A RANS-VoF Numerical Model to Analyze the Output Power of An OWC-WEC Equipped with Wells and Impulse Turbines in A Hypothetical Sea-State

Wave energy is a renewable source with significant amount in relation to the global demand. A good concept of a device applied to extract this type of energy is the onshore oscillating water column wave energy converter (OWC-WEC). This study shows a numerical analysis of the diameter determination of two types of turbines, Wells and Impulse, installed in an onshore OWC device subjected to a hypothetical sea state. Commercial software FLUENT®, which is based on RANS-VoF (Reynolds-Averaged Navier-Stokes equations and Volume of Fluid technique), is employed. A methodology that imposes air pressure on the chamber, considering the air compressibility effect, is used. The mathematical domain consists of a 10 m deep flume with a 10 m long and 10 m wide OWC chamber at its end (geometry is similar to that of the Pico’s plant installed in Azores islands, Portugal). On the top of the chamber, a turbine works with air exhalation and inhalation induced by the water free surface which oscillates due to the incident wave. The hypothetical sea state, represented by a group of regular waves with periods from 6 to 12 s and heights from 1.00 to 2.00 m (each wave with an occurrence frequency), is considered to show the potential of the presented methodology. Maximum efficiency (relation between the average output and incident wave powers) of 46% was obtained by using a Wells turbine with the diameter of 2.25 m, whereas the efficiency was 44% by an Impulse turbine with the diameter of 1.70 m.


Introduction
Many studies have recently been developed to make the wave energy conversion feasible, since it is a renewable energy source that has a large amount of available energy, estimated in 32 000 TWh/year in the oceans (Mørk et al., 2010). However, researchers have been investigating a safe device that has the capacity to convert energy efficiently and inexpensively. The main difficulties of this task are related to characteristics of the wave energy, which is random and seasonal. Besides, there are extreme events that can damage the structure.
Oscillating Water Column wave energy converters (OWC-WEC's) are the most studied ones, due to their construction and maintenance advantages. In the last decades, several analytical (Evans and Porter, 1995;Zheng et al., 2019), experimental (Ning et al., 2016;Wang et al., 2018) and numerical (Teixeira et al., 2013;Vyzikas et al., 2017) studies have been carried out to investigate the performance of OWC devices. Onshore OWC devices, generally designed to be inserted in breakwaters or built on the coast, require low cost of maintenance by comparison with offshore floating OWC's. These devices are composed of an air chamber, valves, ducts, a turbine and an electric generator. The front wall of the chamber is partially submerged (Falcão, 2010) and incident waves make the water flow into the chamber through an opening below the submerged part of the front wall. Therefore, the water free surface inside the chamber is forced to oscillate, inducing air exhalation and inhalation that activate the turbine in the duct connected to the atmosphere. The electric generator is coupled with the turbine axis in order to convert mechanical energy into electrical one. Nowadays, there are some OWC prototype plants in several regions, such as the Pico Island, Azores, Portugal, the Islay Island, Scotland and Mutriku, Spain (Falcão, 2010).
Typical turbines to be used in OWC chambers can be classified into Reaction and Impulse turbines. The most used turbine for this application is the Wells turbine, which belongs to the reaction turbine group. It is relatively simple and self-rectifying, i.e., it rotates in one direction, regardless of the flow direction (Falcão, 2010). Valves are often coupled in series or in parallel to enable optimal turbine efficiency and safety. In the Impulse turbine, there are two rows of guide vanes on both sides of the rotor, to consider the bidirectional flow; they can cause important aerodynamic losses that reduce its efficiency (Falcão and Henriques, 2016). The maximum efficiency of the Impulse turbine is generally lower than that of Wells turbine. However, the Impulse turbine can lead to higher efficiency in certain sea states; besides, unlike the Wells turbine, the stall phenomenon, in which the turbine performance suddenly drops at a high angle of attack, does not occur. Several investigations to analyze the output power of an OWC-WEC at prototype scale by using numerical models based on RANS-VoF (Reynolds-Averaged Navier-Stokes-Volume of Fluid) equations have been carried out (Liu et al., 2009;Conde et al., 2011;Didier et al., 2011;López et al., 2014;Bouali and Larbi, 2017). These types of models take into account complex phenomena, such as viscosity effects, wave breaking and both nonlinear wavestructure interaction and hydro-aerodynamic coupling; besides, scale effects may be avoided. However, weaknesses of methodologies employed by these researchers are: (1) air flow is incompressible; and (2) an orifice or porous media is used to represent damping imposed by the turbine in the air inside the chamber. The latter prevents air damping from being related to real turbine characteristics, since the performance analysis fully depends on the correct representation of the turbine pressure loss.
