Experimental Optimization of the Propeller Turbine Performance Using the Response Surface Methodology

Abstract

The growing global energy demand necessitates a shift towards sustainable sources to mitigate environmental issues and ensure energy security. This work explores the design and optimization of propeller-type hydrokinetic turbines to efficiently harness renewable energy from water currents. Through experimental testing and regression modeling, the research aimed to maximize the power coefficient ($C_p$) by determining the optimal values of the number of blades (Z) and the turbine diameter to hub diameter ratio ($d/D$). By correcting for experimental biases, the study elucidates the importance of factors such as the blockage ratio and turbine configuration on its performance. A second-order polynomial regression model, which was validated through analysis of variance, determined that when Z and $d/D$ were set at 4 and 0.15, respectively, the optimal value for Cp was 53.62%. These findings provide valuable insights for optimizing hydrokinetic turbine efficiency, contributing to the advancement of renewable energy technologies.

1. Introduction

Energy plays a crucial role in the operation of systems that cover human needs, including food, shelter, work, and transportation [1,2,3]. Across the globe, many individuals suffer profound lifestyle changes as subsistence economies give way to industrial or service-based economies [4,5]. This transition in the global economic structure implies a greater demand for energy to meet the needs of the developing industrial sector, which is subsequently accompanied by a significant increase in population. According to projections, by 2050 the world will host nearly 9.6 billion inhabitants [6,7]. In this regard, energy consumption projections indicate that by 2040, energy demand will grow at an average annual rate of 1.1% [8].

Given the potential future energy scenario and energy demand projections, the looming energy deficit raises concerns in the international community, primarily regarding the energy sources required to meet current and future resource needs [9,10]. Traditionally, demand has been primarily met by fossil fuels (coal, oil, and natural gas), which are considered versatile, as well as easy to handle, transport, and store. Additionally, these fuels possess high energy concentrations, meaning that a significant amount of energy can be stored in a relatively small container for long periods [11]. Nevertheless, the adverse effects caused by combustion processes associated with the use of these fuels are well known today, focusing on the notable emission of greenhouse gases (GHG) that strongly contribute to the greenhouse effect on the planet and leading to destabilized temperature conditions on Earth.

With all the challenges outlined above, the production of energy in the short and medium terms requires a greater utilization and harnessing of resources derived from sources whose renewal cycle follows natural processes that occur indefinitely, sustainably, and with minimal environmental impact on ecosystems. Therefore, the solution lies in the energy offered by nature through water, wind, sunlight, volcanic power, and wave force. All these mentioned energies are commonly referred to as renewable energies, which fundamentally differ from fossil fuels in their diversity, abundance, and potential for utilization anywhere in the world [12]. Despite their generation potential, the main drawback of harnessing these sources on an industrial scale lies in the high variability of the primary source, which is associated with the geographic availability of the resource [13].

Hydrokinetic energy is a renewable energy generated from the movement of water in rivers, streams, or tides, without needing large constructions or altering water flow [14]. Unlike solar or wind energy, water currents are more consistent and predictable, making hydrokinetic energy suitable for many locations worldwide [15]. A hydrokinetic turbine is specifically designed to harness the kinetic energy from these water flows, as opposed to turbines in hydroelectric dams, which rely on pressure differences.

Hydrokinetic turbines are primarily composed of a rotor or impeller and a generator, with the rotor being the most crucial component of the generation system. There are various types of hydrokinetic turbines and different principles for harnessing the energy contained in the fluid. The most general classification depends on the axis orientation relative to the flow of the fluid. Accordingly, hydrokinetic turbines are classified according to a vertical and horizontal axis (VAHT and HAHT, respectively) [16,17]. VAHTs have a rotational axis that is perpendicular to the water flow, with the blades rotating in a plane parallel to the flow. Examples include Darrieus and Gorlov turbines, where the blades are designed to capture energy from water flow coming from multiple directions. VAHTs are less sensitive to the direction of water flow, do not require alignment with the water current to function effectively, and tend to have lower energy efficiency compared to HAHTs [15]. HAHTs have a rotational axis that is parallel to the water flow. They operate similarly to horizontal-axis wind turbines, where the turbine blades rotate in a plane perpendicular to the flow. A common example of this type of turbine is the propeller turbine, where water flows through the blades, causing them to rotate. HAHTs require precise alignment with the water current to operate optimally and may be more susceptible to damage if the direction of water flow changes abruptly [14].

