The Pendulum Wave Energy Converter is basically formed by a channel of parallel walls with a reflector wall on the back (caisson) and a pendulum a distance ...
Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Kitakyushu, Japan, May 26~31, 2002 Copyright 0 2002 by The International Society of Offshore and Polar Engineers ISBN l-880653-58-3 (Set); ISSN 1098-6189 (Set)
Numerical Modelling of the Pendulum
Conference
Ocean Wave Power Converter using a Panel Method
M Alves, A. Brito-Melo and A.J.N.A. Sarmento MARETEC, Instituto Superior Tecnico 1049-001 Lisboa, Portugal
The device consists essentially of a channel of parallel walls with a reflector wall on the back (caisson) and a pendulum (flap) at the entrance of the inside caisson being able to perform rotational motion around a horizontal axis, by the action of the waves. This oscillatory movement enables the production of energy.
ABSTRACT The Pendulum Wave Energy Converter is basically formed by a channel of parallel walls with a reflector wall on the back (caisson) and a pendulum a distance from the back wall being able to perform rotational motion around a horizontal axis, by the action of the waves. This oscillatory movement enables the production of energy. In this study the hydrodynamic analysis of the pendulum device is performed in the frequency domain by using the 3D radiation-diffraction panel code AQUADYN to compute the hydrodynamic coefficients and the diffraction momentum amplitude. This study focuses on the analysis of the influence of some geometric parameters on the pendulum performance as well as the effect of imposing constraints to the oscillation motion.
This wave energy device can be classified as an Oscillating Body with only one degree of freedom (the oscillation around a horizontal axis in the mean water free surface plane parallel to the backward reflective wall). Many oscillating systems have been proposed for conversion of wave energy. General results were established by Mei (1976), Evans (1976), Newman (1976) and Count (1980) where theoretical models have been developed for the hydrodynamic performance of wave energy devices and analytical results have been presented for simple geometries. However, for complex geometries, numerical models are required to obtain the hydrodynamic coefficients. The boundary element method has revealed to be adequate to deal with such devices. In this paper we illustrate the application of the three-dimensional radiation-diffraction panel model, AQUADYN (Delhommeau, G. 1987) to the pendulum device. This code based on classic linear water wave theory and potential flow was developed at Laboratoire de Mecanique des Fluides, Ecole Centrale de Names for the study of floating bodies’ hydrodynamics in the frequency domain. The equation of motion of the pendulum is expressed as a function of the complex amplitude of the angular motion, which can be determined through the amplitude of the diffraction momentum and hydrodynamic coefficients of added mass and damping, obtained from the numerical code.
KEY WORDS: Pendulum Device, Wave Energy Converter, AQUADYN, frequency domain, radiation-diffraction, spring and damping coefficients, capture width, angular displacement.
INTRODUCTION The Pendulum Ocean Wave Power Converter was invented and studied by Muroran Institute of Technology Japan: “A pendulum (being a moving body) absorbs wave energy as pendular motion excited by wave force in resonance with a water chamber of a caisson”. PendularDevice Hydraulic Pu p 7
From the pendulum equation we deduced the expression for the angular displacement amplitude as a function of the hydrodynamic coefficients and torque diffraction amplitude. Through the angular displacement we determine the mean power absorption by the device and the capture width, which gives an evaluation of the system efficiency. From the expression of the optimal mean power absorption we obtain the optimal coefficients of the mechanical spring and damping moments. The results show the influence of the distance between the pendulum and the back wall on the hydrodynamic curves of the device and energy absorption efficiency, as well as the effect of constraining the angular motions of the pendulum.
Fig. 1 Schematic Sketch of the Pendulum wave Energy Converter
655
MATHEMATICAL
FORMULATION
where g is the gravitational acceleration, a the complex amplitude of the angular motion and y1 the unit normal vector on the body surface defined as pointing outwards from the body. Also a suitable radiation condition of outgoing waves at infinite distance is required for 4 It is
Governing Equations We consider a Pendulum device formed by a bottom standing caisson with horizontal U-shape and a vertical plate placed between the caisson aperture and back wall, oscillating as a pendulum due to incident wave motion. We take a Cartesian co-ordinate system with x, y on the undisturbed free surface plane, z positive upward and z=-h as the horizontal sea bottom (see Fig. 2). We denote the surface of the fixed caisson by S, and the pendulum surface by S,
usual to consider the problem of the interaction between waves and the device decomposed in three contributions, requiring that Eq. 2 to 6 are satisfied:
where 4, and @d are the complex amplitude of the incident wave and scattering
potentials,
respectively,
and
diffraction potential. The velocity given in the following form: Incident wave
X
~cos[k(xcos~+ysiiz~)-ut]
h
The third component
.