When investigations are developed at small scales, air compressibility inside the chamber is insignificant. However, air compressibility effects can highly influence the evaluation of energy conversion by the OWC at real scale (Sheng et al., 2013). A few number of researchers (Viviano et al., 2016;Pawitan et al., 2019) developed experimental studies of the performance of OWC devices at large scales. This hindrance can be solved when small OWC models have different scales of underwater and above-water (air) chamber volumes, by using a scale based on the Froude's similitude law in the case of underwater volume (cube of the scale factor) and a scale with one less dimension (square of the scale factor) in the case of the air chamber volume (Falcão and Henriques, 2014;Elhanafi et al., 2017). Additionally, there are few studies that consider compressibility effects in numerical investigations of OWC devices by means of RANS numerical models, such as Thakker et al. (2003), Teixeira et al. (2013), Elhanafi et al. (2017) and Simonetti et al. (2018). In general, they concluded that the effect of neglecting air compressibility results in an overestimation up to 12%-15% in the pneumatic power efficiency.
RANS−VoF models have high computational cost when the air flow is compressible. Therefore, analyses of different scenarios (fundamental to have an adequate OWC design) can be prohibitive, since they demand simulations of many cases. Gonçalves et al. (2020) have recently investigated the influence of compressibility effect on the air inside the OWC chamber at prototype scale, by using commercial software FLUENT ® . The methodology was proposed by Torres et al. (2016) which takes into account actual air damping of the Wells turbine air. Water and air flows are considered incompressible in the numerical model, but, at every instant, an air pressure condition (effect due to the turbine) is imposed on the top air boundary of the chamber, considering the effect of air compressibility. This air pressure condition is based on an analytical equation that considers the isentropic transformation of the air and effects of Wells and Impulse turbines. Results show that air compressibility effects can diminish the predicted OWC efficiency up to about 20% in both Wells and Impulse turbines.
This study aims at determining, by means of a RANS−VoF numerical model, the optimal diameter of both Wells and Impulse turbines in a hypothetical sea state in an OWC device at prototype scale. The methodology developed by Gonçalves et al. (2020) is used for imposing air pressure boundary condition on the top boundary of the OWC chamber considering the air compressibility effect and damping of Wells and Impulse turbines.

Case study
The case study consists of a 10 m deep flume with horizontal bottom and an OWC device at its end (Fig. 1). The pneumatic chamber is 6.0 m high, 10.0 m wide and 10.0 m long. Its front wall is 0.5 m thick and is 2.5 m submerged.
In this study, a hypothetical sea state is considered to determine the optimal diameter of both Wells and Impulse turbines. Wave climate is characterized by the occurrence frequency of each sea state, described by its root mean square height (H) and energy period (T). It consists of three wave periods (6, 9 and 12 s) and four wave heights (1.00, 1.37, 1.75 and 2.00 m), as shown in Table 1 (Torres et al., 2018). Paulo R. F. Teixeira et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 760-771 761 P w The power of each incident wave that must be extracted can be evaluated by its time-average, , as follows (Dean and Dalrymple, 2000): ρ w where is the water specific mass, g is the gravitational acceleration, L is the wavelength obtained by the linear dispersion equation (Dean and Dalrymple, 2000), k = 2π/L is the wave number, h is the depth and A is the chamber width, which is 10 m in this study. Table 1 shows the incident wave power and the available power considering the occurrence frequency of the proposed hypothetical sea state. The available power is the power of the incident wave taking into account its occurrence frequency. The total available power of the hypothetical sea state is 201.9 kW.

