IMPACT OF BLADE AND SOLIDITY ON THE PERFORMANCE OF H-DARRIEUS HYDROKINETIC TURBINES BY CFD SIMULATION

Purpose: In the present work, the 2D design of H-Darrieus turbines with a diameter of 900 mm and NACA blades 0018, 0025, 2415, and 4415 has been carried out for the solidity values of 0,5; 1 and 1,5. In order to know its maximum performance. Method/design/approach: The 2D simulations were developed with the ANSYS® FLUENT package in the transient state, varying the angular velocity (ω) for a peak velocity ratio (TSR) of 1 to 7 and the SST K-ω turbulence model, for a constant water flow rate of 1 m/s. This in order to know the results of the torque (Nm) generated and thus calculate the power coefficient (Cp). Results and conclusion: The NACA 0018 blade reached a power coefficient of 0,604 for a solidity of 0,5, followed by NACA 2415 blade at the same solidity with a maximum Cp of 0,594. On the other hand, the NACA 0025 blade for solidity of 1 reached a maximum Cp value of 0,570, while the NACA 4415 profile with a solidity of 1,5 obtained a maximum value of 0.495.


INTRODUCTION
Incorrect patterns of production and consumption of energy, water and resources are the main cause of environmental problems (Matiolli & Lunkes, 2022).Due to the growing energy demand, alternative and renewable energies, such as hydro, wind, solar, and geothermal among others, have become an important source to supply this demand (Abdalrahman, Melek, & Lien, 2017).Giving way to the development of devices capable of transforming this type of energy, such as hydrokinetic turbines, which have been arousing interest in researchers, seeking to achieve higher yields with lower implementation costs (Patel, Eldho, & Prabhu, 2017).One of the models highlighted in recent years is the Darrieus turbine, which has been studied experimentally and numerically, seeking to improve the characteristics of the operation from geometric modifications, such as the shape of the blades, the solidity and diameter of the rotor.These devices were initially developed for the use of wind energy, reaching experimental efficiencies of 0,35 (Pico Saltos, G; Pico Saltos, R; Mendoza, 2018), given the limit set by Betz at 0,593 for this type of turbine in its wind application.
Research related to renewable energy sources has been growing over the last few years and this growth has remained stable over the last few decades, with room for future growth (Freitas, Verbiski, Velásquez, & Alves, 2022).Some researchers have concentrated their studies on knowing the impact that the shape of the blade has on H-Darrieus turbines, since this element has a direct interaction with the fluid (Lam & Peng, 2016).The blades have been classified into two groups: symmetrical and asymmetric blades, mentioned by Hashem & Mohamend (Hashem & Mohamed, 2018).Hashem & Mohamend developed a numerical study of a Darrieus wind turbine testing its performance with different blades.The blades selected for their study were, the symmetric NACA 0015, 0018, 0021, and S-1046 blades and asymmetric blades NACA 63-415, 63-418, 63-421, those of the FX family, the DU and finally the RIS blades.The result of their study reported that the maximum values of power coefficient (Cp) were given with the blades S1046, FXL-142K and RISoA1-24, whose values were 0,3463; 0,3385 and 0,3324 respectively.Affirming that symmetrical blades perform better compared to nonsymmetrical profiles (Hashem & Mohamed, 2018).Batista et al. they addressed in their work the incidence of the geometric shape of the blade for a wind turbine, elaborating an experimental assembly for a Darrieus turbine with a wing blade other f helical shape.Developing several models with the NACA 0018, 0020, 4418, 4420 blades and an innovative blade called EN0005 whose rope curvature is more pronounced than in the asymmetric blades.They compared turbine performance under real-world conditions.Concluding that the blades that presented the best performers was the EN0005 with a maximum power Cp of 0,37; followed by the NACA 0018 blade with a value of 0,34 (Batista, Melício, Mendes, Calderón, & Ramiro, 2015).Like Bastista et al., Sengugta et al. developed an experimental and numerical study with the NACA 0018, EN0005 and S815 blades for an H-Darrieus rotor wind turbine.Reaching experimental results that confirm what was said by Batista et al. with respect to the EN0005 blade, which reached a maximum Cp of 0,16 for a Tip Speed Ratio (TSR) value of 1,6.The numerical results of the study show that