d qm=qFis water density ratio),e1=2 for time, ande1=2d for the time derivative of pressure dp=dt.

We consider first that the geometric similarity is extended to the part of the device located above

water level, i.e. Vm=VF e3. This would allow realistic representation of water motion inside thechamber and provide information on how to avoid green water from reaching the air turbine. Eq.

(9)shows that full dynamic similarity, i.e. Cm CF, requireskm kF(equally efficient turbines at bothscales) andpat;m=pat;F ed. The latter condition, concerning atmospheric pressure, is obviously impos-sible to satisfy in practice if the length scalee is not close to unity.

Now we assume more realistically that the atmospheric pressure is the same at both scales

pat;m pat;F. We further assume that the pressure oscillation p is much smaller than the atmospheric

Fig. 6. Model testing of a cylindrical fixed-structure OWC in a wave flume (Instituto Superior Tcnico, Lisbon, 2013). To

appropriately reproduce the air compressibility effect in the chamber, the top of the tube is connected by a pipe to an air

reservoir placed above.



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pressure, which would exclude the more energetic sea states at full scale. If the Froude scale ratios for

flow rate q, pressure oscillationp and its time derivative dp=dtare as above, we are left with

Cm

CF

VmVF

kFkm

e2d: 10

The correct representation of air compressibility effects in the chamber requires Cm CF. Then, weobtain for the air chamber volume ratio

VmVF

kmkF

e2d1: 11

In model testing at large scale (say about 1:4th to 1:3rd) performed in the sea, it is d 1 (equalwater density). If in addition an appropriately scaled-sized turbine is used to realistically simulate

the full-sized machine, it iskmffi kF(if the model turbine is not too small). Then, from Eq.(11), we haveVmVF

e2: 12

A result identical to Eq. (12)was first obtained in[15], based on a frequency-domain analysis of thehydrodynamics and a linearized isentropic assumption. It later appeared in other papers [16,18,19].

It shows that the scale ratio for air chamber volume should be e2, rather thane3. Failure to meet thiscondition may result in substantial errors in the conversion to full scale of the experimental data at a

smaller scale (see the numerical example in Section 6).

In testing at scales smaller than about 1:8th, the turbine is likely to be simulated by an orifice or by

a window covered by porous material. In such cases, as mentioned above, we have a process at con-

stant temperature, and the polytropic exponent iskm 1:Eq.(11)becomesVmVF

1

kFe2d1: 13

If the model testing takes place in wave tank or flume filled with fresh water, which is the most fre-quent situation at such scales, the water density ratio is d qm=qFffi 0:97 .

If the OWC device is of fixed structure (possibly bottom-standing), it is not difficult to satisfy con-

dition (13) at small model scale. One simple way of achieving that is to connect the air chamber of the

model to a rigid-walled reservoir of air of appropriate volume. This procedure was adopted in the

model testing of the bottom-standing OWC installed in 1999 on the island of Pico, Azores, Portugal,

as reported in[18], and was also adopted in the model testing shown inFig. 6. If however a floating

OWC device is to be modelled at small scale, this could introduce difficulties because the reservoir is

likely not to be small compared with the devices size, and possibly is a lot larger. If the reservoir is

fixed to the floating device, the stability and the dynamics may be severely affected. The alternative

is to have a flexible pipe connection, which however would introduce undesirable forces and

moments.It may be interesting to investigate whether the failure to meet condition (13) in small-scale testing

could be compensated by an appropriate design of the turbine simulator. We will do that in a simpli-

fied way, and consider only regular waves. Besides, we assume small wave amplitude and a linear sys-

tem, so that frequency-domain analysis may be employed. The assumedly linear turbine is

represented by the relationshipw=qat Kp, whereKis a real positive proportionality constant. Tak-ing into account the polytropic pressure-density relationship (6), the time derivative dqch=dt thatappears in Eq.(7) may be written as

dqchdt

qch

kp pat

dp

dtffi

qatk pat

dp

dt: 14

Then, Eq.(7) becomes

Kp V0kpat

dp

dt q; 15

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where the chamber air volume Vwas replaced by its value V0 in the absence of waves, and qch wasreplaced byqat .