Ψ Φ Π
The dimensionless parameters of a turbine are the pressure coefficient, , the flow rate coefficient, , and the power coefficient, , given by (Torres et al., 2016): where is the referential specific mass, N is the rotational speed, D is the turbine diameter, Δp is the air gauge pressure and Q t is the volumetric flow rate of the turbine. Relations among these dimensionless parameters depend on the type and the construction characteristics of the turbine that is linear in the Wells turbine and quadratic in the Impulse one under ideal conditions and at low and intermediate values of flow coefficient (Falcão et al., 2013).
The Wells turbine characteristic relation, k tW , is given by (Torres et al., 2016;Falcão et al., 2013;Falcão and Justino, 1999): where is the relation between the flow rate and the pressure coefficients in the Wells turbine; this relation is approximately constant (K W =0.680 3 at the Pico's plant) (Falcão, 2004). k tW under study ranges from 40 to 240 Pa·s/m 3 . Fig. 2 shows curves of versus , and efficiency ( ) versus of the Wells turbine used at the Pico's plant (Falcão and Justino, 1999). It considers practical values of turbine diameter and rotational speed (Torres et al., 2016). Ranges of the rotational speed and turbine diameter are restricted by maximum (ND/2) and minimum (50% of the maximum value) blade tip speeds, which are 150 m/s and 75 m/s, respectively (Torres et al., 2016).
In the Impulse turbine, the turbine characteristic relation (k tI ) is the relation between pressure and flow rate, which is nonlinear (Falcão et al., 2013). It is given by: Ψ/Φ Ψ where K I = is the relation between pressure and flow rate coefficients in the Impulse turbine; this relation is approximately constant (K I =19.37 in the case of the Impulse turbine studied by Thakker et al. (2009)). In this study, k tI ranges from 0.8 to 8.8 Pa·s 2 /m 6 . Fig. 3 shows curves of  (Falcão and Justino, 1999).
versus and efficiency ( ) versus of the Impulse turbine studied by Thakker et al. (2009).
The available pneumatic power (P p =ΔpQ t ) of Wells (P pW ) and Impulse (P pI ) turbines is expressed by the following equations:

RANS−VoF model
Two-dimensional equations that represent an incompressible fluid flow, continuity and momentum are given by: where i, j = 1, 2, t is the time, p is the pressure, is the specific mass, g i represents components of gravity acceleration and is the viscous stress tensor. RANS equations, based on the decomposition of the instantaneous velocity and pressure fields of the Navier-Stokes equations into mean and fluctuating components, and the subsequent time-averaging of the set of equations, are used. This process introduces Reynolds stress terms associated with turbulence. The standard turbulence model is used for relating Reynolds stresses to mean flow variables and closing the equations.
Free surface flow motion is defined by the VoF method (Hirt and Nichols, 1981), which is based on the transport equation of the volume fraction, given by: α where the volume fraction, , is a scalar that takes values 0 in the air, 1 in the water and 0.5 in the position of the free surface.

Numerical model
The ANSYS Inc. (2016) software applies a Finite Volume technique to discretize RANS and VoF equations, where variables are defined in the center of each control volume.