the S815 blade reached a maximum Cp value of 0,18; while the EN0005 blade reaches a value of 0,16 for a TSR of 2,4 (Sengupta, Biswas, & Gupta, 2016).This shows the importance of studying the shape of the blade, since this element affects the performance of the turbine.Another of the studies carried out in this regard is the one presented by Saini & Saini (2018) carrying out a numerical investigation to determine the maximum Cp delivered by a hydrokinetic turbine type H-Darrieus of a rotor of 300 mm in diameter, using the NACA 0010, 0018 and S-9000 and the S-1046 blades.The study concluded that the most efficient profile for a rotor with a diameter of 300 mm under conditions of constant speed of 1,5 m/s, was the profile S-1046 with a maximum Cp of 0,42; result similar to that achieved by Hashem & Mohamend (Hashem & Mohamed, 2018) in its analysis on the Darrieus wind turbine.
Gosselin et al.Numerically investigated the behavior of an H-Darrieus wind turbine against the effect has the change of blade and rope length on its performance.To do this, they elaborated profiles with solidity values from 0,1372 to 0,5486.And NACA blades 0012, 0015, 0020, and 0025; TSR values were worked in a range of 1,5 to 6,5.One of the results of the study reported that the maximum cp value is 0,37 with solid values of 0,5489 by simulation, depending on the NACA 0015 blade (Gosselin, Dumas, & Boudreau, 2016).Marsh et al.They carried out a numerical study of the H-Darrieus hydrokinetic turbine to look at the incidence of the shape of the hydrodynamic blade and its rope length, designing the Darrieus turbine with the NACA 0012 blades with a solidity of 0,43 and the NACA 63(4)-021 blade with a solidity of 0,31; the results obtained from the study determined that the performance of the NACA 0012 blade was 0,27 and for the NACA 63(4)-021 was of 0,12.Obtaining a conclusion that both the shape of the blade and the solidity affect the performance of the turbine (Marsh, Ranmuthugala, Penesis, & Thomas, 2015).Dai & Lam developed an experimental and numerical study in a Darrieus hydrokinetic turbine, testing the incidence of solidity in a range of 0,59 to 1,64 for a NACA 0025 blade for a 900 mm diameter rotor.Obtaining experimental results of Cp for a solidity of 0,59 of 0,14 and for the solidity value of 1,64 of 0,20.Thus concluding that when you have a constant rotor diameter, you reach better cp values increased solidity (2009).Shiono et al. experimentally studied the incidence of solidity on the performance of the Darrieus turbine in a water channel, using solidity values from 0,108 to 0,537 for a NACA 63(3)-018 blade, with a constant fluid velocity of 1 m/s, in order to observe the behavior of the turbine when varying this parameter.The result of the study determined that the power coefficient of the turbine decreases as the solidity increases, thus contradicting what Dai & Lam said, since for a solidity of 0,109 its maximum efficiency is 0,23 and for a solidity of 0,537 of 0,12.Thus, further studies should be carried out on this topic to corroborate these results achieved experimentally (Shiono, Suzuki, & Kiho, 2000).Lopez et al. approached a numerical study for the Darrieus type water turbine to determine its behavior with the NACA blades 0015, 2415 and 4415, reaching a maximum torque coefficient (Ct) of 0,12 for NACA 0015 and 4415.For the NACA 2415 profile a maximum Ct of 0,14, reporting that the geometric shape of the profile has an effect on the performance of the turbine.

THEORICAL FRAMEWORK
It is necessary to deepen in improving the performance of the H-Darrieus type turbine, making the most of its ability to obtain energy from water.Therefore, carry out studies to improve performance from the modifications of the rotor design, contributing to improve the Cp from the combination of geometric parameters such as blade type and solidity.The main objective of this work is to evaluate the benefits of the H-Darrieus turbine and study how they behave when making geometric modifications, to obtain a better performance.To meet this objective, a 2D model of the turbine will be designed, along with its domain; the blades selected in the present work were NACA 0018, 0025, 2415 and 4415 (López, Meneses, Quintero, & Laín, 2016).With solidity values of 0,5; 1 and 1,5, values found in the ranges worked by (Dai & Lam, 2009) (López et al., 2016) at a Flow rate of 1 m/s.The selected data were taken from the tracking of results in the literature Cardona-Cárdenas et al. (2021).

