Assessment of H-Darrieus Hydrokinetic Turbines by Configuration with a Blocking Plate Using a Design of Experiment Approach

2.1 Theoretical framework

Available kinetic energy P_Tin the water is estimated using Eq. (1) [23]:

$P_T=\frac{1}{2} \rho A U^3$     (1)

where, ρ is the density, A is the projected rotor area, and U represents the free flow velocity. As the turbines do not have the capacity to extract all the available energy from the water, it is necessary to consider the power coefficient C_P, to estimate the power that is extracted by the shaft in an interval of a water current that passes through the cross-section A of the rotor [24], as shown in Eq. (2).

$P_{{extracted }}=C_P\left(\frac{1}{2} \rho A U^3\right)$   (2)

Solving C_Pfrom Eqs. (1) and (2) gives the power coefficient as:

$C_P=\frac{P_T}{\frac{1}{2} \rho A U^3}$   （3）

where, P_T is the mechanical power, where M is the momentum and $\omega$ the angular velocity of the turbine shaft. Additionally, the moment coefficient C_m is the amount of mechanical energy that can be obtained from the turbine per unit of kinetic energy of the water passing through it.

$C_M=\frac{M}{\frac{1}{2} \rho A U^2}$ （4）

Another dimensionless parameter is the solidity σ which describes the relationship between the front surface of the blades and the surface of the rotor disk. Eq. (5). shows its definition, where N is the number of blades, c is the chord length and R is the radius of rotation of the turbine.

$\sigma=\frac{N c}{R}$  (5)

2.2 Design of experiment and control surface

For the testing development, DoE: General Full Factorial Design was proposed, where an ANOVA analysis method was implemented. ANOVA allows for the statistical analysis of how independent variables affect a dependent variable. The DoE was conducted with two factors: solidity and position of the blocking plate. Where each factor was tested at two and three levels, respectively. The experiment was carried out with a repetition.

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Figure 1. Schematic view of domains and boundary conditions

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Figure 2. Main dimensions and configuration of control surfaces

Three specific cases were configured in the ANSYS R2024® Space Claim module: Case A serves as baseline case without blocking plate, while Case B and Case C have the blocking plate at different distance from the rotor. Figure 1 shows the dimensions of the background grid as demonstrated by Munoz et al. [25] and the contour conditions which are the same for all the configured cases. Figure 2 illustrates the rotor and blocking plate design parameters.

Table 1 and Table 2 show the parameters considered, their symbology and the values taken by the DoE, according to the case, which was granted for the design of the rotor and blocking plate. Furthermore, plates were located according to Patel et al. [19] the positions are taken based on the diameter of the rotor. On the other hand, solidity was taken from the case of the previous study [25].

Table 1. General design parameters

Table 2. Design of experiments

2.3 Mesh generation and boundary conditions

The computational domains were performed in ICEM CFD. Structured grids were created with quadrilateral elements with separate domains to implement the overset method. Overset allows you to control the cells of each component and their respective movements. Figure 3 (a) shows the background grid which the blocking plate that corresponds to the stationary domain of the model, in the same way, it has refinement in the part where the rotating domain is located. Figure 3 (b) shows the rotor mesh which is configured as a rotary domain and Figure 3 (c) shows the discretization of the blade, it is configured as a rotary domain, but its movement is relative to the rotor.

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Figure 3. Discretization of control surfaces: a) Background grid mesh; b) Rotor mesh; c) Blade mesh

The metrics of the meshes made are illustrated in Table 3, which are in the acceptable ranges according to the ANSYS® user manual [13, 26, 27]. According to these references, the determinant must be greater than 0.3 and the aspect ratio must be between 1 and 300 depending on the type of simulation to guarantee an adequate mesh quality. These values guarantee the accuracy and stability of the results in the simulations carried out.