Despite the various existing technologies that harness hydrokinetic energy to generate electricity, they often lack the necessary conditions to achieve optimal efficiency and maintain a good cost–benefit ratio. Currently, there are very few studies conducted on propeller-type hydrokinetic turbines, as it is a technology in its early stages [18].

This work focuses on propeller hydrokinetic turbine design optimization, whose performance relies on the interaction between the fluid flow and the blades. The primary objective is to maximize the flow energy conversion into mechanical energy at the turbine’s shaft. To achieve this, the ideal geometric parameters that enable the generation of a desired electrical power were determined, considering factors such as fluid velocity and losses associated with system efficiency. Furthermore, an optimization methodology based on a full-factorial experimental design was utilized to assess the effect of two geometric factors: the rotor diameter (D) to hub diameter (d) ratio ($d/D$) and the blade number (Z) on the turbine’s power coefficient ($C_p$). Our experimental study was conducted in order to obtain an optimal design that maximizes the $C_p$ of the hydrokinetic turbine. This will enable the construction of larger and more efficient models to meet energy needs, including at the domestic level.

2. Methods and Materials

2.1. Turbine Principles

Hydrokinetic turbines play a crucial role in promoting sustainability by converting the kinetic energy from water currents into renewable electricity. By optimizing rotor design, these turbines can be made more efficient, leading to numerous environmental and economic advantages. Enhanced rotor efficiency allows for the use of fewer materials in their construction, which reduces the environmental footprint by decreasing the need for raw material extraction and processing, thereby lessening habitat disruption and pollution.

Incorporating eco-friendly materials and advanced manufacturing techniques into the optimization process further mitigates environmental impacts. Local manufacturing can significantly cut transportation emissions and reduce the overall carbon footprint of production while simultaneously supporting local economies and creating jobs.

Moreover, optimized turbine designs often result in longer-lasting and more reliable equipment, which means less frequent maintenance and replacements are needed. This durability helps to minimize waste and conserves resources over the turbine’s operational lifetime.

Hydrokinetic turbines operate under the same principles as wind turbines. They are equipped with blades that capture the moving water kinetic energy. The blades are assembled concentrically and securely attached to a shaft. When the water stream impacts the blades, they begin to rotate. The design of the blades is optimized to capture the maximum amount of water kinetic energy. The rotor rotational motion is transferred to a generator through the shaft. The generator converts the rotation mechanical energy into electricity [19]. Their design begins with sizing the rotor, starting from the available power (P) of the turbine, as described by Equation (1) [20]:

$$P={\displaystyle \frac{\rho A{V}^3}{2}}={\displaystyle \frac{\rho \left(\pi R^2\right)V^3}{2}}$$

(1)

where $\rho$ is the fluid density, A is the area of the cross-sectional swept by the blade, and V is the velocity of the fluid. For HAHTs, A is equal to $\pi R^2$, where R is the radius of the rotor. The output power ($P_{out}$) is given by Equation (2) [20]:

$$P_{out}=T\omega$$

(2)

where T is the moment generated on the surface of the blades due to the flow of fluid around them, and $\omega$ is the angular velocity. The relation between available and output power is the $C_p$, which can be determined by Equation (3) [20]. This variable accounts for the hydraulic efficiency of the turbine. According to the one-dimensional actuator disc theory proposed by Betz, the value of this coefficient has an ideal maximum limit of 16/27 or 59.26% [21,22]. The $Cp$ is also used to compare different designs and configurations of hydrokinetic turbines. A high value of $C_p$ indicates a high efficiency in energy conversion and is desirable to maximize electricity production. Therefore, designers and researchers are constantly seeking to improve the $C_p$ value to make hydrokinetic turbines more efficient and cost-effective [23].

$$C_P={\displaystyle \frac{P_{out}}{P}}={\displaystyle \frac{2T\omega }{\rho \left(\pi R^2\right)V^3}}$$