Parametric Equations
The hydraulic performance of an H-Darrieus type turbine can be characterized with its power coefficient (Cp) by Ec. (1), this relates the mechanical power on the shaft, product of torque (T) and angular velocity (), with the hydraulic power of the flow that depends on its density (ρ), the cube of its speed () and the projected area of the turbine against the flow (), this is calculated as the area of a rectangle of turbine diameter and height dimensions.
A fundamental parameter to characterize the operation of these turbines is their tip speed ratio (TSR or ), this is defined by the Ec. ( 2) that relates the linear speed of the tip of the blade, which, in turn, results from the product of the angular speed of the rotor with its radius (R), with the linear velocity of the flow.
Traditionally, Cp curves against peak speed ratio are reported as the characteristic curves of hydrokinetic turbine performance, and this is how the results are presented at the end of this numerical study.
On the other hand, the predominant geometric characteristic in the description of an H-Darrieus turbine is the solidity (), Ec. (3), which relates the product of the number of blades () with the length of the rope (), with the radius of the rotor, these being the parameters that determine the shape and size of the device.For this reason, it becomes an important tool to analyze the performance of the H-Darrieus type turbine.For the present 2D study, the ANSYS® CFD FLUENT® resolver (v.21.1) was available, with a workstation with 48 GB of RAM and a 2,67 GHz Intel(R) Xeon(R) processor.

Turbine Modeling
Initially, the modeling of each of the profiles presented in the Figure 1 was carried out.(Benchikh Le Hocine, Jay Lacey, & Poncet, 2019), (Benchikh Le Hocine, Poncet, & Lacey, 2020), through a series of points obtained with the online tool AirFoilTools®, after entering the type and length of string, the selected types can be seen in the figure and the strings were 75, 150 and 225 mm (López et al., 2016).
Initially, the modeling of each of the profiles presented in the .Aerodynamic blades considered.

Source: Prepared by the authors (2023).
For the configuration of the control volume, data taken from (López et al., 2016) and (Lain & Osorio, 2010).An external diameter of the rotational domain of 1200 mm with an internal diameter of 600 mm were used, while the blades were located in the middle of said domain, that is, on a diameter of 900 mm, which would be comparable to the diameter of the turbine.To simulate a projected area over the flow of 0.9 m2 a height of 1 m is assumed for the turbine rotor, and to simulate turbines with solidity values of 0,5; 1 and 1,5 (Dai & Lam, 2009), (Lain & Osorio, 2010), (López et al., 2016), the rope lengths were varied as described above.These turbine dimensions were used for all 4 blades.The dimensions of the outer stationary domain were five times the diameter for the with and eight times the diameter for the length.The turbine is located three diameters from the Inlet.Thus, the domains used in the calculations are as seen in the Figure 2. The boundaries of the domain consist of a velocity input at 1 m/s, a relative pressure outlet at 0 Pa and two movable walls with velocity equal to the fluid inlet velocity (Meneses, López, & Lain, 2013).In addition, two interfaces between the outer part of the rotational domain and the inner part of it.By recommendations of the referent, a value of 10% is added as an initial condition to the intensity of the turbulence of the working fluid (López et al., 2016).In the ANSYS Geometry® module, the geometry described in the Figure 2 was developed ®.There the names were assigned to the borders of the domain.