Table 3. Grid details and metrics

The simulation was carried out in the ANSYS Fluent®, where the system was evaluated in the transient state with the standard turbulence model. According to the study of Munoz et al. [25], a convergence criterion of 1E-4 was configured. The total simulation time is 10 s with a time step of 0.005 s [26]. Table 4 summarizes the conditions configured in the simulation.

Table 4. Simulation parameters

k-ε Realizable turbulence models were used in this study because they correctly predict the behavior of turbines, and their computational performance compared to other turbulence models available in solvers. Moreover, the k-ε Realizable model improves upon the standard k-ε model by incorporating a more physically accurate formulation for turbulent viscosity and an improved dissipation rate equation, allowing it to better capture flow features such as boundary layer behavior and rotational flows [27-32]. The transport equations for k and ε in this turbulence model are given by Mohamed et al. [33]:

$\begin{gathered}\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_i}\left(\rho k U_j\right)=\frac{\partial}{\partial x_i}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right) \frac{\partial \kappa}{\partial x_j}\right] \\ +\cdots G_k+G_b-\rho \varepsilon-Y_M+S_k\end{gathered}$    (6)

$\begin{array}{r}\quad \frac{\partial}{\partial t}(\rho \varepsilon)+\frac{\partial}{\partial x_j}\left(\rho \varepsilon U_j\right)=\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_{\varepsilon}}\right) \frac{\partial \varepsilon}{\partial x_j}\right] \\ +\cdots \rho C_1 S_t-\rho C_2 \frac{\varepsilon^2}{k+\sqrt{v \varepsilon}}+C_{1 \varepsilon} \frac{\varepsilon^2}{\kappa} C_{3 \varepsilon} G_b+S_t\end{array}$      (7)

where,

$C_1=\max \left[0.43 \frac{\eta}{\eta+5}\right]$   （8）

$\eta=S \frac{k}{\varepsilon}$    (9)

$S=\sqrt{2 S_{i j} S_{i j}}$   (10)

where, $G_k$ is the turbulent kinetic energy due to the velocity gradient and $G_b$ is the turbulent kinetic energy due to the buoyancy force. $Y_M$ is also the contribution of pulsatile diffusion to compressible turbulence. The total dissipation rate $\sigma_k$ and $\sigma_{\varepsilon}$ are the turbulent Prandtl numbers $k$ and $\varepsilon$, respectively, $S_k$ and $S_{\varepsilon}$ are user-defined source expressions.

Numerical study was carried out implementing statistical tools such as DoE-ANOVA. General factorial experimental design with two factors was configured with two and three factors. The study was conducted in ANSYS Fluent ® R2024, six configurations were simulated with a replica. The performance of each rotor was determined in terms of power and momentum coefficients, and it was concluded that:

Using Minitab® software, the statistical analysis of the experiment was carried out with two factors: solidity and blocking plate distance; the response variable analyzed was the power coefficient. From the statistical analysis it was found that both factors are significant to improve the power coefficient, likewise, it was found that the lowest solidity of 1.35 with case C (blocking plate at 0.665 m) is the best configuration. These data coincide with the numerical analysis carried out; on the other hand, the behavior of the total moment coefficient of the turbine was also analyzed and it was found that there is also a significant improvement in the minimum moment coefficient generated, which can directly impact the capacity of self-starting of the turbine and improving its cyclic behavior.

This research concludes that the DoE-ANOVA method is an effective strategy for understanding the effect of a variable on the behavior of the hydraulic and mechanical properties of H-Darrieus turbines. Where the use of statistical analysis along with simulation tools are shown as powerful tools to solve the turbine design problem, where the number of possible solutions to be evaluated is reduced in order to find an adequate solution. These methods help ensure that simulation results are not only accurate but also reliable, by distinguishing between real physical effects and random noise or numerical errors.

In future studies, it is possible to consider a regression model that allows considering the variation of the TSR and the different positions and geometries of the locking plate, with the aim of minimizing the negative torque. It is also possible to develop a model that considers all the physical components of the HDHT in a three-dimensional model that allows the validation of the computational data in experimental configurations.