(3)

2.2. Turbine Design

The prototype of the propeller hydrokinetic turbine was designed for a V of 0.5 m/s and a D of 0.12 m. The initial model in this study corresponds to the experimental model used by Romero-Menco et al. (2024) [24]. Romero-Menco et al. (2024) utilized CFD simulations and experimental tests to ascertain the optimal values of the skew ($\phi$) and rake ($\gamma$) angles for maximizing the $C_p$. The highest $C_p$ value was 0.4571 and was achieved when $\gamma$ and $\phi$ were −18.06° and 13.30°, respectively. The model consisted of a hub and three blades. The diameter of the hub was 0.048 m ($d/D=$ 0.20). The initial model is shown in Figure 1.

Another parameter to consider in the design of propeller turbines is the blade pitch. The blade pitch refers to the angle of the blades relative to the direction of the water flow. It determines how the angle of approach affects lift and, consequently, the efficiency of the turbine. For the current prototype, the blade pitch was set to zero, as per the specifications detailed in Chica et al. (2017) [25]. This setting simplifies the design and focuses on optimizing other parameters.

The performance of a propeller can be significantly influenced by its propeller hub, which plays a crucial role in shaping the water flow distribution around the blades. The hub dimensions and design have a direct impact on the influence of water flows from and to the propeller blades on the propeller’s efficiency in generating thrust [26]. A well-engineered hub can minimize turbulence and resistance, facilitating smoother water flow over the blades and, consequently, enhancing efficiency. Conversely, a poorly undersized or designed hub can lead to energy loss and resistance increase, thereby diminishing the overall propeller performance [27,28].

With this initial baseline design of the propeller and the geometric parameters of interest, various geometric configurations of the propeller were studied to find the optimal configuration providing the highest $C_p$. To find this optimal configuration, an optimization methodology was employed to define the design factors of this type of turbine and to study their influence and interaction with the response variable. The selected optimization methodology was the response surface methodology (RSM), which allows modeling and analyzing problems where multiple factors simultaneously affect one or more response variables [29].

The RSM was chosen due to its robustness in handling complex, nonlinear relationships between design parameters and performance metrics [30]. It is particularly advantageous for experiments where the interactions between factors are crucial to understanding system behavior, as in the case of hydrokinetic turbine design. Other optimization methodologies, such as genetic algorithms (GAs) and particle swarm optimization (PSO), are also widely used for similar problems. While GA and PSO explore large, multi-dimensional design spaces and finding global optima, they often require higher computational resources and may converge more slowly in the absence of precise model guidance [31,32]. In contrast, the RSM focuses on building an approximate model of the response surface using a relatively smaller number of experiments, making it efficient and practical for this application. Additionally, the RSM provides clear insights into the interaction between design variables, which can be more challenging to extract from population-based methods like GAs and PSO. Therefore, the RSM was considered the most suitable methodology for this research due to its balance between computational efficiency and the ability to capture critical factor interactions that influence turbine performance.

There are several experimental designs that allow for discerning the effect of various factors on the response variable at the same time. In this work, the full-factorial design was selected. As independent factors of the experimental design, the geometric parameters $d/D$ and Z were selected. The selected geometric factors were assessed at three levels (low, medium, and high levels corresponding to −1, 0, and +1, respectively), as summarized in

Table 1. Values of the independent parameters used for the propeller turbine optimization.

Independent Factor	Values
	−1	0	1
$d/D$	0.15	0.20	0.25
Z	2	3	4

.

For a full-factorial design of two factors and three levels per factor, the number of experimental models or runs (N) is given by Equation (4) [33]:

$$N=n^k$$

(4)

Table 2. Design matrix and 3D views.