k − ε In this study, the algorithm SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) is used for the pressure−velocity coupling and for solving the set of discretized equations. A second-order implicit scheme for time integration is used. Both phases are incompressible and the HRIC (High Resolution Interface Capturing) capturing method (Ferziger and Peric, 1997) is used for defining the free surface. The PRESTO! (PREssure STaggering Option) method is used for spatial discretization of pressure (classically used for wave propagation modeling in FLUENT ® ), while the interpolation scheme of third-order MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) is adopted to momentum and turbulence. The turbulence model is also employed with standard parameters. Relaxation coefficients are 1 for pressure, momentum and volume fraction, and 0.8 for turbulence kinetic energy and turbulence dissipation rate. Fig. 4 shows the scheme of the computational domain, whose wave flume is 2L long and 10 m deep. The OWC device is placed at the end of the flume. The incident wave generation is imposed by a UDF (User Defined Functions), applied to the wave maker boundary. Velocity component profiles, which are related to time and depth according to the linear wave theory (Dean and Dalrymple, 2000), are imposed and the corresponding free surface position is defined  Paulo R. F. Teixeira et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 760-771 by the volume fraction value (0 for air and 1 for water). An active absorption technique is imposed at the wave generation Teixeira et al., 2017) to eliminate re-reflection of waves on the flume by using the methodology proposed by Schäffer and Klopman (2000), which is based on the linear shallow water theory. This methodology has shown that it works relatively well even when used for waves outside the shallow water range Teixeira et al., 2017). The active absorption allows the use of a relative small length of numerical wave flume (low computational cost) and acquisition of long-time series of variables.
The non-slip condition is imposed on walls of the structure and the bottom of the wave flume. The atmospheric pressure is applied to the top boundary of the wave flume.
Pressure loss due to the turbine (see Section 3.3) is imposed as a boundary condition on the OWC chamber top boundary by means of UDF Teixeira et al., 2013;Torres et al., 2016), based on the relation between the pressure and the volumetric flow rate of the turbine (Eqs. (3) and (4) for Wells and Impulse turbines, respectively). Pressure loss boundary condition is imposed at each instant based on the flow rate of the previous instant. By considering that a 2D flow hydrodynamic mathematical model is employed in this study, the air flow rate computed by the FLUENT software is per unit width. Therefore, to take into account the real value of the volumetric flow rate, it is multiplied by the width of the OWC chamber being 10 m in this study (Torres et al., 2016). Thus, the real pressure effect inside the OWC air-chamber (pressure effect at the chamber top boundary due to the turbine) can be considered.
ε Small values of turbulence kinetic energy (k = 10 −6 m 2 /s 2 ) and dissipation rate ( = 10 −6 m 2 /s 3 ) are imposed on the wave maker boundary, the top boundary of the wave flume and the top boundary of the chamber, following Lin and Liu (1998).
Free surface level at rest (at 10 m depth), null velocity components, hydrostatic pressure on the water and atmospheric pressure on the air are initial conditions. Turbulence kinetic energy and dissipation rate are 10 −6 m 2 /s 2 and 10 −6 m 2 /s 3 , respectively.
The time step is equal to T/600 and six non-linear itera-tions per time step enable to reduce the residue by at least two orders of magnitude which are enough to obtain good accuracy in wave propagation and wave-structure interaction Conde et al., 2011;Teixeira et al., 2013;Mendonça et al., 2018). The computational domain is spatially discretized by using regular meshes of quadrilateral cells (Fig. 5). There are 125 vertical cells; 50 of them are located in the free surface variation zone, taking into account the maximum wave height. There are also 390 horizontal cells, with maximum size of 70 cells per wavelength, while, around the chamber, the mesh is refined (Teixeira et al., 2013;Mendonça et al., 2018). The number of cells ranges from 40 379 to 49 792, since it depends on the wave characteristic. It should be highlighted that there is no interest in capturing boundary layers on the bottom and walls, because they have insignificant influence on wave energy extraction, since the boundary layer size is small in comparison with the domain dimensions. Besides, the boundary layers formed by oscillating flows are smaller than the ones formed by uniform flows.

Aerodynamic effect of the turbine
The interaction among incident waves and the OWC device has predominantly a 2D hydrodynamic behavior. Therefore, a 2D numerical model, that demands lower computational cost than a 3D model, is used. However, the calculation of the flow rate of the turbine takes into account the width of the chamber (Teixeira et al., 2013), that is 10 m in this study. Consequently, real effects of the air pressure due to the turbine and its action on the free surface inside the OWC chamber are considered. This methodology has been validated and applied to OWC-WEC's by Teixeira et al. (2013), Conde et al. (2011 and Mendonça et al. (2018).