Since the system is linear and the incoming waves are regular, we may adopt the frequency domain

analysis and write p p0eixt andq Q eixt, wherep0and Qare complex amplitudes andxis radianfrequency. Equation(14)takes the form

KF QFp0;F

ixFV0;FkF pat

; 16

where the subscriptFindicates full-sized prototype. We recall thatKFis real positive for the full-sized

turbine, and thereforeQF=p0;F is complex and there is a non-zero phase difference vF argQF=p0;Fbetween flow rateqFdisplaced by the OWC and pressure oscillation pF, except ifV0;F 0;as shouldbe expected. This phase difference should be the kept unchanged in the model, if hydrodynamic sim-

ilarity is to be conserved.

We recall that the Froude scale ratios forq,pand x are e5=2; e dand e1=2, respectively. Then, we maywrite, for the model,

Km Qmp0;m

ixmV0;mk pate7=2d1 QFp0;F

ie1=2xFV0;FkF pat: 17

From Eqs.(16) and (17), it is not difficult to show that the proportionality constantKm for the turbine

simulator cannot be real positive except if

V0;mV0;F

kmkF

e2d1: 18

Not surprisingly, this is similar to condition (12) obtained above. If a different ratio is to be adopted,

namely V0;m=V0;F e3 for perfect geometrical similarity, dynamical similarity requires a mechanismexhibiting an appropriate phase difference between pressure head and flow rate, which, although con-

ceivable, would be a much more complex device than an orifice or a plug of porous material.

4. Aerodynamic modelling of air turbine

We assume now that the full-sized turbine and its model are geometrically similar. Dimensional

analysis applied to the compressible flow through a turbomachine (gas turbine, compressor) requires

that three dimensionless quantities take equal values between the model and the full-scale prototype

for full aerodynamic similarity: Reynolds number, Mach number and pressure ratio (see[11]). In our

case, during inhalation, the pressure ratio is

K p pat

pat: 19

We saw that Froude similarity requires thatpm=pF ed, whereeis length scale and d qm=qFis waterdensity ratio. Then, it is Km KF only ifpat;m=pat;F ed, a condition involving atmospheric pressurethat cannot be satisfied in practice if the length scale eis not close to unity.

Since full aerodynamic similarity is not compatible with Froude similarity, we look for an approx-

imation in turbine dimensional analysis. To do that, we assume, as above, that oscillations |p| in air

pressure are small compared with atmospheric pressure pat, and so changes in air density may be

neglected. This is equivalent to ignoring Mach number effects. In the case of approximately incom-

pressible flow, the application of Buckinghams theorem yields the following relationship for the flow

through the turbine (see[11])

U FUW;Ret; gFgW;Ret; N FNW;Ret; 20

where

W p

qatX2D2

; U w

qatXD3; N

Yt

qatX2D5

; RetXD2

mat: 21



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Here,w is mass flow rate through the turbine,qatandmatare, respectively, density and kinematic vis-cosity of air under atmospheric conditions, X is the rotational speed (in radians per unit time), Dis the

turbine rotor diameter and Ytis the aerodynamic torque on the turbine rotor. In Eqs.(20) and (21), W

is dimensionless pressure head,g is aerodynamic efficiency, U is dimensionless flow rate, N is dimen-sionless aerodynamic torque, and Ret is a Reynolds number. FunctionsFU,Fg andFN are the same for

model and prototype, since these are assumed geometrically similar.Note that bearing friction torque is in general relatively small and does not follow aerodynamic

similarity laws. For these reasons it will be ignored here.

We assume first that WmWF, for pressure head coefficient, and Um UF, for flow rate coefficient.Later we will investigate whether these conditions are compatible with Re t;m Ret;F .