Run 1	Run 2	Run 3
$d/D=0.20$ Z = 4	$d/D=0.15$ Z = 2	$d/D=0.25$ Z = 4
Run 4	Run 5	Run 6
$d/D=0.20$ Z = 3	$d/D=0.15$ Z = 4	$d/D=0.25$ Z = 3
Run 7	Run 8	Run 9
$d/D=0.25$ Z = 2	$d/D=0.15$ Z = 3	$d/D=0.20$ Z = 2

lists the values of the independent parameters for the nine runs and a view of the models. Mathematically, the current optimization problem was represented by Equation (5):

$$\begin{array}{c}\hfill \mathrm{Maximize}{C}_p\\ \hfill \mathrm{Subject}\mathrm{to}:\\ \hfill 0.15\le d/D\le 0.25\\ \hfill 2\le Z\le 4\end{array}$$

(5)

Once the treatments were defined, and to determine the effect of the different design parameters on the propeller’s performance, experimental tests were conducted in a water channel.

3. Experimental Test

3.1. Manufacturing via 3D Printing

Before starting with the experimental tests, it is necessary to manufacture each of the models. The propellers were fabricated using fused deposition modeling (FDM), a widely used 3D printing technique in which the material is heated and extruded through a nozzle, layer by layer, to create the desired geometry. FDM was chosen for its accessibility and the ability to produce high-strength parts with precise dimensional accuracy. The use of 3D printing is a manufacturing process that creates 3D objects layer by layer from a digital model. The use of 3D printing utilizes a wide variety of materials, ranging from plastics and resins to metals and ceramics, to construct objects with complex shapes and customized structures. Since the complete piece was too large to print at once, it was decomposed into four distinct parts—the hub, the fixation cap, and the blades—as illustrated in Figure 2. Each of these parts was designed and prepared individually in CAD software, ensuring they fit perfectly once printed and assembled.

The material chosen for the manufacturing of the rotors was PETG. PETG, or glycol-modified polyethylene terephthalate, is a type of engineering plastic widely used in additive manufacturing, especially in 3D printing [34]. It is derived from the commonly known PET (polyethylene terephthalate), used in water and soda bottles, that has been modified with glycol to enhance its properties, such as temperature resistance and durability. PETG is recognized for being a versatile material that combines the characteristics of polylactic acid (PLA) and acrylonitrile butadiene styrene (ABS), offering excellent mechanical strength, good layer adhesion, and resistance to heat and chemicals. Its strength and durability make it suitable for withstanding the forces and tensions associated with the rapid rotation of propellers [35,36]. The mechanical properties and printing parameters of PETG are shown in

Table 3. Mechanical properties and printing parameters of PETG.

Mechanical Properties
Tensile strength	60 MPa
Elongation at break	6.80%
Impact resistance	10 kJ/m2
Density	1.27 kg/m3
Printing parameters
Extrusion temperature	220–260 °C
Printing speed	150 mm/s
Bed temperature	60–80 °C

. The printer used in the manufacturing process was the Creality K1 Max printer. The infill percentage was adjusted to 100% to balance strength and material usage, and a layer height of 0.2 mm was chosen to ensure good surface quality. These parameters were selected to optimize both the mechanical performance and material efficiency of the blades.

In Table 2, the printed models of the 9 test models are depicted.

3.2. Data Acquisition System

This system consists of a torque sensor with an encoder and a direct current (DC) motor, as observed in Figure 3. The Futek torque sensor (TRS 605-FSH02052) is a rotary torque sensor with a measurement range of 0–1 Nm. This type of sensor is used to measure the torque applied between two rotating shafts without the need for physical contact with them. The built-in encoder provides information about the angular position of the shafts, allowing for real-time precise measurement of the rotary torque. The DC motor used in this setup is a 6 V DC motor with a gear reducer, which has a torque rating of 1.8 kg-cm. The DC motor was connected to a power source that supplied both current and voltage, thereby increasing the load on the motor. The motor was configured to rotate in the opposite direction to the rotor. As the load increased, it generated sufficient braking force to slow down the rotor, allowing for accurate measurement of the torque produced by the turbine at various rotational speeds.