The OWC chamber is considered an open system, because it allows the transition of mass through its boundaries. The control volume (in the OWC air chamber) is limited by the free surface, walls and the turbine, in which Eq. (3) (Wells turbine) or (4) (Impulse turbine) is used to impose pressure (p) as the boundary condition on the chamber top boundary by using a UDF Didier et al., 2011). To take into account the effect of the compressibility of the air inside the chamber, this study is based on a thermodynamic analysis of the air. It considers an isentropic process and produces a mathematical relation that involves volume, specific mass and pressure inside the air chamber with the volumetric flow rate of the turbine, as follows (Josset and Clément, 2007;Sheng et al., 2013;Gonçalves et al., 2020): where and are the specific masses of the air inside the chamber and of the atmosphere, respectively; and are the air pressure and its rate; and are the volume of the air inside the chamber and its rate; and Q t is the flow rate of the turbine. is the adiabatic expansion coefficient (1.4 for ideal gases) and is 1 for inhalation and 0 for exhalation. V cVc Eq. (9) is discretized by using the first order finite difference approximation. Analysis of time step convergence shows that this discretization is adequate and the criterion validated for wave propagation and incompressible air, Δt =T/600, can be used. Thus, the calculation of pressure to be imposed on the top boundary of the OWC chamber at each instant by means of UDF is based on monitored variables Q t , and of a previous instant. Gonçalves et al. (2020) have compared results of this compressible model with those obtained considering the air compressible and by using a porous media to represent the turbine damping. This methodology allows the air compressibility inside the chamber to be considered without the direct use of the energy equation to close the problem and to maintain a control on the turbine and relief valve by using UDF.

Maximum average output power of the Wells turbine
The maximum average output power obtained in each curve related to the turbine diameter and the sea state characteristic (H and T) is considered to calculate the total average output power expected in this hypothetical sea state. The composition of the average output power takes into account the occurrence frequency of each sea state.
Damping of the Wells turbine on the chamber is represented by its turbine characteristic relation, which depends on the turbine diameter and the rotational speed (Eq. (3)). Thus, the pair (D, N) that provides the best performance in certain sea state must be investigated simultaneously. The Turbine Diameter Optimization (TDO) model (Torres et al., 2018) is employed to reduce the number of simulations of the FLUENT ® software to obtain the optimal pair (D, N) that provides the maximum average output power. The TDO model determines responses of any Wells turbine in the case of a specific incident regular wave (H, T) on an OWC device, based on the response that another Wells turbine gives to the same incident wave. Therefore, if the air pressure inside the chamber of an OWC device is obtained by means of a RANS−VoF model (in this study, FLUENT ® ) by using a specific k tW of the Wells turbine, TDO model allows computing the average output power of any turbine with diameter D and rotational speed N. ω The TDO model considers a harmonic movement of a solid piston with the same frequency of the incident regular wave to describe the free surface inside the OWC chamber of the device. Eqs. (9) and (10) are discretized by using a first-order scheme (Teixeira et al., 2013). A harmonic response (with wave angular frequency ) of the absolute pressure of the air inside the OWC chamber can be calculated by means of Eqs. (9) and (10) for an arbitrary oscillation amplitude inside the chamber (a c ). In the TDO model, a compound amplification factor (CAF), that relates the total oscillation amplitude of the OWC device (a T ) with the amplitude of the incident wave (a w ), is considered quite constant. The total oscillation amplitude, a T , is defined as the sum of a c and a p , where a p is the amplitude of the pressure oscillation in terms of water column . Based on the time series of the air pressure inside the OWC chamber obtained by a RANS−VoF numerical model for a certain incident wave, the TDO model is calibrated by calculating CAF, which can be used for any turbine characteristic relation k tW . In this calibration process, the TDO model uses the time series of the air pressure inside the chamber to calculate the time-average pneumatic power (Eq. (5a)) obtained by the RANS−VoF numerical model. Afterwards, discretized Eqs. (9) and (10) are used for determining both the value of the gauge pressure and the timeaverage pneumatic power for a given a c . It is calculated iteratively until the correspondent average pneumatic power is the same as the one obtained by the RANS model. Finally, the CAF value is calculated.