Turbine Meshing
The Meshing of the turbine model was carried out with the ICEAM® module of ANSYS®, developed in two parts: The first part was developed the meshing of the fixed or stationary flow domain, adopting an unstructured mesh composed of rectangular elements as shown in Figure 3a.The number of nodes that was used to develop the square part of the domain was 30 and 40, for the part of the outer diameter 40 nodes and 50 nodes were used to form the diagonals of the diameter.Next, for the internal diameter, 50 nodes were used both for the diagonals and to give shape to the circumference as shown in Figure 3b.The meshing of the stationary part the same for the twelve turbine models that were evaluated.To ensure an appropriate density of elements, the elements were placed in such a way that by increasing the number of nodes a proximity between the stationary and rotational domain was guaranteed, achieving a faster convergence (Abdalrahman et al., 2017).The second part was the meshing of the rotational domain, adopting an unstructured mesh of rectangular elements.Figure 4 shows the blocks made for the meshing of the blades, 120 nodes were used for the diagonals, 80 nodes for the farthest parts of the blades.For the diagonals near the blade, 60 nodes were used, significantly increasing the number of elements in the contour of the blades.Using the theory of the flat plate boundary layer presented in Ec. (4), relating the product of y + , to the dynamic viscosity of the fluid (μ) divided the product by the density of the fluid (ρ) and the friction rate of friction (To calculate the size of the boundary layer with a value of ρ, V and τ).y + equal to 1 at a growth rate of 1,2 over the entire surface of the blade wall, in order to ensure the capture of the phenomenon that occurs on the surface of the blade (Maître, Amet, & Pellone, 2013).

Mesh Independence study
A mesh independence study was carried out to determine which mesh would result independent of the size of the element.A change was made in the size of the mesh elements, both in the stationary and rotational domains, having a refinement in the contour of the blades.
To continue with the independence study, the mesh was exported to the FLUENT® solver, where the following parameters were configured: The water with default properties such as working fluid, turbulence model SST k-ω (Marsh et al., 2015) and the SIMPLE solver (semiimplicit pressure linked by equations) that uses second-order equations for greater precision in the terms of convection, that is, in the equations of momentum, turbulent kinetic energy and turbulent dissipation rate.In addition, the input velocity of the fluid was configured 1 m/s with a turbulence intensity of 10% (López et al., 2016),then the angular velocity to rotate the rotational domain (mesh motion), this value was calculated with the Ec.(2) from a TSR=3.Subsequently, the interfaces were configured to simulate the rotational movement of the turbine and thus delimit the model between its stationary domain and the rotational domain.For the simulation of six complete turns of the rotor in the transient state, a time step of 0,005 (s) was selected (López et al., 2016) and a number of steps of 1131 calculated with Ec. (5) and a residual value of 10 -3 .This in order to ensure that the system stabilizes and reaches convergence.Table 1 it presents the results of torque generated at the exit of the turbine as a parameter of determination, to know the independence with respect to the size of the mesh element.In addition, the number of nodes of each mesh and the value of the relative error are presented as a selection criterion, this error is considered acceptable in a range less than 0,02 (Le Hocine et al., 2020).Based on the results presented in Table 1.mesh 2 was selected, to begin with the CFD simulation study for each profile.8

Setting Up the Experimente
The configuration of the simulations was performed in FLUENT® by changing the rotational speed of the rotor for each of the TSR values (Tunio et al., 2020) (Benchikh Le Hocine et al., 2019)(Benchikh Le Hocine et al., 2020) as presented in Table 2, for each configuration the number of time steps from Ec. ( 5) was calculated.With the values of TSR () and rotational speed described in Table 2. simulations were performed for each of the selected profiles.

