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Ocean Engineering 33 (2006) 847-877www.elsevier.com/locate/oceaneng
Laboratory experiment on heaving body with
hydrauHc power take-off and latching control
T. Bjarte-Larsson, J. Falnes*
Department of Physics, Noiwegian University of Science and Technology, NTNU N-7491 Trondheim, Norway Received 4 April 2005; accepted 6 July 2005
Available online 14 October 2005
Abstract
A heaving axisymmetric fioating body is tested with sinusoidal incident waves in a wave channel. It is connected to the piston of a pump, and it may be latched by an electromagnetic mechanism. Experimental results are compared with a hnear mathematical model, for heave response, hydrodynamic parameters, absorbed wave power and converted hydraulic power. Heave resonance occurs at 1.1 Hz. For sub-resonant frequencies, latching control results in a significant increase in heave response and in absorbed, as well as converted, power. Hydraulic energy capture increases by a factor of 2.8 or 4.3 for frequency 0.75 or 0.5 Hz, respectively.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Enhanced power capture; Hydrodynamic parameter determination
1. Introduction
Among several possible methods to capture wave energy the use of heaving bodies has been proposed for this purpose. In order to convert the captured energy into a useful form of energy, hydrauHc machinery may be applied, where at least one cylinder-and-piston pump utilises the oscillatory vertical motion of the body (bodies) with respect to some force-resisting reference. One option is to use an anchor on the sea floor as such a reference (Ambli et al., 1977; Budal and Falnes, 1980; Hicks and Pleass, 1986; Nielsen, 1986). Another possibility is to use an immersed submerged body, such as e.g. a horizontal plate
* Corresponding author. Tel./fax: -t-47 735 97710. E-mail address: johannes.fahies@ntnu.no (J. Falnes).
0029-8018/$ - see front matter © 2005 Elsevier Ltd. A l l rights reserved. doi:10.1016/j.oceaneng.2005.07.007
and its associated added mass of water, to realise the necessary force reaction (Bergdahl et al., 1979; Hopfe and Grant, 1986).
In order to increase energy capture by a power buoy, a control method ('latching control') has been proposed, where the relative motion of the buoy with respect to the said reference is kept latched during certain time intervals of the oscillation cycle (Budal and Falnes, 1980; Falnes and Lillebekken, 2005). An object of the control is to achieve an oscillatory velocity that is in phase with the excitation force from the incident wave.
During the last two decades, many sea tests have been earned out on wave-energy converters (WECs) of the type with oscillating water columns (OWCs) and pneumatic power take-off (B0nke and Ambli, 1987; Ohno et a l , 1993; Koola et al., 1993; Heath et al., 2001; Falcao, 2001; Masuda et al., 1996). Because of the significant compressibility of air, the method of control by latching is not well suited for such a WEC of the OWC type. For this reason it has been proposed to install a floating body on top of the inner water surface of the OWC, in order to replace the pneumatic power take-off by a hydraulic one (Lillebekken et al., 2000). A reduced-scale model of a shore-based WEC of this type has recently been tested, somewhat extensively, i n a wave channel (Bjarte-Larsson et al., 2002). A purpose of the present work is to investigate a model of a similar WEC, but with a different geometrical shape. We have chosen an axisymmetric body, because such a shape is thought to be more suitable for future extension to a WEC for offshore deployment. The investigated model is tested in the frequency range of 0.5-1.5 Hz. For a model scale of
1:25 the coiTesponding frequency range is 0.1-0.3 Hz for a fuU-scale WEC.
2. Laboratory model and experimental set-up
As indicated in Fig. 1, the floating body is shaped as a vertical cylinder with a hemispherical bottom. Its diameter is 2r^= 140 nun and the total height is 0.25 m. This axisymmetric body has a coaxial bore of diameter 2ri = 42 mm, and it can slide up and down along a vertical fixed cylindrical strut of diameter 40 mm. The strut is made from aluminium, and the inner surface of the bore is plexiglas (perspex). The strut is erected on a, 40 mm thick, horizontal rectangular plate of length 330 nun and width 325 mm. The plate, and hence also the strut, is fixed to the structure of a wave channel of width d= 0.33 m, as indicated in Fig. 2. In order to keep viscous losses at a minimum, the two 325 mm wide plate edges are of semi-cylindrical shape (cf. Fig. 1). In both figures it is also indicated how the heaving body is connected to the piston of a pump. Also two water reservoirs and two check valves are indicated in Fig. 1 (but omitted i n Fig. 2, as well as in Fig. 3). During upstrokes of the body and piston some water is discharged into the upper reservoir. During down-strokes some water is drawn from the lower reservoir into the pump. Thus during each osciflation cycle, wave energy is converted to potential energy of upflfted water. To avoid taking into account the variation of the upper water reservoir's level, water is discharged at a fixed level just above the water surface (cf. Fig. 1). In the analysis we neglected variation in the water level in the lower reservoir, as the water surface (0.06 m^) here is five times larger than in the upper reservoir. During experiments the level difference A//between the two reservoirs was varied from 0.03 to 0.8 m. (It was found that a level difference of A//=(Aiï)min = 3 cm was sufficient for achieving proper
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 849
1
V
C 3
Fig. 1. Wave power buoy pumping water to a higher level. The axisymmetric buoy, which is connected to the piston of a pump, is vertically moveable along a coaxial strut, running thi-ough a bore in the buoy. The strut is fixed to a submerged, horizontal, rectangular plate. Water is drawn from the lower water reservoir through the lower check valve during a downstroke of the piston, and discharged through the upper check valve into the higher reservoir during an upstroke.
functioning of the two check valves.) The piston diameter is 16 mm, and the maximum stroke length is 129 mm. For lubrication purposes some non-foaming washing-machine soap was added to the water in this closed-loop hydraulic system.
The upper side of the submerged plate was 0.475 m above the bottom of the wave channel. The water depth was in the range of 0.73-0.74 m. (It varied because of continuous evaporation and occasional refilling of water.) The draft of the buoy was 0.16 m at equilibrium, corresponding to a mass of m= 1.92 kg for the displaced water, as well as for the buoy including the two bars, the traverse and the pump piston. Because of the limited stroke of the pump, the heave amplitude of the buoy could not exceed 65 mm. Thus the hemispherical part of the buoy was completely submerged, even at its uppermost heave position.
In the wave channel, the distance is 10 m from the wave-generating flap to the top of the wave-absorbing parabolic beach. This top is 742 mm above the bottom of the wave channel. The vertical symmetry axis (at the origin, x = 0 , of our chosen coordinate system) of the installed buoy (Fig. 3) is at a distance 5.6 m from the wavemaker. The six two-wire resistive wave sensors indicated in Fig. 3 are placed in the middle of the wave channel at coordinates (from left to right) -3.022, -2.214, -1.847, 1.377, 2.179 and 2.557 m (with a measuring uncertainty of approximately ± 2 mm). Moreover, in order to check the phase
Fig. 2. Front view of the axisymmetric power buoy model installed in a 0.33 m wide wave channel. A vertical coaxial strut is connected to a submerged plate and to an upper plate, which is fixed to the top of the frame structure of the wave channel. On this latter plate, which is the lower one of two fixed platfoms, two operable electromagnets are mounted. When they are activated, two vertical bars running thi'ough holes in the platform are latched. At the same time, also the buoy is latched, because the bars are connected to the buoy at then lower ends, as well as to one traverse at then top ends. The upper platform caiTies the cylinder part of the pump. The piston rod, which runs through a hole in this platform, has its upper end connected to the piston and its lower end to the said traverse. Thus, the buoy, the two bars, the traverse and the piston oscillate together.
of the wave at x = 0 , a pressure transducer was used to measure the, pressure inside the upper closed end of a vertical thin (4 mm outer diameter) glass tube that was placed abreast the buoy with its open lower end 0.10 m below the still-water level. The signal from this transducer was used as input to the algorithm for determining best instants for unlatching. The heave oscillation was measured by an inductive position transducer and also by an accelerometer, both of which produced input signals to the algorithm used in
r. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 851
—7 ^
1
1
Y( 1
© ® © ®
Fig. 3. Side view of the power buoy (3) in a narrow wave channel with wavemaker (1) and absorbing beach (4). The water-surface elevation is measured by two sets (2) of three resistive two-wire probes, one set in front of and one set behind the power buoy.
order to determine proper instants of latching. Another transducer was used to measure the pressure i n the pump fluid above the piston. Finally, a two-wire resistive sensor recorded the amount of water discharged into the upper reservoir, which had a water-surface area of 0.0117 m^, about five times less than the corresponding area in the lower reservoir. The produced potential energy was calculated as the weight of uplifted water multiplied by the vertical difference between the top of the outlet tube in the upper part of the higher reservoir (Fig. 1) and the average water level i n the lower reservoir.