Therefore, CAF establishes a constant relation between a T and a w for a particular incident wave characteristic and, consequently, for each different k tW , a c and a p can be iteratively calculated by Eqs. (9) and (10). When the relief valve is opened to maintain the chamber pressure under its critical value, p in the turbine and, consequently, a p , are kept constant, in agreement with the methodology described by Torres et al. (2018).
The algorithm to obtain the average output power of Wells turbines with a range of diameters D and maximum blade tip speeds ND/2 for an incident wave with wave period T and height H consists in the following steps: (1) simulate the study case by the RANS model, taking into account k tw = 100 Pa·s·m −3 ; (2) determine the amplification factor CAF by means of the TDO model; (3) calculate the turbine characteristic relation of each N (Eq. (3)); (4) calculate iteratively Δp, by Eqs. (9) and (10) (TDO model), that satisfies the calibrated CAF; (5) calculate the average output power based on the performance curves of the turbine (Fig. 2).

Maximum average output power of the Impulse turbine
Unlike the Wells turbine, the Impulse turbine characteristic relation depends only on its diameter (Eq. (4)). Therefore, in this case, the rotational speed is the one that provides the highest efficiency, taking into account the performance curve of the turbine (Fig. 3c). The algorithm to obtain the average output power consists of the following steps: (1) simulate the study case by the RANS−VoF model, taking into account k tI (Eq. (4)) related to each turbine diameter (D); (2) calculate the optimal output power and the correspondent rotational speed based on the performance curves of the turbine (Fig. 3c).

Results and discussion
Time series of mean free surface elevation inside the OWC chamber, mean pressure, and power by using Wells and Impulse turbine are shown in detail in an incident wave T=9 s and H=1.0 m. Afterwards, performance analyses of both turbines in a hypothetical sea state and a comparison of the output powers of both turbines regarding the turbine diameter are carried out.
4.1 OWC device response by using Wells and Impulse turbines in an incident wave of T = 9 s and H = 1.0 m Fig. 6 shows time series of free surface elevation and air pressure inside the chamber, pneumatic and output powers of the Wells turbine with k tW =100 Pa·s·m −3 and D=2.25 m in the case of an incident wave of T=9 s and H=1 m. The turbine characteristic relation k tW =100 Pa·s·m −3 was adopted because it is an intermediate value in the range adopted in this study (from 40 to 240 Pa·s·m −3 ). Besides, the turbine diameter D=2.25 m allows it to be operated in the range of the blade tip speed (from 75 to 150 m/s). The mean free surface elevation inside the chamber (Fig. 6a) has almost harmonic behavior, with oscillation amplitudes of 0.55 m and −0.50 m. The mean air pressure (Fig. 6b) also has similar behavior, with the amplitude of exhalation higher (3.79 kPa) than that of inhalation (−3.29 kPa) for a classical observation in OWC devices. This behavior affects the one of pneumatic and output powers (Fig. 6c) that reach peaks in exhalation (143.7 and 112.9 kW, respectively) higher than those in inhalation (108.1 and 83.9 kW, respectively). Time-average pneumatic and output powers are 61.3 and 43.9 kW, respectively, which correspond to the use of 72% of the available pneumatic power. Fig. 7 shows time series of mean free surface elevation and mean air pressure inside the chamber, pneumatic and output powers of the Impulse turbine with k tI =2.8 Pa·s 2 /m 6 (intermediate value in the range of k tI adopted in this study) and D=1.70 m (according to Eq. (4)) in the case of an incident wave of T=9 s and H=1 m. Although the mean free surface elevation inside the chamber shows almost harmonic behavior, the mean air pressure shows nonlinearities. Oscillation amplitudes of the mean free surface elevation inside the chamber are 0.56 m and −0.53 m. Mean air pressure amplitude is 4.21 kPa in exhalation and −3.30 kPa in inhalation. Pneumatic and output powers reach higher amplitudes in exhalation (177.3 and 124.6 kW, respectively) than in in-  halation (109.9 and 90.9 kW, respectively). Time-average pneumatic and output powers are 60.2 and 45.7 kW, respectively, which correspond to the use of 76% of the available pneumatic power.