The pendulum motion subject to the oscillatory waves is given by the following equation
the
following
of the
boundary
conditions
v2(b=o,
in fluid domain
(2)
g-+0,
onz=O
(3)
,
action of incident
(10)
in which 1 represents the torque, 0 the angular displacement of the pendulum and I is the moment of inertia for the pendulum. According to the linear theory the torque is decomposed in the hydrodynamic, hydrostatic and mechanical components. The hydrostatic component is given by:
> satisfies
is the complex amplitude
Pendulum equation
IB=l
$b
in Eq. 7, &,
of oscillation.
potential in the form
amplitude
(9)
velocity potential for each oscillation mode of the body, referred to as the radiation potential. In the present case 4~ refers to only one mode
Under the assumption of incompressible and inviscid flow with irrotational motion we introduce the velocity potential satisfying the Laplace equation. In accordance with the linear theory the boundary conditions are linearized and all oscillatory quantities are assumed to be time-harmonic with angular frequency u). Thus we express the velocity
complex
,
w2 = gktanh(kh)
Fig. 2 Plan view of the Pendulum wave Energy Converter
when
to the wave is
F
::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: :::::::::::::::
4D= RelQ(.r,y,Z)r-‘“‘I
is refers
of the incident
where p is the incident angle of the waves, A is the wave amplitude, h the water depth and k the wave number given from the dispersionrelation:
Z
:j:j:j:j:j:j:j: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: ::::::::::::::: :::::::::::::::
4, +@d
potential
1est =-; hat
- Pwater bb20
>
where the parameters b, t and a are, respectively, the length, thickness and width of the pendulum and p its density. The mechanical component of the torque is expressed in the following form: 1met = -Lko -L&j
,
(12)
in which Ld and Lk, both positive, represent the effects of a spring and respectively. Applying the Bernoulli expression the dampb, hydrodynamic component may be expressed by:
-= a4 0, az
656
where a is the complex amplitude of the oscillation angle given by Eq. 16. Introducing Eq. 16 in Eq. 20 we obtain the mean absorbed power as a function of the mechanical coefficients Ld and Lk
where Ap is the pressure difference between opposite sides of the pendulum due to the flow which is predominantly horizontal. From Eq. 1 and Eq. 7 the hydrodynamic torque in Eq. 13 can be decomposed in two contributions: a diffraction torque contribution due to the incident wave action
2 Ebs
Ldrf
1 2 = 2” Ld
(21)
Lk-i~d-iwD-(I+I,d)w2-Lest
Ld,f = imp
(14) The capture width is defined to be the proportion
of available power per
unit crest length of the incident wave, FW being extracted by the device and a radiation torque contribution, due to the oscillatory motion, being expressed by a term proportional to the angular velocity and a second term proportional to the angular celerity:
1rad = -I,,&t)
-D&t)
aoR ynzdS
=p
%bs L cup =pw
From Eq. 21 it can be shown that the mean absorbed power is maximal when the mechanical coefficients are:
(15)
11 SP
(22)
In this expression I,, represents the added mass and D is the damping coefficient. By replacing in Eq. 10, the Eqs. 12-15, we obtain, from the pendulum equation, the complex amplitude of the angular motion, Ldf Lk- +‘mat-Pwate,
h%b2-
(16) m2(%d) I
The terms Lk -l/2
(23)
(pmat - pwater)gtab2
-i&J
+Ld )
in which
the subscript
o refers
to optimal
Lz =D
if Lk =Li,
a
known
well
conditions,
result.
in which
Introducing
the
mechanical coefficients, given by Eq. 23, in Eq. 21 and Eq. 16 we obtain, respectively, the optimal mean power absorption and the corresponding modulus of the angular displacement amplitude in the following form
, (I + I,d ) and (D + Ld)
represent, respectively, the spring effect, inertia effect and global damping. The coefficients I,, and D are defined, from Eq. 1 and Eq. 15, in the following form
(24)
(17)
(25) In linear wave theory only small amplitudes of the angular displacement can be considered. This, in the numerical calculations a maximum angular displacement is imposed. Thus, it is necessary to recalculate the mechanical coefficients that maximize the energy extraction for the wave frequencies when a maximum angular displacement is imposed. Since the spring coefficient, Lk, does not depend on the angular displacement, its optimal value is still given by Eq. 23. The mechanical damping coefficient, Ld, is given from Eq. 16 imposing the maximum value CX~to the angular displacement:
Energy Absorption The mean absorbed power by the device corresponds to the mean power, due to the hydrodynamic effort during a period of time T. Thus, it can be written as
(26) Since the contribution for the energy absorption work realized by the damping, it follows
is given only by the where the subscript i refers to the imposed conditions in the angular displacement. In this case the optimal absorbed power will be
1 -1 P abs = -J,2 2
(20)
657
I
(27)
The energy absorption results to be present assume: (i) a zero value for the mechanical spring coefficient in the frequency range where this coefficient is negative (a restoring force always acts in the opposite sense of the movement). (ii) The mechanical spring and damping coefficients are taken to be positives and constants (with values chosen such that the maximum angular displacement of the pendulum does not exceed, respectively, 15”or 30°, within the selected frequency range.