3.3. Experimental Setup

The experimental tests were carried out in the laboratory of the Alternative Energy Research Group at the University of Antioquia, which features a recirculating water channel with a cross-section of approximately 0.350 × 0.265 m and a length of around 5 m. The recirculating water channel (Figure 4) comprises a suction tank (1) and a discharge tank (11) connected by a suction pipe (2) and equipped with two gate valves (3 and 9). Additionally, there is an eccentric reducer of 10 × 6 inches for suction (4), a centrifugal pump (5), and a motor (6). The discharge side includes a concentric reducer of 8 × 5 inches (7), a check valve (8), and a discharge pipe (9). The selected centrifugal pump is the GRUNDFOS model NK 125-200/176-154 EUP A1F2AE-SBAQE, with a hydraulic capacity of 1200 GPM @ 9.8 m. The pump connections feature DN 150 × 125 DIN flanges, while the mechanical seal consists of carbon/silicon/EPDM, and the casing and impeller are constructed of cast iron. The motor specifications include a power rating of 15 HP and an operating speed of 1800 rpm. The water flow in the channel was controlled using speed controllers connected to a programmable logic controller (PLC), allowing for precise flow rate adjustments. A target flow velocity V of 0.5 m/s was set, which was verified using an handheld flow meter (model FLOWATCH^®, Geneva, Switzerland ), with an accuracy of ±0.2%. The water level was measured manually using a tape measure. The velocity and water level were taken at multiple points along the channel to ensure consistent flow conditions. In the turbine characterization process, three experimental tests were conducted for each of the configurations described in previous sections. During each test, the turbine’s rotational speed, torque, and power output were measured using a torque sensor (TRS 605-FSH02052), with a sampling frequency of 100 Hz to capture high-precision data. The data acquisition system (DAQ) was an IHH500 Pro module, ensuring reliable data logging and analysis throughout the tests.

The experimental setup utilized for the optimization of the propeller turbine is depicted in Figure 5. The channel was utilized to test the 9 fabricated experimental configurations. It is important to note that the water used in the experiments was clean and free of sediments, ensuring accurate measurement of the turbine’s performance. Additionally, the system’s pumps and tanks were designed to facilitate feedback, allowing for the efficient reuse of water and minimizing waste during the testing process.

4. Results

In Figure 6, the results obtained for the nine test models as a function of the blade tip speed ratio ($TSR$) and $C_p$ are shown. The $TSR$ coefficient is the ratio between the tangential speed ($\omega R$) at the blade tip and the actual fluid speed (V), as expressed by Equation (6) [37,38]:

$$TSR={\displaystyle \frac{\omega R}{V}}$$

(6)

The solid red line in the graph marks the Betz limit. According to this limit, the kinetic energy maximum percentage that can be converted into mechanical energy has been determined at approximately 59.3% [39]. When water flows through the channel towards the turbine, the channel walls may interfere with the flow, causing some of the water to be blocked or diverted before reaching the turbine [40]. This can result in a nonuniform flow distribution and affect the accuracy of experimental measurements, as evidenced in Figure 6, where some of the models exceeded the Betz limit [41]. To correct this effect, adjustments to the experimental data are necessary. This may involve applying corrections to compensate for the influence of the channel walls on flow velocity [42,43]. Without these corrections, the experimental data could be biased and may not accurately reflect the actual performance of the turbine under operational conditions.

The blockage ratio ($BR$) is calculated using the projected area of the turbine within the test section rather than the actual surface area of the turbine itself. This distinction is crucial, as the projected area represents the portion of the flow that the turbine intercepts in the testing environment, which directly influences the blockage effect. However, for reference,

Table 4. Surface area values for each model.

Run	$\mathit{d}/\mathit{D}$	Z	Surface Area [m2]
1	0.20	4	0.04242
2	0.15	2	0.02259
3	0.25	4	0.04306
4	0.20	3	0.03317
5	0.15	4	0.04171
6	0.25	3	0.03421
7	0.25	2	0.02537
8	0.15	3	0.03215
9	0.20	2	0.02393

has been included with the surface area values for each model.