In the present experiments only sinusoidal incident waves were used, with frequencies ƒ in the range of 0.5-1.5 Hz. Usually, for each experimental run, we made records during 64 s and analysed the interval from 30 to 62 s. With a sampling frequency of 32 Hz, we recorded 2048 samples, and analysed 1024 of them. I f we then chose wave frequencies that were a multiple of (1/32) Hz, we could apply fast Fourier transform.
Disregarding near-field wave components near the power buoy, near the wavemaker and near the beach, the complex amplitude of the wave elevation in the wave channel may be written as
77_ = Ae"'^- + Bê^ and 77+ = Ce"^^' + öe^^^' (1)
for the regions x < 0 and x > 0 , respectively. The angular frequency (jj = 2v:f and the angular repetency (wave number) k are related by, (x)^ = gk tanh(^/z), where h is the water depth and g the acceleration of gravity. The complex quantities A, B and C represent the incident wave, the reflected wave and the transmitted wave, respectively. The two latter waves are resulting from diffraction, as well as radiation, from the buoy. The quantity D, which represents reflection from the beach, is negligible for a good wave absorber. Observe that i f the wave-channel width, J = 0 . 3 3 m, is less than one wavelength A = 2TU/A:,
then the wave outside the near-field region—as i n Eq. (1)—may be considered as a two-dimensional problem (Srokosz, 1980; Falnes, 1984). In our case this requirement corresponds to a frequency less than 2.1 Hz, a condition that is amply satisfied; the wavelengths used in the experiments exceed even the double channel width.
The following procedure was used to obtain experimental values for A, B, C and D. In order to apply EFT analysis, a time interval coiTesponding to an integral number of periods was chosen from each time series obtained from the two-wire resistive wave sensors on
either side of the buoy in the wave channel (Fig. 3). Fourier analysis showed that the waves were essentially sinusoidal, as intended. The second and third harmonic amplitudes were usually found to be approximately 1/20 relative to the fundamental amplitude. Higher hai-monics were even more negligible. Knowing the fundamental frequency component ^_ at two different locations JCi and X2, say (xi, X2<0), then A and B are obtained by application of the first of equations (1). Since we have even a third wave sensor in the region x < 0, we have three possibilities to find the values for the complex quantities A and
B. We utilise this redundancy to calculate a weighted average for the quantities.
Analogously, using the time series from the thi'ee wave sensors in the region x > 0, we find a weighted average for the complex quantities C and D. Experimental values of the (time-averaged) wave power absorbed by the model may then be obtained from the formula
p PlRm^lA\^-lB\'-\Cf + \D\\ (2)
where p is the mass density of water, and Dikh) = tar\h(kh) + kh-kh tanh\kh) is a depth function that equals 1 for deep water—see e.g. pages 73 and 77 in the textbook of
Falnes(2002).
Defining r = D / C as the complex reflection factor of the beach, we found that \r\ was i n the range of 0.01-0.1, depending on the amplitude and frequency of the wave and on the actual water depth h. This means that less than 1% of the wave energy incident on the beach is being reflected. Thus, the last term | D p i n Eq. (2) is negligible compared with the thi-ee other terms. We do not, however, neglect D i n all of the subsequent analysis. On the contrary, we may consider D to represent an additional incident wave coming from the opposite direction. Then the complex amplitude of the excitation force may be written as
=f^(A + D), where /e is the excitation-force coefficient for the heave mode of this
symmetric buoy.
3o Mathematical modelling
Assuming that linear theory is applicable, we may write
B = rA+a~ü + TD and C = TA + a^ü + TD, (3)
where the dimensionless complex reflection and transmission coefficients T and T, respectively, are due to waves diffracted from the power buoy. When the buoy osciUates, it also radiates waves, which are represented by the above two terms d^u and d^ü, where ü is the complex velocity amplitude of the sinusoidally heaving body. The complex quantities a7 and at are the far-field coefficients for waves radiated in opposite directions. Because of the syrometry of the heaving buoy, we have in this case = = a^, say. [Observe that for a real non-symmetric case, the two terms TA and TD in Eq. (3) should be replaced by
T^A and T^D, respectively. In general, T" and T^ may have different phases, but
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 853
The principle of conservation of energy implies that
\T\' + \r\' = l - \ a j \ ' (4)
where kvoP is the fraction of wave energy lost by viscous effects when u = 0 . For irrotational flow in ,an ideal .fluid, Newman (1976) derived some general reciprocity relations, which for our case of a symmetric body may be written as follows. Between the above two diffraction coefficients we have the two relations
l r p + | r| 2 = i and r r + r r = o, (5) where the asterisk denotes complex conjugate. Further, between diffraction and radiation
coefficients we have the reciprocity relations
(T + r)a: = a, and = ^
-^r-and also the following relation that expresses the radiation resistance R, in terms oia, or/e,
_ p_lDmd
2 ^ ^ ^. ,2_
(7)
The latter of the two relations (6) is the weU-known Hasldnd relation. These two-dimensional relations (3)-(7) are applicable even for our three-two-dimensional geometry, provided d<X = 2n/k (Srokosz, 1980; Falnes, 1984), a condition that is fulfilled for the present experimental investigations. The optimum heave^ oscillation amplitude, at which the absorbed wave energy attains maximum, is Wopt = Fe/(2i?r)—see e.g. Eq. (3.46) or (6.9) in the book of Falnes (2002). Introducing the complex heave-amphtude-response ratio C = = w/(iwA), we may write
j f l _ pg'D(kh)d (8)
where Eq. (7) was used in the last step.
From the two relations (5) it foflows tiiat
l r ± r |
= 1, which means that Tand T may be represented in the complex plane as two sides of a right-angled triangle where the hypotenuse equals 1, and we may write {T+ F) = e'^ say, where Ó = Z ( r + F) is the phase angle of (T+T). Hence, the phase angles of T and T differ by TZ/2. Denoting the phase angle of the excitation-force coefiicient as Z/e = /c, we may w r i t e / e = l/ek". Then, according to relations (6) we havee . ^ = ( ^ + ^ ) = ^ = ^ = e^^^ (9)
Taking the square root yields two possible phase angles K = 5/2 or /c = 5/2 ± . From the Haskind relation it follows that the phase angle of a, is Aa,=K±Tz. For our symmetric buoy, the Froude-Kriloff force coefficient is real and positive, while the diffraction force coefiicient (in the long-wavelength approximation) has a small positive imaginary part. Hence, we would expect to find a positive but small angle K. In other words, we would expect to find an approximately real and positive excitation-force coefiicient/e to be m
the first quadrant of the complex plane, and hence a, in the third quadrant with a phase angle Zar=/c-7T. In the point-absorber approximation T~l and r==0. Hence, for our buoy we would expect 7 to be close to the positive real axis with a magnitude (module) slightly less than 1, and T to be close to the positive imaginary axis, but much smaller in magnitude.
Experimentally, we have observed that the heave oscillation of the buoy is rather closely sinusoidal, in spite of non-hnear elements in the power take-off. The reaction force of the pump piston is far from sinusoidal; this force is, however, small in comparison with other dynamical force components acting on the buoy. This observation may be considered as a justification for assuming that the system may be approximately modelled by means of the linear dynamic equation
-w^(m + m, + mp)s + icj(R^ + Rf + Rp)s + Ss = F^, (10) where s=u/(ico) is the complex amplitude of the buoy's heave excursion s(t) {u is the
complex amplitude of the heave velocity u = s). Further, S= 137.6 N/m is the hydrostatic stiffness corresponding to the water-plane area 0.0140 m^ of the buoy, m = 1.9 kg is the mass of the buoy and i7i^^m,((o) is its hydrodynamic added mass. A n additional dynamic mass mp = 0.65 kg caused by the water in the pump cylinder and in the hoses, has to be included, except for cases where the pump is disconnected from the buoy. Moreover, R,=
Rr(M) and Rf are the radiation and loss resistances, respectively. We introduce the intrinsic
mechanical resistance
Ri ^ R,+Rf (12)
and the intrinsic mechanical reactance
= oom + a)m,((o) —S/co Q2)
In our Hnear model, R^ in Eq. (10) represents the effective load resistance. Thus, the term
iojRpS is the complex amplitude of the force that acts on the pump piston.