4.2 Average output powers of the Wells turbine in the hypothetical sea state In each sea state, the average output power of the Wells turbine depends on the turbine diameter (D) and the rotational speed (N) (Eq. (3)). Therefore, it is possible to construct a curve of average output power versus rotational speed for each turbine with diameter D to find its maximum average output power in this sea state. Fig. 8 shows curves of average output powers of the Wells turbine for a range of turbine diameter (from 1.50 to 2.75 m) and rotational speed in the case of an incident wave of T=9 s and H=1.0 m. Determination of the range of the rotational speed in each curve considers the maximum blade tip speed (ND/2) of 150 m/s and the minimum of 75 m/s (50% of the maximum value). In this case, the maximum average output power is 46.3 kW, that occurs at D=2.00 m and N=113.6 rad/s.
The maximum average output powers of each sea state (T, H), average output power ( ) and efficiency ( ) of the hypothetical sea state at different diameters of the Wells turbine are shown in Table 2. The turbine with diameter D=2.25 m achieves the best performance at . However, turbines with diameters ranging from 2.00 to 2.50 m exhibit differences of output powers in only 4%. Table 3 shows the rotational speed correspondent to the optimal efficiency of a turbine diameter at each sea state. The optimal rotational speed decreases as the turbine diameter increases, whereas it increases as wave height and period increase, due to increase in wave energy. At T=12 s, optimal rotational speeds are restricted by the maximum blade tip speed of 150 m/s for every turbine diameter. Fig. 9 shows efficiencies in each sea state (H, T) at different diameters of the Wells turbine. They are the relation between maximum output power ( Table 2) and incident wave power (P w ) of each sea state, shown in Table 1. In general, efficiency decreases with the increase of wave period. At T=6 s (Fig. 9a), efficiency decreases with the increase of wave height. Maximum efficiency (58.8%) occurs for H=1 m and D =2.25 m. In every wave height, maximum efficiencies are found in diameters from 2.25 to 2.50 m. At T=9 s (Fig. 9b), efficiency decreases with the increase of wave height in small turbine diameters (from 1.50 to 2.25 m); however, this behavior inverts in larger diameters. Maximum efficiency is 49.3% (lower than that at T=6 s), which occurs when H=1 m and D=2.00 m. In every wave heights, maximum efficiencies are found in diameters from 2.00 to 2.25 m. At T=12 s (Fig. 9c), maximum efficiency is 41.7% (H=1 m and D=1.75 m). The higher the wave height, the larger the turbine diameter (from 1.75 to 2.25 m) to obtain the maximum efficiency. This analysis of efficiency in each individual sea states allows observing that different op-P tPt /P w  timal diameters are obtained. Therefore, the optimal diameter can change according to the sea state distribution under investigation, since they depend on the distribution of the occurrence frequency.