RESULTS AND DISCUSSION
To observe the behavior of the flow passage through the turbine and in direct contact with the blades, the speed and pressure contours for each of the evaluated profiles are presented, in order to know the phenomena that are generated in the turbine.The color scale indicates the values reached by the turbine when it is in operation, the red color indicates the maximum value of speed and pressure, while the blue color indicates the minimum values as shown in Figure 5.The contours of the turbine with NACA 0018 blade were presented for the maximum value of Cp that was reached in TSR=2 as well as for NACA 0025 and 4415, there it is observed how the maximum speed values are being given in the head and the exit of the blades, reducing their speed until reaching almost zero.In addition, it is observed how the maximum pressure value is being generated in the A1, for the A2 it is generated on the front of the blade and for the A3 it is given on the inner side of the blade.The faces contrary to the maximum pressure values show minimum values thus generating the lifting force necessary for the turbine to rotate (Saini & Saini, 2018).For NACA 2415 the maximum cp value was reached with a TSR=3,5.Below are the results of the turbine output torque (Nm) obtained from the simulation and the mechanical power on the (W) axis calculated from it, data that was used for the calculation of the Cp from the calculation of the hydraulic power of the flow according to the configuration parameters of the study.The calculation of the Cp was performed by Ec. (1).Since the torque and speed generated are not constant in the turbine, the calculation of the Cp was extracted from the average values obtained in the last cycle or turn that the turbine performed.
Figure 6 reports the results for NACA profile 0018.With a solidity of 0.5, TSR values 1 to 7 were evaluated with an intermediate value of 3,5.Between TSR 3 and 3,5 there is a maximum, being typical parabolic behaviors, but the Betz limit was exceeded, this may be due to the simplifications and idealization of the numerical model, which in an experimental or real assembly would not occur, but it is also possible that the solidity value 0,5 is constructively high and not recommended.In TSR 7 a negative torque was found, a phenomenon associated with the excessive increase in the angular speed of the rotor, at which time the flow of water acts in the opposite way slowing its rotation.But, with solidity 1, when finding negative torque in TSR 5, TSR 6 and 7 were not simulated, and with solidity 1,5 TSR 5 was not simulated because it found negative torque in TSR 4. This range of TSR is commonly used to observe the behavior of hydrokinetic turbines (Benchikh Le Hocine et al., 2019).The behaviors remain typical, parabolic, with a maximum now shifting towards TSR 2. The Betz limit is not exceeded, but it is evident that as the solidity increases between 0,5 and 1,5 the Cp decreases, which contradicts the literature and shows that the solidity values evaluated are very high for the construction of turbines that possibly would not operate properly.Figure 7 reports the results for the NACA 0025 blade.The TSR range, as in the previous study, was established to find negative torque associated with excessively high angular velocities.Thus, with solidity 0.5 TSR was evaluated between 2 and 6 with maximum between TSR 3 and 4, with solidity 1.0 TSR was evaluated between 1 and 4 with maximum in TSR 2 and with solidity 1.5 TSR was evaluated between 1 and 3 with maximum in TSR 2. As for the TSR associated with the maximum value of Cp there is agreement with the previous study for each solidity, this time the Betz limit is not exceeded, in general the Cp values for NACA 0025 are lower than for NACA 0018.In the previous study the Cp was decreasing with the increase in solidity, now there is an increase from 0,5 to 1 with subsequent decrease between 1 and 1,5; being again the minimum in the greater solidity.Where the TSR range for solidity of 0,5 was 1 to 7 with an intermediate value of 3,5; this in order to observe in maximum Cp reached by the turbine under the configuration conditions, as was done in the study of the NACA 0018 blade.The same coincidence was presented in the range of TSR with the other solidity values.Reading the parabolic behavior of the results, and the range of RRT associated with the maximum Cp, the similarity in the results is preserved, but, contrary to the previous study, the lowest value is reported for the intermediate solidity of 1; even so, Cp values remain higher for NACA 0018.Again, a rape of the Betz limit with the solidity of 0,5 is reported.
Figure 8 presents the results of the simulations carried out with the NACA 2415 blade.Where the TSR range for solidity of 0,5 was 1 to 7 with an intermediate value of 3,5; this in order to observe in maximum Cp reached by the turbine under the configuration conditions, as was done in the study of the NACA 0018 blade.The same coincidence was presented in the range of TSR with the other solidity values.Regarding the parabolic behavior of the results, and the range of TSR associated with the maximum Cp, the similarity in the results is preserved, but, contrary to the previous study, the lowest value is reported for the intermediate solidity of 1; even so, Cp values remain higher for NACA 0018.Again, a rape of the Betz limit with the solidity of 0,5 is reported.The results presented in Figura 9, for NACA 4415 blade, were evaluated in a range of TSR 2 to 5 for solidity values of 1and 1 to 4 for solidity of 1,5; with maximum Cp in TSR 2 as usual, while for solidity of 0,5 the range of TSR was from 1 to 7, with maximum Cp in TSR 4 that differs from the previous blades.All The Cp respect the Betz limit, and their behavior with respect to solidity is similar to that of NACA 2415 when the minor presented with solidity 1.In general, it can be seen that the Cp values for each of the simulated blades achieve greater results with the symmetric profile NACA 0018 and the asymmetric NACA 2415 with low solidity (0.5), but the symmetric profile reports a slightly higher value, also the negative torque is higher, this may be because the NACA 0018 blade has more mass (inertia) than the NACA 2415 blade.The most robust blades, the symmetric NACA 0025 and the asymmetric NACA 4415, showed lower Cp, with the symmetric blade with solidity 1 being the highest reported among them.It follows that independent of the symmetry of the blade better efficiencies can be obtained with thinner profiles, but the symmetric profile reports higher Cp.This can also be evidenced by the fact that increasing the solidity increases the inertia of the rotor and, in most cases, the Cp with solidity 1,5 was lower than the Cp with solidity 0,5.