From Eq. (10) it follows that the heave velocity has a complex ampHtude
u = icos = F. e
Rp + Ri + i(X; + cjmj (13)
Resonance occurs for a frequency/o = wo/27r at which the total reactance vanishes, Xtot =
(omtot~S/bi=Xi + (ijmp = 0. Thus, the resonance condition is
^ 2 ^ J _ ^ S .
° mtot m + m,(a)o)+mp ^^^4) A t this frequency the heave velocity ü is in phase with the heave excitation force F^.
In this linear model, the useful hydraulic power may—as shown in more detail by Bjarte-Larsson et al. (2002)—be expressed as
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-, 855
where
-(Y / I F l^21,0|2 . ( 1 6 )
P,=\RtW and P,=\r,\C^\^ ^^^^
Here P is power associated with wave radiation, and P, is power loss (resulting from dissipative effects like friction) associated with the motion of the buoy. The power absorbed from the waves is P a = ^ e - ' P r - ^ ^ ,
The excitation force is = / e ( A + D) = / e A c p . We shall consider the complex-valued CD = ( 1 + Ö /A) as a coiTection factor. Because it varied during the time our experiments were performed, we prefer to set Fe = / e A in the theoretical derivations in the following paragraphs However, when we compare experimental results with this theoretical model we shall coii-ect measured values of Ü and Pp by multiplication with a factor of l/c^ and
1 / | C D I ' , respectively. Observe that | C D P = 1 + 2 R e ( D / A ) + | D / A p has a term that is of first order in DIA unless this ratio happens to be purely imaginary.
It is convenient to normafise the power relative to the incident wave power
We shall adopt this last approximation, because the error is of second order i n | D / A | ,
corresponding to a relative eiTor less than 1 % . Now, in terms of the heave-response-amphtude ratio iCl = \slA\ = \Ü/(coA)\, we write the relative absorbed power as
^ = a\K\\/l-bK\^-c\K\^ + K ^^^^ Pi
and the relative useful hydrauhc power as
Pi where a = colFeA|/(2Pi) = 2aj%\/[pg^Dmd] ^^^^ b = (wX,ot/ifel)' = [(w'm,ot - 5 ) / i f e l ] ' (22) c = w ' P j A p / ( 2 P i ) = 2io'R,/ [pg^Dmd] (^^^ . = CO%\aW2P0 (24)
and where Av = A f = A h = 0 according to the above ideahsed theory. However, i n order to improve fitting of theoretical curves to experimental results, we may find it convement to introduce A „ Af and Ah as ad hoc empüical positive correction terms. The parameter A^ represents viscous loss of wave power for the situation when the buoy is not osciUatmg
{Ü = 0). A possible choice is to set Ay equal to an experimentally determined value of the
quantity la^oP appearing in Eq. (4). The parameter Af represents additional energy loss by non-linear friction not taken care of by the friction resistance Rf. It includes also effects of friction between the cylinder and piston of the pump. Finally, Ah represents hydraulic losses including hysteresis losses in the rectifying check valves and in the hoses.
When (for w < WQ) latching control is applied to achieve optimum oscillation phase
(Ü in phase with F^), Eqs. (19) and (20) are still applicable, provided we set b = 0, and
provided and hence also ü and s, are interpreted as the con'esponding first harmonic components of the periodic heave oscillation (Budal and Falnes, 1980). A l l experimental results reported below are obtained with sinusoidal input to the wave generator. However, measured waves and, in particular, heave oscillation (even when latching control is not apphed) show slight deviation from sinusoidal performance. For this reason, we Fourier-analyse measured signals when we determine experimental amplitudes. Thus, the quantities Ü, s, A and used above are, from an experimental point of view, to be understood as first harmonics.
4. Heave response and resonance
In this section we present experimental results, firstly from hnarity tests, and secondly from measurements of the frequency response of the heaving buoy. When the pump is disconnected from the buoy, we would, from previous experience (Budal et al., 1979; Falnes and Lillebekken, 2005), expect to find reasonable linearity when the relative motion between the buoy and the suiTounding water has amplitude that is less than the minimum radius of curvature of the wet body surface. To check this linearity, we performed measurements as presented in Fig. 4 for frequency ƒ = 1.25 Hz. Most of the plotted points for \s/A\, marked as crosses ( X ) , appear to he close to a fine, thi-ough the origin, having a steepness of \s/A\ « 3.4. However, since two-wire sensors may produce rather inaccurate wave measurements for low waves, the first three to four plotted points deviate substantially from this inclined fine. (Concerning the first plotted point, one might suspect also that the relatively very low value 1^1 = 3.4 mm for |A| = 2.0 mm, is influenced by static friction between the buoy and the stnrt.) It seems to be more difficult to achieve accurate measurements for the wave phase than for the wave ampHtude. Excluding the measurements for |A| < 8 mm, the remaining plotted cross points ( X ) for Z (s/A) lie in the range of - 8 5 to - 7 5 ° at frequency ƒ = 1.25 Hz.
The diagrams in Fig. 5 show results from linearity tests when the pump is installed. I t appears that the check valves, which are non-linear components, do not introduce serious non-Hneaiity into the response. For not too small wave amplitudes, the plots for | f | seem to approximately fit a fine with \s/A\ steepness 1.1, 1.5, 2.0 and 0.5 for frequencies 0.5, 1, 1.125 and 1.375 Hz, respectively. The coiTesponding phase angles A (is/A) are roughly - 2 0 , - 6 0 , - 1 0 0 a n d - 1 3 5 ° , respectively.
At resonance the oscillation phase is optimum (u/F^ is real), and a maximum fraction of absorbed wave energy is to be expected. The method of latching to achieve optimum phase is practically applicable for frequencies below resonance. In order to find resonance, we measured the frequency response for the floating body with the pump piston disconnected.
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 857 f=1.25 Hz 50 40 -£ 30 -30 20 10 -0 - 5 0 r 6 0 7 0 N 8 0
-A
8 10 12 14 16Fig 4 Linearity tests when the piston pump is disconnected from the model. Amplitude 1^| (upper diaram) and phase Z.S (lower diagram) of heave position versus incident wave amplitude |A| for frequency/-1.25 Hz.
Results shown i n the left-hand part of Fig. 6 indicate a maximum response \Ü/A\ = 27 s'^ at a frequency near 1.25 Hz. Analysing five runs with |A| > 9 mm at this frequency gives average and standard deviation as follows, |M/A1 = (26.7 ± 0 . 7 ) s ^ and Za^/A)= 10 7 + 4 7° Observe that possible systematic errors and calibration errors have not been taken into account here. (Such errors may be most significant with the wave measurements.) Runs taken at various frequencies between 1.242 and 1.258 Hz mdicate that resonance occurs at frequency ƒ=/oi = o.oi/27r = (1.252 ± 0 . 0 0 3 ) Hz when the pump is disconnected. Thus, we may conclude that at this frequency the phase angle between
^f^j^ and A is /< = icoi= Z/e(woi)== 10°. Moreover, since the pump is disconnected, and
hence Rp^O, we may write Eq. (13) as
U . S
A " ' " ^ A " P i + i X i
(25)
where the intrinsic reactance Zi = wm ± W » T , ( W ) vanishes at resonance. Hence, the added
mass at w = W Q I is
mAcooi) = S/OJI -m = (2.22-1.92) kg = 0.30 kg (26)
Moreover, it fofiows from Eq. (25) that /e(fa^Oi)l ^ Pi(Woi) (x>S T max = 27 s -1 (27) max
Since we know that \f.\<S-137 N/m, Eq. (27) provides an upper bound to the intrinsic resistance, Pi(a)oi) <5/(27 s - ^ ) < 5 . 1 N s/m. In accordance with the left-hand part ot
f=0.5 Hz f=^ Hz f=1.125 Hz f=1.375 Hz
Fig. 5. Linearity tests when the piston pump is installed. Amplitude \s\ (upper diagram) and phase As (lower diagram) of heave position versus incident wave amplitude \A\ (in mm) for frequencies ƒ =0.5, 1.00, 1.125 and 1.375 Hz. The water-reservoir-level difference was neghgible (3 cm) during these linearity tests.
Fig. 6, \ü/A\> ( l / V 2 ) | M M U a x in a frequency interval W / / 2 T C= / i = Wu/27r, the so-called resonance bandwidth, w h e r e / i « 1.19 Hz a n d 1 . 3 1 Hz. Let us now, as a reasonable approximation within this narrow frequency interval, neglect frequency variation ofR, (and hence of Rd, of m, and o f f ^ . T h e n X - . ^ - R ^ at frequency/i a n d X ^ ^ R ^ at frequency ƒ„. Accordingly, Z (Ü/A) = + 7u/4 = /CQ, + 45° at f , and Z (Ü/A) ^ KQ^
-n/4 = KQi - 4 5 ° at ƒ„. From equation (25) it now follows that
and
^i(woi) 27r(/u - / i ) ( m + m,{iDQ{)) 1.7 N s/m
l/e(woi)l = IwMljnaxPi ^ (27 s ^)i?i = 45 N/m.