4.3 Average output powers of the Impulse turbine in the hypothetical sea state In each sea state, the average output power of the Impulse turbine only depends on the turbine diameter (D) (Eq. (4)). Fig. 10 shows the average output power of the Impulse turbine within a range of turbine diameter and correspond-ent rotational speed in the case of an incident wave of T=9 s and H=1.0 m. In this case, a turbine with diameter equal to 1.70 m and rotational speed of 57.0 rad/s reaches the maximum average output power (45.7 kW). P tPt /P w Maximum average output powers of each sea state (T, H), average output power ( ) and efficiency ( ) of the hypothetical sea state at different diameters of the Impulse turbine are shown in Table 4. Complementarily, Table 5 shows the rotational speed that provides the maximum output power for each turbine diameter. The turbine with diameter D=1.70 m achieves the best performance, at P t /P w =44%. It operates with rotational speed ranging from 53.7 to 88.7 rad/s. However, turbine with diameter of 1.90 m reaches almost the same relation P t /P w . The optimal rotational speed has the same behavior shown by the Wells turbine: it decreases with turbine diameter and increases with wave height and period. Rotational speeds are lower than those of the Wells turbine. Fig. 11 shows efficiencies (relation between maximum output power and incident wave power of each sea state) in each sea state (H, T) at different diameters of the Impulse turbine. Similar to Wells turbine, efficiency decreases with the increase of wave period. At T=6 s (Fig. 11a), efficiency decreases with the increase of wave height. Maximum efficiency (58.1%) occurs for H=1 m and D=1.70 m. In every   wave height, maximum efficiencies are found in diameters from 1.70 to 1.90 m. At T=9 s (Fig. 11b), efficiency decreases with the increase of wave height up to turbine diameters of 1.90 m and there is a behavior inversion for larger diameters. Maximum efficiency is 48.6% at H=1 m and D=1.70 m. In every wave height, maximum efficiencies are found in diameters from 1.70 to 1.90 m. At T=12 s (Fig. 11c), maximum efficiency is 40.1% (H=1 m and D =1.50 m). The higher the wave height, the larger the turbine diameter (up to D=1.70 m) to obtain the maximum efficiency. As observed in the analysis of efficiency of Wells turbine, the optimal diameter to be used in an OWC devicē P tPt /P w   . 11. Efficiencies in each sea state (at T = 6 s (a), 9 s (b) and 12 s (c)) in different diameters of the Impulse turbine.
Paulo R. F. Teixeira et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 760-771 769 depends on the distribution of the occurrence frequency of each sea state.
4.4 Comparison of average output powers of Wells and Impulse turbines in hypothetical sea statē P t /P wP t /P w P t /P w Efficiency ( ), relation between average output power and incident sea state power, versus turbine diameter (D) in the hypothetical sea state by using Wells and Impulse turbines, is shown in Fig. 12. The highest performance occurs at turbine diameters D=2.25 m for Wells turbine and D=1.70 m for Impulse turbine. The Wells turbine had slightly higher average output power ( =46%) than the Impulse turbine ( =44%) in this hypothetical sea state. The Impulse turbine had a smaller diameter than the Wells turbine and a lower rotational speed: for Impulse turbine with D =1.70 m, the rotational speed ranges from 53.7 to 88.7 rad/s; and for the Wells turbine with D=2.25 m, it ranges from 88.9 to 133.3 rad/s. It shows that the range of diameter that provides high output power of the Impulse turbine is larger than that of the Wells turbine.

Conclusion
This study showed a methodology to determine the diameter of the turbine that provides the highest performance of an onshore OWC-WEC by using Wells and Impulse turbines in a sea state. The FLUENT ® software was applied to a hypothetical sea state and the TDO model was used for defining the optimal output power and the correspondent rotational speed of a Wells turbine based on the performance curves of the turbine, taking into account the action of relief valves.
The analysis of the response of the OWC device in the hypothetical sea state shows that the highest performances occurred when turbine diameters were D=2.25 m and D=1.70 m (Wells and Impulse turbines, respectively). The Wells turbine had slightly higher average output power (P t /P w =46%) than the Impulse turbine (P t /P w =44%) in this hypothetical sea state. However, the Impulse turbine had smaller diameter and lower rotational speed in relation to those of the Wells turbine. Moreover, different chamber geometries and economic aspects, such as operational and installation costs, must be considered to find the best solution.