CONCLUSIONS
In the present work, the comparative evaluation of four hydrodynamic profiles was carried out for an H-Darrieus turbine at a constant water flow rate of 1.0 m/s under different TSR values.The numerical results of the study of Cp against TSR are reported.This in order to observe the natural behavior of this type of turbines, with maximum values of Cp between 0,38 and 0,61 and the negative minimum values that correspond to excessively high rotational speeds.From the point of view of turbine design this event must be kept in mind, because by increasing this type of opposing forces, greater will be the loads on the blades and central axis of the turbine.
The typical parabolic behavior is evidenced with maximum values in the evaluated range of TSR between 1 and 7 for the four (4) blades, in some cases it was necessary to evaluate in intermediate value of 3,5 and other higher TSR values in order to see the maximum behavior and when the Cp values begin to decrease to where negative values are found.Maximum Cp that exceeded the Betz limit require experimental validation.The claim that increasing Solidity increases Cp as stated in the literature cannot be corroborated, there were cases that gave similar results to those reported by Shiono et al. (Shiono et al., 2000), so studies, both numerical and experimental, should be deepened in wider ranges of solidity to determine a correlation between the parameters.
From the results obtained from the four modeled blades, it is obtained that the NACA 0018 blade and 2415 presented better values of maximum Cp for the solidity of 0,5; while for the solidity of 1 the blades that reached maximum values of Cp were the symmetric blades NACA 0018 and 0025, these results in accordance with what was reported by Hashem & Mohamend (Hashem & Mohamed, 2018) who stated that symmetrical blades perform better in terms of power coefficient, compared to asymmetric blades.While for the solidity of 1,5 the NACA 2415 and 4415 blades reached higher maximum Cp values, compared to the other two blades evaluated.These results were achieved in a range of TSR values of 2 to 4 with a mean value for NACA 0018 blades and 2415 of 3,5.
fluid dynamics (CFD) is one of the most widely used tools for solving and analyzing the behavior of fluid flow around a body with the help of computing machines.

Figure 2 .
Figure 2. 2D geometry of the turbine and the Boundary condictions used.Source: Prepared by the authors (2023).

Figure 3 .
Figure 3. (a) Blocks of the mesh model for stationary domain.(b) Meshing of the stationary domain.Source: Prepared by the authors (2023).

Figure 4 .
Figure 4. (a) Mesh model blocks for the rotational domain.(b) Meshing of the rotational domain.Source: Prepared by the authors (2023).

Figure 5 .
Figure 5. Contours of speed and pressure around the turbine blades, for the Cpmax.Source: Prepared by the authors (2023).

Figure 8
Figure 8 presents the results of the simulations carried out with the NACA 2415 blade.Where the TSR range for solidity of 0,5 was 1 to 7 with an intermediate value of 3,5; this in order to observe in maximum Cp reached by the turbine under the configuration conditions, as was done in the study of the NACA 0018 blade.The same coincidence was presented in the range of TSR with the other solidity values.Reading the parabolic behavior of the results, and the range of RRT associated with the maximum Cp, the similarity in the results is preserved, but, contrary to the previous study, the lowest value is reported for the intermediate solidity of

Table 2 .
Experiment settings: Angular velocity and number of steps for different TSR.