The surface areas of the turbines exhibited considerable variation across the different models, ranging from 0.02259 m² to 0.04306 m². This variation was primarily driven by differences in key turbine parameters, such as the diameter ratio ($d/D$) and the number of blades (Z). Although the blockage factor was determined using the projected area rather than the surface area, understanding these surface area differences is crucial for assessing the overall hydrodynamic behavior of the turbines. Turbines with larger surface areas are likely to capture more energy due to increased interaction with the flow, which can be beneficial for maximizing power output. However, this comes at the cost of potentially greater hydrodynamic resistance, which could negatively impact efficiency and increase structural stress. On the other hand, turbines with smaller surface areas may experience reduced resistance, leading to more streamlined flow and potentially lower operating costs, but they might also capture less energy, limiting their overall performance. Additionally, these variations in surface area could influence the flow dynamics around the turbines, affecting wake formation and turbulence intensity. Larger surface areas might generate stronger wakes, which could impact downstream turbines in an array, while smaller surface areas could contribute to a more stable and uniform flow field.

According to the $BR$ definition (Figure 7), where the BR is >10%, the wall effects cannot be ignored [44]. With a projected area of 0.045 m² and a test area of 0.0927 m², the BR comes out to 48.77%; hence, a correction procedure is required.

In this regard, Pope and Harper’s method [45,46] was used. This method obtains the blockage effect by multiplying the input flow V with the correction factor, as expressed in Equation (7) [45,46]. Here, $U_c$ is the corrected velocity (m/s):

$$U_c=V(1+{\displaystyle \frac{1}{4}}BR)$$

(7)

Figure 8 shows the corrected results obtained for the nine test models. In turn,

Table 5. Experimental results obtained in the matrix domain.

Run	$\mathit{d}/\mathit{D}$	Z	${\mathit{C}}_{\mathit{p}}$ Experimental [-]	${\mathit{C}}_{\mathit{p}}$ Model [-]	Residuals	TSR Experimental [-]
1	0.20	4	0.3936	0.4111	−0.0175	3.7692
2	0.15	2	0.3089	0.3034	−0.0054	4.7743
3	0.25	4	0.4770	0.4646	0.0126	4.2717
4	0.20	3	0.4044	0.3682	0.0361	4.2717
5	0.15	4	0.5412	0.5362	0.0049	4.2717
6	0.25	3	0.4033	0.4290	−0.0257	4.0204
7	0.25	2	0.2743	0.2611	0.0131	3.5179
8	0.15	3	0.4756	0.4860	−0.0104	4.2717
9	0.20	2	0.1743	0.1929	−0.0186	5.7794

lists the highest $C_p$ values for each configuration and their corresponding $TSR$ values. The maximum $C_p$ was achieved through treatment 5, leading to a $C_p$ equal to 0.5412 at a $TSR$ value equal to 4.2717.

In examining the other treatments, several configurations demonstrated promising $C_p$ values. For instance, treatment 3 achieved a $C_p$ of 0.4770 with the same $TSR$ of 4.2717 as treatment 5, highlighting that minor adjustments in design parameters can lead to competitive performance. Meanwhile, treatment 1 and treatment 4, with $C_p$ values of 0.3936 and 0.4044, respectively, exhibited lower efficiency, suggesting that the parameter combinations used in these cases may not optimize the turbine’s performance as effectively.

Additionally, the residuals column, from Table 5, indicates the differences between the experimental and modeled $C_p$ values, providing a measure of model accuracy. For example, treatment 5 showed a minimal residual of 0.0049, reinforcing the reliability of the model in predicting performance for this configuration. In contrast, treatments 6 and 7 presented larger residuals, indicating potential discrepancies between the model predictions and experimental outcomes, which could be explored further to refine the optimization model.

5. Discussion

Based on the corrected data obtained, a regression model was developed. This model was used to analyze the relationship between ($C_p$), Z, and $d/D$, and it was also utilized for estimating the value of $C_p$ based on the values of the independent factors evaluated. Equation (8) represents a polynomial regression model, where y refers to $C_p$; ${\beta }_0$ is a constant; ${\beta }_{11}$ and ${\beta }_{22}$ correspond to the linear coefficients related to the conside independent factors; ${\beta }_{11}$ and ${\beta }_{22}$ represent the quadratic coefficients; and ${\beta }_{12}$ describes the interaction effects of the independent factors to be assessed.