(28)
(29) When the pump is installed on the model, the frequency response, as described above, w i l l be modified, not only because of additional friction, viscous losses and intended load associated with conversion of energy into useful potential hydraulic energy, but also because the water in the hoses represents substantial additional inertia. The additional mass mp that we introduced in the dynamic Eq. (10), depends on geometrical parameters, such as length and cross section in various parts of the pump, including cylinder, valves and water hoses. We define as a coefficient such that {llA)mp\ü\^ equals the time average (during one oscillation period) of the kinetic energy associated with the pump
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 859
w i t h o u t p u m p with p u m p
Fig 6 Resonant response when the piston pump is discomiected from the model (left) and when it is mstalled on the model (right). Plotted points show ampHtude (upper diagram) and phase Z Ü (lower diagram) of the heave velocity relative to the incident wave ampUtude |A|, versus frequency/. Experimental pomts marked as dots (• are taken from the linearity tests shown in Fig. 4 (without pump) and Fig. 5 (with pump). Other expenmental points, marked as crosses ( X ) , represent measurements taken at various selected frequencies, but without performed linearity tests.
(observing that parts of the hoses beyond the valves have zero flow during half of the cycle) In this way we have estimated to be 0.65 kg. Accordingly (if we adopt the value of added mass m,=0.30 kg found at frequency/oi= 1.252), we should expect resonance at the frequency
/o = ^(m + m,)/(m+m,+mp) - ^2.22/(2.22 + 0.65) 1.252 Hz
= 1.10 Hz. (30) As seen in Fig. 5 and in the right-hand part of Fig. 6, we measured the heave response
for frequencies 0.5, 1,1.125 and 1.375 Hz. We found the largest value of \u/A\ « 13 s at frequency 1.125 Hz, with Z (Ü/A) in the range of - 1 0 to - 1 5 ° . A t resonance, however, we should expect to find Z(^^/A)|ƒ=ƒ„ = /CQ = Z/e(wo) in the range of 0-10°. Accordingly, we might state that resonance occurs for a frequency shghtly lower than 1.125 Hz—m agreement with the numerical result of Eq. (30).
From the right-hand part of Fig. 6 it seems that the resonance bandwidth is roughly 0 25 Hz, when the water level difference is at its least possible value ( A / 0 m i n = 3 cm, conesponding to a pump resistance Rp=Rpo, say. Proceeding i n analogy with Eqs. (29) and (28) we amve at the following coarse estimate, Ri{a)o)+Rpo~2'K0.25 Hz X (2.22 + 0.65)kg = 4 . 5 N s / m and l / e ( w o ) l - 1 3 s-^X4.5 N s/m=59 N/m. It seems that we may conclude that [/el is in the range of 55-65 N / m at the resonance frequency/o ~ 1.1 Hz.
Moreover, assuming—cf. Eq. (28)—that is in the range of 1.5-2 N s/m, we have found the minimum pump resistance Rpo to be i n the range of 2-3 N s/m when the level difference between the two water reservoirs is negligible ( A i ï = 3 c m ) . I t might be interesting to compare this experimental result with the pump resistance for an ideal pump with piston area A Ipiston,
„ , 2p^ALst„„Ai/ A / f AH
^plideai= 77 = (1-26 N / m ) ^ = (1.26 N / m ) ^ , (31)
T^m \u\ (jj\s\
which is obtained by equating the potential energy 2pg\s\A\pi,i^j,AH, stored during one oscillation cycle, with (Tz/cj)Rp\u\^—cf. Eq. (15). For a typical value |^| =^ 30 mm (Fig. 5), with ƒ = 1.125 Hz and with AH= 30 nun, we find an ideal pump resistance of 0.2 N s/m, which is much less than the experimentally found i?p Q. I f (with the same values for co and l^l ~ 30 nun) AH is increased to 0.4 m, then PpLeai is increased to 2.4 N s/m.
The pump has two check valves, and it works against essentially constant pressure, which is, however, different for the up and down strokes. This may lead to a slight offset of the mean heave position from the equilibrium position of the buoy. This possible slight offset was not studied quantitatively. In spite of the non-linear functioning of the pump, the response is essentially hnear (as indicated in e.g. Fig. 5). Moreover, in agreement with this observation, the motion of the fioating body and the pump piston is approximately sinusoidal. There is no strong generation of higher harmonics in the heave oscillation of the buoy. Fourier analysis of measurements of waves and heave motion showed presence of small second and third harmonics (typically less than 6% compared to fundamental amplitude), but more negligible higher harmonics. For instance, with a water level difference AH=OAm and a first harmonic (fundamental) frequency 0.75 Hz, then we found first, second and third harmonics for wave amphtude to be 35, 2, and 1.2 mm and for heave ampfitude to be 18, 0.8 and 0.7 mm, respectively. Thus, this supports our assumption that the osciUation system is essentially linear. Even at the near-resonance frequency/o = 1.125 Hz, where the non-linear pump resistance—cf. Eq. (31)—is expected to be of greater relative dynamical significance (as the mechanical reactance Xi + ojiUp is rather small), we find very littie higher harmonics in the heave oscillation. A reason for this is, of course, the resonance (maximum heave response) at the first harmonic frequency, fo. When the heaving body is used for converting wave energy into useful potential energy represented by water pumped into an elevated reservoir, we experience that the wave ampHtude has to exceed a certain minimum value before heaving oscillation can occur, and that this minimum wave amplitude is larger for larger values of the water level difference AH. This fact is clearly seen in the response resuHs presented in the diag rams of Fig. 7. For wave frequency 0.75 Hz, the limiting minimum wave amplitude for onset of heave motion is increasing from 4 to 13 mm, approximately, when AH increases from 0.2 to 0.8 m. Clearly, the heave-response-ampHtude ratio |Cl = \s/A\ decreases with increasing Ai?, as was to be expected, but it also increases with increasing wave amplitude | A | . Similar observations are made at other frequencies too.
Observe that in Figs. 4-7 each experimental value of the independent variable (complex Ü or s) has, in accordance with previous remarks (in Section 3), been reduced to the ideal case Z) = 0 by dividing the measured variable by C D = 1 + Ö / A .
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 861 A H = 0 . 2 m A H = 0 . 4 m E E 4 0 2 0 \A\ [ m m ] 2 0 \A\ [ m m ]
Fig. 7. Heave-amplitude response at frequency ƒ =0.75 Hz when the piston pump is installed. Amplitude | j | of heave position versus incident wave ampUtude |A| for various values (0.2, 0.4, 0.6 and 0.8 m) of the water-reservoir level difference A/ï.
A l l experimental results presented so far, apply for the case without latching control. Latching control introduces more significant higher harmonics in the heave oscillation, not as a result of inherent non-linearity in the system, but because the system is no longer time invariant. Higher harmonics are created by periodic latching and unlatching operation, rather than by non-linearity. When the frequency of the incident wave is significantly smaller than the natural frequency (approximately 1.1 Hz), a larger heave response is attainable with than without latching control. This is demonstrated in Fig. 8 for a wave frequency 0.5 Hz, when the water-level difference is AH =0.20 m. In this particular case, latching control results in an increase, by a factor of approximately five, in the first-harmonic heave amplitude.
5. Hydrodynamic parameters from wave measurements
It is difficult to make accurate measurement of water waves. Nevertheless, we have attempted to utilise wave measurements in order to determine experimentally the hydrodynamic parameters entering into the above Eqs. (4)-(9).