We introduce the notations NX Xm=XF and ND Dm=DFfor the scale ratios for rotational speedand rotor diameter, respectively. We assume that atmospheric conditions are the same for the model

and the prototype.

From Um UFwe have

NXN3D

wm

wF

: 22

Eq.(7)may be rewritten as w C 1qatq, whereqis the volume flow rate displaced by the oscillat-ing-water-column motion and C is a dimensionless quantity, defined by Eq.(8), that is related to the

spring-like air compressibility effect in the chamber. If this effect is adequately simulated in the

model, then Cm CF and wm=wF qm=qF. We recall that the conditions for this to apply are definedin Section 3. The Froude scale ratio for the volume flow rateq is e5=2:Eq. (22)becomes

NXN3D e

5=2: 23

Condition WmWF, combined with pm=pF e d for Froude similarity, gives

N2X

N2D e d: 24

This, together with Eq.(23), yields

DmDF

e d1=4; 25

Xm

XFe1=2 d3=4: 26

Now we easily find

Ret;mRet;F

e3=2 d1=4: 27

Sinced is equal, or close, to unity, no equality in terms of Reynolds number is possible in the air flow

through the turbine. Eqs.(25) and (26)give the scale ratios for turbine size and rotational speed if a

geometrically similar turbine is used in the model testing of the OWC converter.

It may be of interest to investigate the effects of variations in Reynolds number on the efficiency of

self-rectifying air turbines, particularly variations between prototype and model. We consider first the

case of impulse turbines, that are geometrically and aerodynamically more similar to conventional

turbines than Wells turbines are. Typical values for the blade tip speed XD=2 of full-sized air turbines

of impulse type in various sea states range between about 50 and 100 m/s. Typical rotor diameters D

of full-sized impulse turbines are between about 0.7 and 2.0 m. Taking mat 1:47 105m=s at 15C,we find, for the Reynolds number Ret XD2=m, the values 4:7 106 for a small turbine in mild seas,

and 2:7 107

for a large turbine in energetic seas. There is extensive information in the literature onthe effects of the Reynolds number on the performance of conventional turbomachines; only a few

results are presented here. In centrifugal pumps, the efficiency becomes independent from the Rey-

nolds number for values ofXD2=m greater than about 4 106, and is affected by a factor of about0.95 or 0.75 if the Reynolds number is reduced to 4 105 or to 4 104, respectively[9]. In the case

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of centrifugal compressors, the critical Reynolds number was again found to be about 4 106 [29]. Foraxial-flow gas turbines, the critical Reynolds number is about 2 105 based on blade chord and flowvelocity, or roughly Ret;crit XD2=m ffi 2 107 for a typical axial-flow gas turbine; for lower Reynoldsnumbers, the loss 1 gshould be corrected by a factor proportional to Re1=5t according to the Ainleyand Mathieson criterion (see [11]). It seems reasonable to conclude that, in the range of Reynolds

numbers within which full-sized impulse-type air turbines operate, the efficiency is practically insen-

sitive to variations in Reynolds number. Since, in model testing, Ret is proportional to e3=2 (assumingd 1, see Eq.(27)), Reynolds number effects may significantly affect the performance of the impulseturbine model if the scale e is small, say less than 1:10th.