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_{12}{x}_1{x}_2+{\beta }_{11}{x}_1^2+{\beta }_{22}{x}_2^2$$

(8)

The quadratic polynomial model allows for capturing not only the linear effects of the variables on the response but also nonlinear interactions and curvatures. Thus, a comprehensive analytical representation of how the $C_p$ is affected by the factors in their analyzed ranges of variation was obtained. It is notable that various regression models can be employed; however, second-order polynomial regression models are commonly used [47,48,49], particularly within the turbine design [33,50,51]. Additionally, in the numerical results, a quadratic curvature in the $C_p$ values was observed, supporting the argument related to the use of a second-order regression model.

Residual normality, independence, and homoscedasticity were the hypotheses checked to know whether the experimental data were represented by the regression model constructed [52]. Residuals determine the difference between the experimental values and those predicted by the model. A model with small residuals indicates that it is fitting the data well, while large residuals may indicate that the model is not fully capturing the relationship between variables or that there is some pattern not captured in the data [53]. The residuals are listed in Table 5.

The normality test of residuals is used to check whether residuals follow a normal distribution. This test ensures that the estimates of the model parameters are reliable and that the inferences made from the model are valid [54]. There are several statistical tests that can be used for this purpose, such as the Shapiro–Wilk, Kolmogorov–Smirnov, and Anderson–Darling tests, among others [55]. These tests compare the residuals to a theoretical normal distribution and determine if there is sufficient evidence to reject the null hypothesis that the residuals come from a normal distribution.

When a normal distribution is not followed by the residuals, it may indicate that the regression model is not suitable for the data or that important aspects of the relationship between variables are being overlooked. In such cases, modifications to the model or analysis may be necessary [54,56]. The p-values associated with Shapiro–Wilk (0.6130), Kolmogorov–Smirnov (0.5359), and Anderson–Darling (0.7784) tests were >0.05; therefore, the data do not contradict the assumption of normality [57]. Figure 9 provides visual evidence of the normality test for the statistical analysis.

The residual independence test is used to verify whether the residuals of a regression model exhibit any correlation or autocorrelation pattern [58]. Autocorrelation in the residuals indicates that the observations are dependent on each other. Typically, the Durbin–Watson test is used to perform this test to examine whether there is any systematic pattern in the residuals over time or observations [59]. When significant evidence of autocorrelation is found in the residuals, this may indicate that the model is not adequately capturing the data structure and that corrective measures should be taken, such as including additional terms in the model or considering a different approach to the analysis [60]. The p-value for the Durbin–Watson test was 0.5857; it can be concluded that the data do not contradict the assumption of independence.

The test of homoscedasticity is used to verify if the regression model residuals exhibit a constant variance across different values of the independent variables; i.e., this test evaluates whether the dispersion of the residuals is constant across the entire range of values of the independent variables [61]. One common way to conduct this test is by analyzing the standardized residuals versus the model-fitted values. If the residuals are uniformly dispersed around the horizontal line in a plot of standardized residuals versus the fitted values, this suggests that homoscedasticity is maintained [62]. Additionally, the Breusch–Pagan test can be employed to formally assess homoscedasticity in regression models. The p-value for the Breusch–Pagan test was 0.2062, so it can be concluded that the data do not contradict the assumption of homoscedasticity.

Once the regression model has been validated by checking these fundamental assumptions for the residuals, it can be confidently used for predictive purposes. The second-order quadratic regression model constructed according to Equation (8) is expressed in Equation (9). The model had a high $R_{adj}^2$ coefficient (0.9201) and a p-value of 0.01713. Therefore, the model resulted in being a highly significant model representing the maximum $C_p$.

$$\begin{array}{c}\hfill y=0.90071-14.42317\left({\displaystyle \frac{d}{D}}\right)+0.53540\left(Z\right)-0.14650\left({\displaystyle \frac{d}{D}}\right)\left(Z\right)\\ \hfill +35.7333{\left({\displaystyle \frac{d}{D}}\right)}^2-0.06617{\left(Z\right)}^2\end{array}$$

(9)

Another way to complement the response surface analysis is by conducting an ANOVA (analysis of variance). ANOVA makes it possible to compare the means of two or more groups and determine if there are significant differences among them [63]. This tool is widely employed across various disciplines, including scientific research, industry, and medicine, among others, for comparing different treatments, interventions, or experimental conditions. ANOVA allows researchers to assess whether observed differences between groups are due to genuine effects or simply random variation [64]. It provides valuable insights into the effectiveness of various factors or treatments under study, helping to guide decision-making and optimize outcomes.