Consider first diffraction experiments where the buoy is kept latched in equilibrium position. We may then apply Eq. (3) with ü = 0. Excluding waves of incident amplitude |A| less than 5 nun (to avoid the most inaccurate wave measurements) and larger than 30 mm (to avoid onset of somewhat non-linear behaviour), we found experimental values for the refiection coefficient F and the transmission coefficient T as shown in Table 1. The table entries give mean and standard deviation for results obtained from several
40
-40
1 1—
\
/Fig 8. Wave (thinner curve) and buoy excursion (thicker curve) versus time, with and without latcW^S control^ The incident wave has frequency ƒ =0.5 Hz and ampUtude \A\ = 12 mm for the case with latching control. A 3 s I n g s e ^ f o n o i ; ^ ^
an equally long section of another run without latching control, where the incident wave had a ^l^S^tly srn^^^ m p L d e |A I i 11 mm. The shown wave curves are signals from the transducer that measures the hydrodynami T e ! ure abre st the buoy (pressure given in mm water head). The 'ringing' (damped short-period osciUation )
Z::Z:TTZ.oZl
c u r v e s l a resuh of vibrations generated by shocks caused by the electromagnets when they operate in order to latch or unlatch the buoy.experimental i-uns. The number of runs with different wave amplitudes ^as 4 for frequency 0.5 Hz, 2 for 0.75 Hz, 4 for 1 Hz and 2 for 1.0938 Hz. The standard deviation does not reflect systematic errors and possible inaccurate calibrations m the measurements. But it gives an indication of how wefl the wave-response hnearity is within the wave-amplitude range 5 m m < l A | < 3 0 m m , and of experimental reproduci-bflity among the present rather few experimental inins. It seems that there is more uncertainty in the phase angles than in the moduli (absolute values) of complex parameters. The last line in the table shows that | r p + | r p < l . Thus, as was to be expected, Eq. (4) matches reality better than the ideal-fluid equation (5) At frequencies 0 5 and I H z , viscous effects result in energy loss of approximately 3 and 11% respectively, relative to the incident wave energy. Moreover, we see from Table 1 that i r + n is somewhat less than 1, and that the angle ( Z r - Z D is somewhat smal er than 90° We observe that T does not differ much from the ideal point-absorber value 1 but it has' a relatively smafl negative imaginary part, and iTl is shghtly smaller than 1.
Table 1
Hydrodynamic diffraction parameters for four different wave frequencies ƒ / ( H z ) \r\ z r irl z r l r + r | z ( T + r ) 0.5 0.75 0.049 + 0.002 84.5 + 2.0° 0.984 + 0.003 - 1 . 2 + 0.3° 0.989 + 0.002 1.6 + 0.3° 0.971+0.006 0.087 + 0.006 81.5 + 2.7° 0.963 + 0.000 - 5 . 1 + 0 . 8 ° 0.972+0.006 - 0 . 1 + 1.2° 0.935 + 0.000 0.150 + 0.007 80.0+1.7° 0.934 + 0.010 - 9 . 1 + 3 . 0 ° 0.948 + 0.009 -0.006 + 2.6° 0.895+0.023 1.0938 0.182 + 0.014 69.4 + 7.6° 0.919 + 0.014 - 7 . 8 + 7.3° 0.974+0.025 2.6 + 5.6° 0.879 + 0.030 Mean values and standard deviations for \r\, ZT, |r|, AT, \T+r\, Z ( r + r ) and | r P + | r P , as obtained from wave measurements when H = 0.
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 863 Table 2
Hydrodynamic parameters F, T and a, for four different wave frequencies/ obtained by least-square fit to the two Eq. (3)—imposing a7 = fl+ = a,, and—using results from experimental inns with u^O
/ ( H z ) 0.5 0.75 1 1.0938 llril 0.045 0.301 0.343 0.294 z r 126.3° 166.9° 62.4° 104.2° iri 0.958 0.668 1.086 0.774 z r - 3 . 8 ° - 7 . 3 ° 3.9° 6.0° l«rl ( S ) 0.0080 0.0478 0.0380 0.0085 I/el (N/m) 95.2 367 197 39.7 Zar+TC= Z / e 61.0° 122.5° 13.8° 75.6°
Entry/e is obtained from a^ using the Haskind relation (6).
Furthermore, while F turns out to have a relatively small positive real part, i t has, as was to be expected, a dominating positive imaginary part. As was also to be expected from point-aborber considerations, | r | | r | , in particular for the lower frequencies.
We have made further attempts to determine hydrodynamic parameters from wave measurements in experiments where the buoy oscillated, ü 0. For a particular wave frequency/, we may then use a set of measured values of A, 5, C, D and ü to obtain a single solution for the three complex parameters T, T and ci, by applying the method of least-square fitting to the two Eq. (3)—imposing aj=at = a,. When using measurements from experimental runs for each of the frequencies 0.5, 0.75,1 and 1.0938 Hz, we obtain values of F, T and as presented in Table 2. (It may be noted that for the frequencies 0.5 and 1 Hz the results presented in Table 2 and in Fig. 5, are based on the same experimental runs.) We observe that results for F and T seem to be less consistent in this table than i n Table 1, where (in contrast to Table 2) these diffraction parameters, as is reasonable, appear to vary monotonically with frequency (in modulus as well as in phase). Also, some values given in Table 2 are definitely too large to be physically acceptable, namely | r | exceeding 1 at frequency 1 Hz and %\ exceeding 5"= 137 N/m at frequencies 0.75 and 1 Hz. Moreover, the values for \f^\ and Z / e derived in Table 2, show an unacceptable non-monotonic behaviour. One should expect [/èl to decrease and Z / e to increase monotonically with increasing frequency. We conclude that we cannot rely on the results given in Table 2. Instead we wish to attempt a different approach of squeezing out results from these experimental runs, and thus obtain some more consistent results for the hydrodynamic parameters. We try to impose some restrictions on the fitting in order to avoid the strong non-monotonic behaviour.
We next use the same experimental results when we impose the condition that the values of F and F are as obtained from the diffraction experiments (with w = 0) and shown in Table 1. Then we use Eq. (3), to find a value for «7 and for af for each experimental m n (with Ü 0). Using the Haskind relation (5), excitation-force coefficients/e corresponding to a^ = a~ and to a^ = af may then be obtained, e.g. as plotted, as crosses ( X ) and circles ( X ) , respectively, in Fig. 9 for frequencies 0.5 and 1 Hz. It may be noted that a significant deviation is found between the results, corresponding to the significant obtained deviation of a'^/af from 1. As we from symmetry consideration should expect a~/af = l, the deviation must be due to experimental inaccuracy. For f r e q u e n c y / = 0.50 Hz, where
H z f=1 H z
300
300
Fig 9 Heave-excitation-force coeiïicient as determined from experiments with diiïerent values of heave velocity amphtude |M| (given in mm/s) by application of the Haskind relation (6), where radiation far-field coefiicients a, (aJ and at) have been obtamed from Eq. (3) with coefiicients F and T as given m Table 1 (determined from diffraction experiments). Resuhs f o r b a s e d on «7 and af are plotted as crosses ( X ) and cncles
(O) respectively. The upper and lower diagrams show experimental values for the modulus %\ (m N/m) and the
phase angle A(f.) (in degi-ees), respectively, for two frequencies ƒ = 0.50 Hz (left) and/=1.00Hz (nght). For frequency ƒ = 0.50 Hz the points plotted as bullets ( • ) and pluses ( + ) show how the points plotted as cncles (O) and crosses ( X ) , respectively, would be displaced i f Z r were changed from - 1 . 2 ° (as in Table 1) to 3 . 1 .
the deviation is strongest, we have tried to change only Z T f r o m - 1 . 2 ° (as in Table 1) to - 3 . 1 ° , while letting | r | and T remain as before. From a new calculation we then obtam results'plotted as bullets ( • ) and pluses ( + ) in Fig. 9. It appears that, as a result of this supposed modest change in AT, aj/aj comes much closer to 1.
This exercise shows that e.g. the experimental value of [fel may be extremely sensitive to the accuracy of measurements. For this reason, we wish to examine how a procedure, as explained in the following, may be applied to the same measurements for obtaining results as presented in Table 3 and in Fig. 10. For each frequency/, we at first, choose a reasonable approximation (e.g. long-wavelength approximation—see below) f o r / e (that is for [fel as weU as for K= Z / e ) . From this choice a con-esponding approximate value for a, follows directly from the Hasldnd relation—the last of Eq. (6). Secondly, inserting this value
a, = a- = at into the two equations (3), we use the set of measurements of A, B, C, D and Ü
to obtain values of T and of T that give a least-square fit to Eq. (3). Finally, inserting these found 'global' values of T and Tinto the two equations (3), we solve these equations with respect to a' and at for each member of the set (that is, for each experimental run). Con-esponding values of/e, obtained from the Haskind relation, are plotted in Fig. 10, for frequencies 0.5 and 1 Hz, with the heave velocity amplitude \ü\ as independent vanable. Values f o r / e obtained from the new values of and at are plotted as crosses ( X ) and circles (°), respectively.