Compared with impulse turbines, and for identical applications, Wells turbines are characterized

by significantly larger values of the rotor diameter D and the rotational speed. The following ratios

were derived in [14] from theoretical considerations:X DWells=X Dimp 2:3, DWells=Dimp 1:4,Ret;Wells=Ret;imp 3:2. This shows that the Wells turbine is characterized by higher Reynolds numberthan turbines of impulse type. On the other hand, Wells turbines, because of their special aerodynamic

conception, are known to be much more sensitive to changes in Reynolds number than more conven-

tional turbines, as explained in[14], and to perform poorly in small model testing (and small flow

velocities), more so than impulse turbines[14]. A review of experimental rigs for testing air turbinescan be found in[14]. Most of the more reliable results for Wells turbines as well as impulse turbines

were obtained with model rotor diameters about 0.6 m and Reynolds numbers about 6 106 to12 106. It is worth mentioning that the highest efficiency values for Wells turbines (about 0.75) wererecently measured in a test rig with rotor diameters about 0.4 and rotational speeds up to 4000 rpm

[27]. Although experimental evidence is scarce, we may say that, at small scales (say e < 1=6, Rey-nolds number effects are likely to be more marked on Wells turbines than on impulse turbines.

5. Turbine rotational speed control

The turbine rotational speed should be controlled to match the energetic level of the sea state. It

should be noted that variations in rotational speed affect the turbine aerodynamic efficiency. Besides,such variations also affect the damping effect of the power take-off system on the OWC and so, indi-

rectly, affect the hydrodynamic efficiency of the wave energy absorption process. The instantaneous

rotational speed is controlled through the electromagnetic torque on the generator rotor. This is

implemented by the programmable logic controller (PLC) of the plant by acting on the power

electronics.

The following control algorithm for irregular waves

Ye aXb: 28

was proposed, based on numerical simulations of different OWC converters, air turbines and sea states

[28,30,31].Here,Ye is the electromagnetic torque on the electrical generator rotor, a is a proportion-

ality constant andbis an exponent. The optimized values a andbwere found to depend on device andturbine geometry, but not (or only weakly) on sea state. Optimized values of the exponent b were

found to range between 2 and 3.

Since the scale factors for Xand Ye are respectivelye1=2 d3=4 ande4 d3=2, we find easily

amaF

e8b=2 d63b=4: 29

The dimensionless exponent bis unchanged by the conversion.

If the bearing friction torque is ignored, the time averaged values of the pneumatic power XYtand

of the electromagnetic power XYeare equal. The instantaneous power difference XYt Ye is equal tothe rate of change of the kinetic energy stored in the coupled rotors of the turbine-generator set oper-

ating as a flywheel. We may write, from basic dynamics,

Yt Ye IdX

dt ; 30



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whereIis the rotational inertia of the turbine and generator rotor set. For dynamic similarity, we eas-

ily find, from the scale factors for torque, rotational speed and time,

ImIF

e5d1=4: 31

When, in OWC model testing, the turbine is simulated by a simple device like an orifice or a porous

plug, it is assumed that the device provides a reasonably good representation of the pressure-versus-

flow-rate curve of the real turbine. It is well known[14,27]that Wells turbines are approximately lin-

ear, i.e. we may write W ffi C1WellsU, whereCWells depends on the turbine geometry but not on its size,rotational speed or fluid density. It is also known that, for self-rectifying impulse turbines, of axial-

flow and radial-flow types, the relationship is approximately quadratic[14,28], i.e. W ffi C1impU2. A plugmade of porous material may exhibit a linear pressure-versus-flow-rate relationshippmffi v1plugwmprovided that it is designed in such a way that the flow through the porous material is laminar. The

relationship for an orifice is approximately quadratic (i.e.pmffi v1orifw2m if the Reynolds number isnot too small. For these reasons, in model testing in wave tank or wave flume, a porous plug or an ori-

fice are frequently used to simulate a Wells turbine or an impulse turbine, respectively.It should not be forgotten that, in most OWC plants, the rotational speed of the turbine is not con-

stant: it varies with sea state and also over the wave-to-wave time scale, depending on rotational iner-

tia and on the rotational speed control strategy and algorithm, as seen above. So it is important to

examine how variations in turbine rotational speed affect the pressure-versus-flow-rate relationship.