From the results listed in

Table 6. Analysis of variance (ANOVA) results.

Term	SS	Df	Ms	F-Ratio	p-Value
Z	0.07142	1	0.07142	68.567	0.00369
$d/D$	0.00486	1	0.00486	4.668	0.11951
$Z^2$	0.00876	1	0.00876	8.407	0.0653
($d/D$)2	0.01596	1	0.01596	15.324	0.02963
(Z)($d/D$)	0.00021	1	0.00021	0.206	0.68070
Residuals	0.00312	3	0.00104

, it can be concluded that Z has a significant effect on the response variable, as it had a very low p-value (0.00369). In turn, $d/D$ appears to have no significant effect, as its p-value was relatively high (0.11951). The interaction between Z and $d/D$ is also not significant, as its p-value was high (0.68070). Nevertheless, the quadratic terms have significant effects, as their p-values were equal to 0.06253 and 0.02963, respectively. Once the construction of the regression model was complete, the $C_p$ optimal value was calculated and resulted to be 53.62% when Z and $d/D$ were 4 and 0.15, respectively. The response surface and the contour plot obtained for $C_p$ can be found in Figure 10.

6. Conclusions

The surging global energy demand, coupled with concerns over climate change and environmental pollution, has propelled the quest for alternative and sustainable energy sources. Renewable energies, such as solar, wind, and hydrokinetic energy, offer viable solutions to meet energy needs while minimizing environmental harm. Hydrokinetic turbines represent a promising technology for large-scale renewable energy generation. Although they do have some environmental impact, they are generally less intrusive compared to traditional hydraulic turbines, which can disrupt aquatic ecosystems and require large-scale dam constructions. For example, hydrokinetic turbines do not require the diversion of rivers or the creation of reservoirs, making them a more environmentally friendly option in many cases.

This comprehensive study delved into the performance assessment of hydrokinetic turbines, employing a combination of experimental testing and regression modeling. The analysis unveiled that while the Betz limit theoretically establishes the maximum achievable efficiency for these turbines, practical considerations, such as blockage effects induced by channel walls, necessitate corrections to experimental data. The employed correction methodology, specifically Pope and Harper’s method, effectively mitigated these effects, enabling a more accurate representation of turbine performance. Furthermore, the developed second-order polynomial regression model yielded valuable insights into the intricate relationship between the turbine design parameters and the $C_p$, facilitating the identification of optimal configurations for maximizing the energy conversion efficiency. The validation of the regression model through rigorous assessments of homoscedasticity, normality, and independence in the residuals ensured its reliability for predictive purposes. The model’s high R² adjusted coefficient and significance level underscore its robustness in accurately representing the observed data. Additionally, an ANOVA complemented the regression analysis by identifying significant factors that affect the turbine performance under study. The analysis revealed that the optimal value for $C_p$ was 53.62% when Z and $d/D$ were equal to 4 and 0.15, respectively. These findings collectively contribute to advancing our understanding of hydrokinetic turbine design and optimization, offering practical guidance for enhancing the efficiency of renewable energy generation from water currents. In addition, this work emphasizes the potential of hydrokinetic energy as a sustainable energy solution and underscores the importance of ongoing research to refine turbine designs for enhanced energy conversion with the aim of contributing to energy sustainability goal achievement.

Future research will focus on conducting additional tests in relevant environments to validate the findings obtained in the laboratory setting. As highlighted, it is crucial to investigate how the optimized hydrokinetic turbine designs perform under real-world conditions, considering various environmental factors such as flow speed, lateral currents, and water quality. These practical assessments will enhance our understanding of the operational limits and applicability of the proposed solutions, ensuring their robustness in diverse scenarios. Additionally, future work will explore the effects of these variables on turbine performance to further refine the design and optimize energy conversion efficiency. This effort aligns with the growing global demand for sustainable energy solutions and aims to contribute significantly to the advancement of hydrokinetic energy technology, thereby supporting the achievement of energy sustainability goals.