increase of the projecting sidewalls and shifted to higher frequencies. This is in accordance with the study presented by Count & Evans (1984) for an idealized device consisting of a rectangular block with a vertical side facing the incoming waves freely oscillating. Thus by changing the pendulum location inside the caisson it is possible to approximate the two peaks in order to enlarge the range of frequencies with higher amplitudes of resonance. 12E+08
NUMERICAL
RESULTS
The following results concern a pendulum geometry: 1 Caisson structure: Sidewalls external length = 27 m External width = 16 m Walls thickness = 2 m 1 Pendulum: Length = 12 m Thickness = 1 m The water depth was assumed to be 10 m.
r &I OE+08 2 ? S80E+07
device with the following
s 560E+07 B ;40E+07 2 e20E+07 Q OOE+OO 02
04
06
08
Wave
2quency
(rad/i)2
Wave
10 12 Frequency(rad/s)
Influence of the pendulum position in the caisson This study focuses on the influence of the pendulum position in the hydrodynamic performance of the device. Three situations were analysed: pendulum placed 5.25 m, 11.5 m and 17.75 m away from the backward wall, to which corresponds a length of projecting sidewalls of respectively 18.75 m, 11.5 m and 6.25 m. Fig. 3 presents the corresponding discretizations. The convergence criterion was fulfilled with a mesh composed by 1265 panels: 712 for the caisson and 553 for the pendulum. Also the use of l-point, 4-point and 9-point Gaussian integration method was investigated and convergence was reached with a 4-point method. It was verified that the solution of the linear system was highly sensitive to errors in the computation of the influence coefficients. This structure requires an accurate method for the computation of the influence coefficients, particularly near the resonance. We found that the major errors in the influence coefficients occur between the lateral panels of the pendulum and nearly panels on the sidewalls of the fixed structure and the convergence is much conditioned by the distance between these panels. A minimum distance of 10 cm was required to get convergence.
21 ZE-01
Y ! ; 6 OE-02 @ ~E 3 OE-02 6 z
OOE+OO 02
04
06
08
Fig. 4 Results achieved with AQUADYN Amplitude Binary (on top) and the Hydrodynamic (below) Referents to the Geometries A, B and C.
for the Diffraction Damping Coefficient
It is of interest to analyse the effect of constraining the angular displacement of the pendulum up to 30” or 15”. Results are presented concerning the amplitude of the angular motion, capture width and the mechanical coefficients of spring and damping. These quantities were determined from the hydrodynamic coefficients and diffraction torque by Eqs. 22 to 27. Results are presented in Fig. 5 and 7 for the case of 30” and in Fig. 6 and 8 for the case of 15”. Comparing the curves concerning the angular displacement (Fig. 5 and 6 on right), we observe that by imposing a lower maximum limit of the angular displacement it occurs a reduction of the range of wave frequencies in which the pendulum may oscillate without restraint. Thus the absorbed energy decreases because the pendulum with a restrained motion cannot extract the maximum energy but only the one available for the imposed angular displacement. We verify in the curves concerning the capture width (Fig. 5 and 6 on left) that if the optimal angular displacement is higher than 15”, the capture width decreases when a limit of 15” is imposed.
A B C Fig. 3 Discretization of the Pendulum Device for three positions (A, B and C) of the pendulum inside the caisson. Fig. 4 shows the numerical results obtained with AQUADYN code for the three configurations, namely the diffraction moment and the hydrodynamic damping coefficient non-dimensionalised by pL5w (L=27 m).
The curves corresponding to the mechanical coefficients of spring and damping (Figs. 7 and S), show a range of wave frequencies where a zero spring coefficient has to be imposed in order to prevent negative values, as the restoring force should act in the opposite direction of the displacement (no negative values are acceptable).
We notice an increase and narrowing of the main peak as the distance between the pendulum and the backward wall varies from 17.25 m to 5.25 m and the main resonance peak shifts to lower frequencies. We also observe a second resonance peak becoming more intense with the
658
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7 OE+Ol
c30E+Ol F
6 OE+Ol
?25E+Ol d 9 42OE+Ol
r ;50E+Ol 0 $4OE+Ol
0 $5E+Ol
230E+Ol
P QlOE+Ol 5
d 20E+Ol
$OE+OO