From the definitions ofUand W(see Eqs.(21)), we may write for impulse turbines

p

w2ffi

1

CimpqatD4; 32

which means that the relationship between pressure headp and flow rate w is not significantlyaffected by changes in rotational speed. The same is not true in the case of Wells turbines, for which

we find

p

wffi

X

CWellsD: 33

We may conclude that, while the damping provided by an orifice, such that vplug CimpqatD4;may sat-isfactorily simulate a real impulse turbine damping independently of variations in rotational speed, a

given porous plug can only simulate a Wells turbine at a fixed rotational speed. In the latter case, it

should bevplug CWellsDX1:

6. Numerical example

It may be interesting to investigate the magnitude of the error introduced if results from model

testing of an OWC device are converted into full-size prototype values when, as is frequently done,

the compressibility effect of the air in the chamber is not appropriately simulated. We illustrate this

with the real case of the bottom-standing OWC plant installed in 1999 on the island of Pico, Azores,

Portugal, and still operational. The plant was tested in wave tank in 1993, at scale 1:35, at the National

Civil Engineering Laboratory, Lisbon[18], and again, in 1994, at scale 1:25, at the University College

Cork, Ireland. In both cases, the air compressibility was simulated by connecting the air chamber of

the model to an air reservoir of appropriate volume.

The plant is shown in cross section inFig. 1.a, and is described in detail in[32]. The chamber cross-

section is square with 12m 12m, and the air volume, in the absence of waves and under mid-tidal

level conditions, is V0 1050m3

: The plant is equipped with a Wells turbine, with guide vanes, ofrotor diameterD 2:3 m:Model tests of the turbine indicated that it is approximately linear and char-acterized by U CWellsWor w CWellsDX1p, with CWells 0:680 .

We use the frequency domain analysis, with regular waves of radian frequency x and amplitudeAw; and assume the pressure on the inner free-surface as spatially uniform, rather than modelling

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the free-surface as a rigid piston. In linear theory, we may write (see[33])q qr qe, whereq is, asabove, the flow rate displaced by the inner free-surface motion, qris the radiation flow rate (induced

by the air pressure oscillations in the absence of incident waves) and qe is the excitation flow rate

(induced by the incident waves if the chamber air pressure were equal to the atmospheric pressure).

In the frequency domain, we write

fp; q; qr; qeg fp0; Q; Qr; Qeg eixt; 34

wherep0,Q,Qrand Qeare complex amplitudes. We also writeQr G iBp0, whereGandBare thefrequency-dependent radiation conductance and radiation susceptance, respectively, andjQej RAw;whereR is an excitation flow rate coefficient. Values of the hydrodynamic coefficientsG,B andR were

computed for the Pico OWC plant with the aid of a boundary-element-method code [34]and are plot-

ted versus frequency x in [30].We may find (see[26])

p0 CWellsD

qatX G i

xV0kpat

B

1Qe; 35

where, as above,k is the polytropic exponent. In our linearized analysis, the instantaneous power Pavaiavailable to the turbine is given by the pressure head ptimes the volume flow rate w=qat. This maybe written as Pavai CWells Dq1at X1p2. Taking into account Eq. (34), we find, for the time-averagevalue ofPavai,

Pavai CWellsD

2qatXCWellsD

qatX G

2 xV0

kpat B

2" #1R2A

2w: 36

Note that this is power available to the turbine, not turbine power output.

Fig. 7 shows curves of the dimensionless power P1;avai PavaiL7=2q1g3=2 versus dimensionlesswave frequency x

x L1=2g1=2. At full scale, the size of the chamber cross-section is LF

12 m

and the wave amplitude isAw;F 1 m. InFig. 7, the solid line represents the real prototype, with poly-tropic exponentk = 1.25 and rotational speed XF 80rad=sec (764 rpm). The chain line represents amodel at 1:4th scale, assumed to be tested in sea water, with a scaled Wells turbine (rotor diameter

Dm 0:575m, rotational speed Xm 160rad=sec, polytropic exponent k 1:25) and a chamber airvolume equal to V0;m V0;Fe3 1050 43 16:4 m3. Finally, the dotted line represents a model at1:35th scale, tested in fresh water, with a porous plug simulating a linear turbine (polytropic exponent

k 1:0) and a chamber air volume equal to V0;m V0;Fe3 1050 353 0:0245m3. Note that inboth models, the chamber volume was chosen as V0;m V0;Fe3, rather than as defined by Eq. (11)for appropriate representation of the air compressibility effect.