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 865 Table 3
Hydrodynamic parameters for four different wave frequencies ƒ. Based on a reasonable choice, for each frequency, of a theoretical estimate of/^ (non-dimensionalised with respect to hydrostatic stiffness S)—and hence of a,, through the Haskind relation (6), values of F and Tare obtained by least-square fit to the two Eq. (3), using results from experimental runs with u =^0
/ ( H z ) 0.5 0.75 1 1.0938 IfMs 0.78 0.64 0.47 0.40 0.8° 2.5° 7.2° 10.3° k r l ( S ) 0.0090 0.0115 0.0125 0.0118 0.0466 0.0615 0.140 0.207 z r 88.6° 89.4° 81.8° 75.4° \T\ 0.986 0.961 0.949 0.912 Z.T - 3 . 1 ° - 5 . 5 ° - 5 . 6 ° - 8 . 7 ° i r + r i 0.985 0.958 0.965 0.956 z ( r - t - r ) - 0 . 4 ° - 1 . 8 ° 2.7° 3.7° | r p + | r p 0.974 0.927 0.920 0.875
Corresponding values for T-hT and |rp-|- |rp ai'e also given.
As the starting point with the two first lines of Table 3, a long-wavelength approximation for the excitation-force coefficient f^, normalised with respect to the hydrostatic stiffness S = pgnirj — r f ) = 137 N/m, has been chosen as
\fJ/S - (l-(2/3)ii,,ak)oxp{-k(l + 2rJ3)} f=0.5 Hz (32) f=1 Hz 300 300
Fig. 10. Heave-excitation-force coefficient / e as determined from experiments with different values of heave velocity amphtude |wl (given in mm/s) by application of the Haskind relation (6), where radiation far-field coefficients a,. (a~ and a f ) have been obtained from Eq. (3) with coefiicients F and T as given in Table 3. Results for/e based on Ö 7 and af are plotted as crosses ( X ) and circles ( O ) , respectively. The upper and lower diagrams show experimental values for the modulus (/èl (in N/m) and the phase angle Z (/e) (in degrees), respectively, for two frequencies/= 0.50 Hz (left) a n d / = 1.00 Hz (right).
[From Eq. (17)—see also Fig. 3—of reference (Bjarte-Larsson and Falnes, 2001)] and
K = Z/e = (u/2)(/cre)' = 90°(/cre)' (33) [From Eq. (3.30) of reference (Kyllingstad, 1982)]. Here 1x33 has been set equal to 1.
Although approximation (32) applies to the deep-water case, we have used it here, but with correct value for k for the actual water depth h ~ 0.74 m. We may observe that the values for r and T obtained in Table 3 deviate much less from the corresponding values in Table 1 than the values in Table 2 do. Fig. 10 shows that we have obtained a better agreement, than in Fig. 9, between the/e values obtained from «7 and a f . Both these figures are based on eight experimental runs for each of the frequencies 0.5 and 1 Hz. CoiTesponding average values and standard deviations for/e are shown in Table 4, together with values also for frequencies 0.75 and 1.0938 Hz, based on two and three experimental runs, respectively. We see that (except for \fJ/S at 1.0938 Hz) the assumed approximate theoretical estimates for \fJ/S and Z/e, given in Table 3, fall within the standard-deviation interval of experimentally obtained values presented in Table 4. (Conceming significance of standard deviation here, similar comments are applicable as for Table 1.) When we compai-e entries for 0 = Z ( r + 7) and /c = Z/e in Tables 1, 3 and 4, it does not appear that ö and 2K are approximately equal. Thus, we have not been able to verify the ideal-fluid relation (9) by means of our (presumably too inaccurate) experimental results.
Table 4 also shows average and standard deviation for values of the radiation resistance
R, obtained from Eq. (7), but with \a,\^ replaced by (|fl7p + \at h/2. The second last hue
of the table gives the non-dimensionalised radiation resistance rr=i?r/^r.N> where we have chosen
_ 2 o^p27zrl f J \ (34)
For / = ( 0 . 5 , 0.75, 1,1.0938) Hz we have i ? r , N = (9-32, 7.78, 6.15, 5.65) N s/m. The last line of Table 4 gives coiTesponding values for the optimum heave-response-amplitude ratio iClopt, which in accordance with Eq. (8) is given as a frequency-dependent coefficient divided by [/el • On the right-hand part of Fig. 10, it may be seen that the plot for M = 52 mm/s is rather untypical among plots for the other runs with frequency 1 Hz. If we leave out this untypical case and analyse the seven remaining mns, we obtain average values and standard
Excitation-force coefficient / e , radiation resistance R, and optimum iieave-amplitude response i C L p t E —
lfe|/(2wKr)], for four different wave frequencies ƒ
/ ( H z ) 0.5 0.75 1 1.0938 I/el (N/m) R, (N s/m) rT=RJRi,n IClopt 0.84 + 0.06 2.9 + 6.7° 115 + 8 1.12 + 0.15 0.120 + 0.016 16.5 + 1.2 0.66 + 0.07 1.1 + 1.7° 90 + 10 1.06 + 0.21 0.137 + 0.027 9.1 + 1.0 0.63 + 0.17 8.3 + 11.2° • 87 + 23 1.56 + 0.72 0.254 + 0.117 5.0 + 1.1 0.52 + 0.06 17.0°+10.0° 72 + 9 1.12 + 0.23 0.198+0.040 4.8 + 0.6 Average values and standard deviations are obtained from results for/g plotted in Fig. 10.
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 867
deviations l/èl = (81 ± 16) N/m, Z/e = 5.4 + 7.7°, i?,=(1.33 ±0.31) N s/m and iaopt= 5.2 ± 1.0, instead of the values given in Table 4 for frequency 1 Hz.
6o Absorbed and converted wave power
Experimental results for absorbed wave power and converted hydraulic power Pp is presented in Fig. 11 for frequency 1.125 Hz, in Fig. 12 for frequency 0.75 Hz and in Fig. 13 for wave frequency 0.5 Hz. The absorbed power Pg is plotted as star (*) points (with full-line curve fits). In accordance with Eq. (2), each point is the power that the heaving body has removed from the waves, based on measurement of incident, refiected and transmitted waves.
The converted hydraulic power Pp has been measured in two altemative ways, denoted as Pp 1 and Pp_2. Fii'stly, we have plotted as square ( • ) points (with dashed curve fits) the pump power Pp i calculated as the time average of the product of the piston's (and floating body's) velocity, the measured water pressure above the piston, and the piston area. Secondly, we have obtained results for the useful hydraulic power Pp^2, which is plotted as
/ = 1 . 1 2 5 H z , p a s s i v e
0 0.5 1 1.5 2
\s/A\
Fig. 11. Power versus lieave ampUtude in non-dimensionalised units, for near-resonance frequency ƒ = 1.125 Hz. Measured absorbed wave power (*) and useful hydraulic power ( • or O—see main text), versus relative osciUation ampUtude ICI = \s/A\. Power is given in units of the incident wave power pg'^D(klï)d\A\'^/{4oj) i n the wave channel. The curves show attempts of curve fitting to the measured absorbed power and useful hydraulic power. The curve drawn as a thicker line coiTesponds to the absorbed-power formula (19), with è = 0, with data for (/el and —and hence for a and c—-fi-om Table 4, and with data for Av = 1 - ((rl^-firj^) from Table 3. The
thinner fully drawn curve for the absorbed power corresponds to data from line 1 of Table 5. The dashed curve and the dotted curve for P^j and Pp,2, respectively,—see main text—correspond to the hydrauhc-power formula (20) with data from Ime 1 of Table 5.
f=0.75Hz, p a s s i v e 0 . 2 | > > > > .