Fig. 7. Dimensionless time-averaged power P1;avai Pavai L7=2q1g3=2 available to the turbine, versus dimensionless wavefrequencyx x L1=2g1=2. Solid line: full-sized prototype. Chain line: model at 1:4th scale in sea water, equipped with modelturbine, and with air chamber scaled as the submerged structure. Dotted line: model at 1:35th scale in fresh water, with porous

plug simulator, and with air chamber scaled as the submerged structure.



16/17

The results plotted inFig. 7show that scaling the air chamber in the same way as the submerged

part of the structure (i.e. V0;m V0;Fe3may introduce significant errors into the results for absorbedpower. The errors are larger for the smaller scale 1:35th, but the representation is only marginally bet-

ter for the much larger scale 1:4th. It should be noted that the errors may change sign depending on

the wave frequency, which can be explained by the phase difference, due to the air compressibility

effect, between the air flow rate displaced by the OWC motion and the air pressure oscillation in

the chamber.

7. Conclusions

Relationships were derived for application in model testing of OWC converters in wave tank or in

sheltered sea waters. The new results concern the aero-thermodynamics of the air chamber and the

aerodynamics of the air turbine.

If the air chamber is to be geometrically scaled as is the submerged part of the device (i.e. the

chamber volume scale ratio is equal to the cube of length scale ratio e), exact dynamic similarity of

the spring-like effect of air compressibility in the chamber would require the atmospheric pressure(or room pressure) during the model testing to be much smaller that the atmospheric pressure at full

scale, which is unpractical or even impossible. For the more realistic case when the atmospheric pres-

sure is the same, then approximate dynamic similarity can be obtained provided that the air chamber

volume scale ratio is equal to km=kFe2d1, whered is water density ratio, and k (km in model andkF in

full-sized prototype) is a polytropic exponent that is equal to c 1:4 for a perfectly efficient air tur-bine, is in the range of about 1.2 to 1.25 for good air turbines, or is simply equal to unity if the lab-

oratory device used to simulate the turbine is a simple orifice or a plug made of a porous material. It

should be noted that in many published papers reporting OWC model testing these similarity rules

were simply ignored (the scale ratio for chamber volume was simply taken equal, or approximately

equal, toe3, the consequence being that substantial errors may have been introduced and overlooked

in the conversion of results from the tested model to the full-scale prototype. That such errors may besubstantial was confirmed by a numerical example. It was found that, if the chamber volume scale

ratio is taken equale3;approximate dynamical similarity can still the achieved, but this would requirethe real turbine to be simulated in laboratory by a mechanism exhibiting a phase difference between

air pressure head and flow rate.

If geometrically similar turbines are to be used at both scales (which in practice would require

model testing at a relatively large scale, possibly about about 1:4th), then the linear scale ratio for

the turbine should be the same as the length scale ratio e for the submerged structure of the waveenergy converter, and the rotational speed scale ratio should be e1=2 (with small corrections if thewater density is not the same).

Finally, scale ratios were derived for application in rotational speed control of the turbine-genera-

tor set.

Acknowledgements

This work was funded by the Portuguese Foundation for Science and Technology through IDMEC,

under LAETA Pest-OE/EME/LA0022 and contracts PTDC/EME-MFE/103524/2008 and PTDC/EME-MFE/

111763/2009, and by Project Offshore Test Station, KIC InnoEnergy, European Institute of Technology.

The authors want to thank the anonymous reviewers for their constructive comments.

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