Fig. 12. Power versus heave amplitude, non-dimensionaUsed as in Fig. 11, but for wave frequency ƒ =0.75 Hz. The upper and lower diagrams are obtained with passive loading and with latching control, respectively. Curves according to Eqs. (19) and (20), with data from Tables 3 and 4 for the thicker-hne curve, and from line 3 of Table 5 for the other curves, aie explained in more detail in the main text.
circle ( O ) points (with dotted curve fits). It is calculated as the time-average rate of the product of the weight of water pumped into the elevated reservoir and the head (level difference) between the two reservoirs. Because of losses in hoses and valves, we should expect to find somewhat lower values for Pp,2 than for P p j . This was also found in our
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 f = 0 . 5 H z , p a s s i v e 869 0.25 Q . 0.15 0.05 0.2 0.4 0.6 0.8 |s/A| 1.2 f = 0 . 5 H z , latching control
Fig 13 Power versus heave ampUtude, non-dimensionalised as in Fig. ll,butforwavefrequency/=0.5 Hz.The upper and lower diagrams are obtained with passive loading and with latching control, respectively. Curves according to Eqs. (19) and (20), with data from Tables 3 and 4 for the thicker-Une curve, and from Une 5 of Table 5 for the other curves, are explained in more detail in the main text.
experimental mns, however, with one exception, as seen in the lower diagram of Fig. 13. The strangeness of this exception probably resulted from a measurement error. Quantities
e and Rf related dnectly to Pp,i and P^^ through Eqs. (20) and (24) will—e.g. i n Tables 5
Table 5
Dimensionless parameters used in Eqs. (19) and (20) for curves approximately fitting experimental points in Figs. 11-13 Line / ( H z ) a b ei e-i Af A h + A f 1 1.125 0.1819 0.0186 0.1791 0.0486 0.1313 0.0476 0.0304 2 1.125 0.1741 0 0.1808 0.0501 0.1281 0.0444 0.0285 3 0.75 0.1252 0.4020 0.0693 0.0163 0.0202 0.0267 0.0304 4 0.75 0.1294 0.4494 0.0680 0.0174 0.0214 0.0242 0.0279 5 0.5 0.0579 0.6429 0.0347 0.0060 0.0069 0.0067 0.0097 6 0.5 0.0688 0.6830 0.0202 0.0078 0.0087 0.0090 0.0119 In the least-square fitting, certain conshaints were used with lines 2, 4 and 6: the resonance constraint è = 0 with fine 2, and the constraint of Eq. (36) with hues 4 and 6. Moreover, also the constraint a > 0.0688 was used with line 6.
The three and two largest values of | Cl plotted in the diagram of Fig. 11 and in the upper diagram of Fig. 13, respectively, are from runs made to obtain the linear test results presented i n Fig. 5. As the amount of pumped water was not measured during these runs, coiTesponding plotted points for the useful hydraulic power Pp^2 are missing i n the diagrams. Below, we are, at first, going to give comments about experimental results plotted in the diagrams. Secondly, in the next section, we shall comment on the curves that are fitted to the plotted experimental results.
For a given incident wave, the relative osciUation amplitude |Cl = l^/Al = \ü/((oA)\, which is the independent variable of the diagrams in Figs. 11-13, is varied by changing the load represented by the pump. As the level difference Ai?between the two water reservoirs is increased, |;^| is reduced. In addition, as Fig. 7 or Eq. (31) indicates, |Cl decreases also i f the absolute heave amplitude (|^| or |w|) is reduced as a consequence of reduced wave amplitude |A|. Wave amplitudes in the range of 10-35 mm were used in the experiments. For frequencies 0.75 and 0.5 Hz, experiments were mn with AH chosen to be 0.20, 0.40, 0.60 or 0.80 m. With wave amplitude |A| i n the range of 15-25 mm, optimum values of
AH for obtaining maximum values of Pp/Pi were found to be 0.60 m for the lower wave
amplitudes and 0.80 m for the higher amptitudes. The optimum AH (for a given |A|) seems to have essentially the same value for both frequencies and for both control modes (passive and latching). A t frequency 1.125 Hz we had (for reasons of more frictional problems and poorer experimental reproducibility) to ran experiments with lower hydraulic head. We chose AH to be 0.12, 0.28 or 0.40 m. With |A| ==0.03 m, the best value of Ai? appeared to be 0.28 or 0.40 m.
Table 6
Values for %\, R^, Rf, X^ot and nitot obtamed from Eqs. (21)-(24) using data from Table 5
Line / ( H z ) I/el (N/m) Ztot (N s/m) '"tot (kg) Rr (N s/m) i?f,i (N s/m) Rf,2 (N s/m)
1 1.125 58.3 1.07 2.90 0.75 2.20 5.94 2 1.125 55.8 0 2.75 0.69 2^27 5.80 3 0.75 102.0 -13.7 3.27 1.36 2.82 3.50 4 0.75 105.5 - 1 5 . 0 3.00 1.45 3.01 3.69 5 0.5 109.5 - 2 8 . 0 5.02 1.01 3.62 4.18 6 0.5 130.3 -34.3 3.01 1.43 4.68 5.24
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 871
Because of the loss resistance in the experiment, we were not able to reach the optimum values I d opt, for maximum absorbed wave power, as given in the last Une of Table 4. The piston pump, with its seals, valves and hoses, caused substantial friction, as can be seen by comparing the right-hand amplitude-response diagram with the left-hand one in Fig. 6. Additional loss was caused by friction between the heaving buoy and the fixed guiding strut. This additional friction loss appeared to be a particular problem at frequency 1.125 Hz, as the buoy at this frequency had a strong tendency to stick against the strat, probably as a result of wave-excited horizontal force components. For this reason, a disappointingly low hydraulic power Pp (cf. Fig. 11) was attained at this, approximately resonant, frequency. The power absorbed from the waves was, however, appreciable. Values of PJPi exceeding 0.40 were measured. Considering Fig. 11 and envisaging an extrapolation of the Pa plot to higher values of |?1 than obtained experimentally, it seems that (with I d approaching IClopt) values in excess of the ideal-situation theoretical maximum of 0.5 for PJPy might have been obtained i f a sufficiently low friction could have been achieved for the buoy's heave motion. This excess may be explained as additional wave absorption resulting from viscous effects. We may observe from Fig. 11, that approximately one tenth of the incident wave energy was absorbed even i f the buoy was not oscillating at all. Actually, the two plotted star (*) points at s/A = 0 in Fig. 11 are obtained from two experimental runs, on which Table 1 is based, i n combination with Eq. (4). Thus these two points, referring to frequency 1.0938 Hz, have been adopted in the diagram, assuming that the corresponding values do not deviate significantly from the ones at 1.125 Hz.
For the two sub-resonance frequencies 0.75 and 0.5 Hz, we measured absorbed power and hydraulic power, for experimental rans without, as well as with, latching control. Results are shown in Figs. 12 and 13. For frequency 0.75 Hz, we observe that values ofPJ Pi, -Pp.i/^i and Pp,2/Pi up to 0.17, 0.05 and 0.04, respectively, have been obtained for the case without phase control, but up to the very significantly higher values 0.4, 0.15 and 0.11, respectively, for the case with latching control. Thus the maximum hydrauhc power has increased by a factor of three, as a result of applying latching control. For frequency 0.5 Hz, the corresponding factor of increase is approximately four; the maximum obtained value for Pp.i/Pi, for instance, increases from 0.026 to 0.11 by applying latching control. This higher factor of increase was to be expected, because frequency 0.5 Hz is more distant from resonance than frequency 0.75 Hz. Comparing the horizontal scales i n the diagrams of Figs. 12 and 13, we note that significantly higher values of | d are obtained with than without latching control, a fact that is more lucidly demonstrated i n the time-series diagram of Fig. 8.
7. Discussion
Concerning curve fitting in the diagrams of Figs. 11-13, we apply Eq. (19) for the plotted values of PJPi and Eq. (20) for the plotted values of Pp,i/Pi and Pp,2/Pi- (Note that when these equations are apphed to latching-control cases, we have to set b = 0.) The thinner curves are obtained by least-square fitting to the experimental plotted points. In each diagram there is one thick curve obtained ftom Eq. (19) with data for coefficients a
and c obtained from Table 4 through Eqs. (21) and (23) and for viscous-loss term Av from Table 3 assuming A v = l - | r p - | r p . Moreover, coefficient b is obtained through Eq. (22) assuming mtot = 3.00 kg.
When least-square fitting is applied to the experimental data, we obtain dimensionless parameters as shown in Table 5 to be applied i n Eqs. (19) and (20). In this case, the relation
c = a^ll, which coiTesponds to reciprocity relation (7), has been imposed. The thinner
curves presented in Figs. 11-13, coiTespond to lines 1, 3 and 5, respectively, i n Table 5. Observe that, for each frequency, all least-square-fitted curves are constrained to have common values for a, b and c (except that b = Q with latching-control results). As we expected to find somewhat larger loss resistance Rf with than without latching control, we allowed, in an initial least-square fitting, the parameter e to take different values for the two modes of control. We then discovered, however, no significant difference between the two obtained values. For this reason the final least-square fitting has been performed with common e values for the two modes of control. Thus each of lines 3-6 in Table 5 applies to the case of passive loading as well as to the case of latching control (except that b = Q with latching-control).
In correspondence with Table 5, experimentally obtained, physical quantities for our dynamic system, as obtained through Eqs. (21)-(24), are given in Table 6. From the found values for a and b, we obtain an experimental value for Z ^ f In order to determine the sign of the mechanical reactance Ztot, we know that it is negative for frequencies below the resonance frequency, which we assume to be 1.10 Hz i n accordance with Eq. (30). It appears that the total mass mjot turns out to be umeasonably large i n fine 3 (f= 0.75 Hz) and, particularly, in fine 5 (/'=0.5Hz), of Table 6. As the frequency decreases substantially below resonance, the total mass has a minor influence on the reactance, and then the experimental determination of oïtot becomes very sensitive to measurement errors. In order to avoid unreasonably lai-ge experimental result for mtot, we made another least-square fitting (cf. fines 4 and 6 of Tables 5 and 6) with the constraint that mtot does not exceed 3.00 kg. The motivation for choosing this value is the following. In connection with Eq. (30), it was estimated that mtot=w + mr+mp = 2.9 kg, where the added mass contributes with a fraction of only one tenth. Thus, possible frequency variation of the added mass may be considered to be of negligible significance. Values of mtot in excess of 3 kg seems to be quite unacceptable. Combining the two Eqs. (21) and (22) gives
b = 2 w X t 1 ' 1 , = [K{<^)laf, (35)
[pg^D{kh)d
where ^ ( w ) is a frequency-dependent factor. I f we set mtot = 3.00 kg, then we have
b = [K3{oj)laf, (36)
where K^itjS) equals 0.0867 and 0.0568 for frequency 0.75 and 0.5 Hz, respectively. Concerning line 6 in Tables 5 and 6, we made the additional constraint that even the largest value—obtained experimentally without latching control—for | d = \slA\ could be used in the curve-fitting process without resulting in an imaginary value for the square root
A / T ^ ^ Ï C F in Eqs. (19) and (20). This constraint isb< iClmL or, in view of Eq. (36), a > ^3(w)|?Lax- For frequency 1.125 Hz the least-square fitting (cf. line 1 in Tables 5 and 6)
T. Bjarte-Larsson, J. Falnes / Ocean Engineering 33 (2006) 847-877 873
resulted in mtot = 2.90 kg, which is (accidentally!) rather close to the estimate (2.22 + 0.65) kg used in Eq. (30). Since resonance occurs for a frequency slightly below 1.125 Hz, fitting with the restriction b = 0 (cf. line 2 in Tables 5 and 6) resulted in a somewhat smaller value than expected for
mtot-In order not to overload the diagrams of Figs. 11-13 with too many different curves, we have chosen not to draw curves coiTesponding to Table 5 hues 2, 4 and 6, respectively. I f drawn, such curves would not deviate significantly from the, already drawn, thinner curves con-esponding to Table 5 hues 1, 3 and 5, respectively. For each frequency, the difference between parameters given in a pair of lines in Tables 5 and 6 represents an indication of inaccuracy i n the obtained experimental results. From our previous estimation of the total mass, however, we consider the values for mtot and X t o t in Table 6 to be more correct i n
lines 1, 4 and 6, than in lines 2, 3 and 5. Concerning the hydrodynamic parameters \ f j and
Rr in Table 6, it is sufficient to discuss |/el since we have applied reciprocity relation (7).
For frequency 0.5 Hz, the result for \ f j found in Table 4 is between the two values given in lines 5 and 6 of Table 6. For frequency 0.75 Hz, Table 4 indicates somewhat lower value for [fel than hues 3 and 4 of Table 6. For frequency 1.125 Hz, lines 1 and 2 of Table 6 indicate values for [/èl in reasonable agreement with a statement given in the text between Eqs. (30) and (31). As was to be expected, the heave excitation-force coefficient [/e decreases with increasing frequency, while the radiation resistance R^, which tends to zero for very low and very high frequencies, appears to have a maximum ~ 1.4 N s/m within the range of investigated frequencies. From the curve fitting we have obtained values (Table 5) for the viscous-loss term A^=l-\r\^-\T\^ in reasonable agreement with results from our diffraction measurements (Table 1, see also Table 3).
The obtained values for the friction loss resistance, as given in Table 6, are, unfortunately, significantly larger than the radiation resistance. As was to be expected, larger values are found for i?f,2 than for Rf^i. I t may be noted that, for frequency 1.125 Hz, the difference /?f,2-Pf,i appears to be particularly large, while Ah turns out to be negative (as is seen by comparing the two last columns of Table 5). This is related to the, previously mentioned, experimental difficulties and poor reproducibility at this particular frequency. The dashed curve in Fig. 11 deviates rather much from the dotted one, because it is strongly infiuenced by the three plotted points from the diffraction experiments, where the amount of pumped water into the upper reservoir was not measured.
From the more consistent results presented in Figs. 12 and 13, i t appears that energy lost in the hydrauhc ducts between the pump piston and the water reservofi-s is much less than energy lost by friction of the oscfilating buoy. Let us assume that the difference i?f_2—Rf,\ conesponds
to a total pressure drop Ap = CtotaiP«^/2 in the pump, valves, ducts and connections in the hydraulic system. Here Ctotai is a numerical factor that could probably be less than 10 for a well
designed hydrauhc system, with optimised valves, bends and connections. For sinusoidal motion, the conesponding hydrauhc loss may accordingly, be written
Pp,i-Pp,2=Ktot^p4\ü\'/6 (37)
where dp is the diameter of the pump piston (Jp = 16 mm). By analysing experimental results presented in Figs. 12 and 13, we find that Ctotai is not less than 40 with our non-optimised hydraulic system. This indicates potential for improvement.
In the experiment, by applying latching control, we have, for sub-resonant frequencies, obtained a substantial increase in absorbed wave power as well as i n converted hydraulic power. For frequency 0.75/0.5 Hz, we were able to absorb a fraction of up to 0.16/0.07 of incident wave energy without control, but a fraction up to 0.4/0.26 with latching control. That this latter figure is significantly smaller than the theoretically achievable maximum of 0.5 may be attributed to the disappointingly large friction forces experienced with our physical model. For frequency 0.75 Hz, the maximum pump-power fraction P p j / A (cf. dashed curves in Fig. 12) increased by a factor of 2.9 from 0.046 to 0.136 when latching control was applied. The maximum hydraulic-power fraction Pp,2/Pi (cf. dotted curves in Fig. 12) increased by a factor of 2.8 from 0.039 to 0.109 when latching control was applied. For frequency 0.5 Hz, the maximum Pp,i/Pi increased by a factor of 4.3 from 0.024 to 0.102 and the maximum Pp JPi by a factor of 4.3 from 0.020 to 0.088.
It should be observed that the radiation resistance R^. depends on the width of the wave channel. In a wider wave channel the radiation resistance, and hence also the resonance bandwidth would be smaller. In our case the bandwidth is dominated more by the friction loss resistance Rf. We may hope that in a well designed full-scale WEC the loss resistance Pf w i l l play a less dominating role than in the present investigated laboratory model. I f the dominance of the loss resistance can be avoided, there w i l l be less deviation than shown in the graphs of Figs. 12 and 13 between results for the absorbed wave power and results for the converted useful hydraulic power.
The subject of viscous energy losses experienced with fixed body (M = 0) has been touched upon several places in this paper. See Eq. (4), the last line in Tables 1 and 3, and the column for Av i n Table 5. Accordingly, the curves for absorbed power in Figs. 11-13 intersect the vertical axis {\s\/A = O), not at the origin, but at a positive value for PJP^. Experimentally, the relative viscous loss increases from approximately 3% to approximately 12% when the frequency increases from 0.5 to 1.1 Hz. As argued below, most of this energy loss may be attributed to attenuation i n our nanow wave channel.
Based on wave measurements, with no body immersed in the wave channel, Hoff (2003) estimated this attenuation a to 0.0052 and 0.0131 m~^ at frequency 0.65 and 1 Hz, respectively. Since the distance between the two wave-sensor triplets (Fig. 3) is A x = 4.4 m, this attenuation should account for a relative energy loss of 1 — e"^""^^' which is 4 and 11% at frequency 0.65 and 1 Hz, respectively. Hence the observed viscous energy loss for Ü = 0 is due to the experimental equipment in our laboratory. Thus the observed loss is not representative for a full-scale system i n the sea.
8. Conclusion
A wave-energy converter is described, i n which the absorbed wave energy is converted to useful energy by means of a hydraulic power take-off. The laboratory model that is investigated in the present experimental study contains an axisymmetric body, which may oscillate along a fixed vertical strut. The diameters of the body and strut are 140 and 40 mm, respectively. The model is placed in a 0.33 m wide wave channel. Means are provided to enable the body to be latched for phase control. The body is connected to a piston pump, which pumps water from a lower level (approximately at the equilibrium
