# Comprehensive Parametric Study of Blockage Effect on the Performance of Horizontal Axis Hydrokinetic Turbines

^{*}

## Abstract

**:**

_{P}) of a three-bladed, untwisted, untapered horizontal axis hydrokinetic turbine. The investigation is based on validated 3D computational fluid dynamics (CFD), design of experiments (DOE), and the analysis of variance (ANOVA) approaches. A total number of 36 CFD models were developed and meshed. A total of 108 CFD cases were performed as part of the analysis. Results indicated that the effect of varying θ was only noticeable at the high TSR. Additionally, the rate of increment of C

_{P}with respect to ε was found proportional to both TSR and $\sigma $. The power and thrust coefficients were affected the most by $\sigma $, followed by ε, TSR, and then θ.

## 1. Introduction

#### 1.1. Hydrodynamic Parameters

#### 1.2. Blockage Effects

#### 1.3. Previous Work

_{P}) and thrust coefficient (C

_{T}) of both technologies were deemed negligibly affected by altering CA compared to altering the $\epsilon $. They also found that the C

_{P}and C

_{T}increased linearly with increasing the blockage ratio when both turbines operate at a specific TSR. Additionally, they indicated that ADM was effective in correcting the C

_{P}and C

_{T}of both axial and low-solidity cross-flow turbines, but that ADM slightly underestimated the correction factor for high-solidity cross-flow turbines, particularly for C

_{T}. They also stated that the performance of both types of turbines was insensitive to the blockage when the dynamic stall is presented (i.e., at low TSR). The same was observed in our previous study [16]; the performance was insensitive to blockage at low TSR, but this was only for the low-solidity (σ ≤ 0.111) small scale HAHkTs. The blockage effects were slightly more noticeable for the higher-solidity HAHkTs at low TSR. According to our earlier work [16], the ADM marginally underestimated the performance of all small-scale HAHkTs across all solidity levels at high TSRs. This overestimation was attributed to the high rotational speeds used by the small-scale rotors to obtain these high TSR values, which caused them to enter the braking state [27,28]. At this brake state, the correction model derivation may not be accurate. Madrigal et al. [17] utilized a shear-stress transport (SST) k-ω turbulence model to examine the effects of the solidity on an off-design performance of an axial water turbine. The solidity was controlled by altering the number of blades; however, blade number alteration has its individual effects regardless of solidity level [13,14]. They claimed that C

_{P}at the peak increased by 2% for each added blade, but the effect of increasing the blade number above a specific reference number was insignificant

_{.}The range of operational TSR shrunk, and the peak shifted to lower rotational speed as the number of blades increased. Kolekar and Banerjee [5] investigated the influence of boundary proximity and blockage on the performance of HAHkTs using experimental and computational techniques. They found that the effective blockage ratio started at 10%. Additionally, the TSR range was extended as the blockage ratio increased from 10% to 42%. Additionally, they observed that the blockage effects increased when the TSR was increased, owing to the increased wake strength and bypass flow. The current study exhibited a similar pattern of behavior. Increased flow velocity increased performance slightly, but only to 0.7 m/s. The C

_{P}vs. TSR curve was shown to be insensitive to flow speeds greater than 0.7 m/s [5] (the speed employed in this study was U = 1.5 m/s). Kolekar and Banerjee [5] investigated the influence of boundary proximity and blockage on the performance of HAHkTs using experimental and computational techniques. They found that the effective blockage ratio started at 10%. Additionally, the TSR range was increased as the blockage ratio rose from 10% to 42%. Additionally, they observed that the blockage effects increased when the TSR was higher, owing to the increased wake strength and bypass flow. The current study showed a similar pattern of behavior. Increased flow velocity increased performance slightly, but only to 0.7 m/s. The C

_{P}vs. TSR curve was shown to be insensitive to flow speeds greater than 0.7 m/s [5] (the speed employed in this study was U = 1.5 m/s). As with [8], Badshah et al. [29] utilized validated CFD simulation to demonstrate that altering the blockage ratios had a negligible effect on the HAHkTs performance when operating at low TSR. Similar to [5,15], they discovered that the effect of adjusting the blockage ratio increased as the TSR rose. Additionally, they discovered that blockage ratios less than 10% are regarded as ineffective for modifying the HAHkTs performance. Schluntz and Willden [30] optimized the design of closely spaced tidal turbines operating in arrays using a coupled blade element momentum (BEM) approach embedded in a Reynolds averaged Navier–Stokes (RANS) model. The optimization technique took into account the blockage ratio effects, which were controlled by varying the lateral tip to tip distance. The improved rotors performed optimally at the maximum blockage ratio (i.e., at smallest lateral spacing). The optimal solidity of optimized rotors was determined to be inversely proportional to their spacing. Local blade twisting was found to be less responsive to changes in local blockage, albeit it did decrease slightly as the space between rotors was decreased (i.e., increasing the blockage). In the current work, validated CFD simulation and design of experiments (DOE) approaches were utilized to comprehensively investigate the effects of different parameters on blockage behavior. The independent parameters considered in the current blockage investigation are the blockage ratio ($\epsilon $), solidity ($\sigma $, varied by changing chord length), tip speed ratio (TSR), and the pitch angle ($\theta $). These parameters affect the blades’ loading and turbines’ performance and will likely affect how the blockage behaves. The objective of this study was to fill a gap in the literature by providing a comprehensive insight into the main and interaction effects of these independent parameters on the thrust and power coefficients of the HAHkTs

## 2. Hydrokinetic Turbine Principle Definitions

_{P}) is a dimensionless parameter that measures the turbine’s performance. It is calculated by dividing the harnessed power by the kinetic energy crossing perpendicularly the rotating rotor

^{3}), U is the flow speed (m/s), and A is the rotor swept area (m

^{2}).

_{T}) is also an important dimensionless factor in turbine design. It is calculated by dividing the thrust force exerted on the rotor (T) by the dynamic pressure force acting perpendicularly on the rotor’s plane of rotation

_{r}) is the angle formed between the sectional chord of the blade and the rotational plane. Pitch angle is another approach to alter generated power in pitch-regulated turbines. Pitch angle is also utilized in adjusting twists along the span of optimized blades.

_{r_rel}) and the sectional chord (c

_{r}) of the blade is termed the local angle of attack (Uα

_{r}) and is given as

_{r}is the angle between U

_{r_rel}and the rotational plane. This local angle, Ø

_{r}, is calculated as follows.

## 3. Computational Fluid Dynamics

#### 3.1. Geometry and Meshing

_{l}) to drag (C

_{d}) ratio [32]. The blades were untwisted and untapered. The untwisted, untapered blades were used to reduce the number of factors altering the turbine performance and consider only those that may affect the blockage behavior. The rotor radius was 15.557 cm (6.125 in.), blade length was 13.970 cm (5.5 in.), and the hub radius was 1.587 cm (0.625 in.). This work considered different chord lengths to study σ effects on the blockage behavior and eliminate the effects introduced if the solidity was varied by altering the number of blades.

#### 3.2. Turbulence Modeling

#### 3.2.1. SST k-ω Model

#### 3.2.2. Moving Reference Frame

_{rel}is the relative velocity defined as ${U}_{rel}=\overrightarrow{U}-\text{}\overrightarrow{\mathsf{\Omega}}\times \overrightarrow{r}$. The centrifugal forces and Coriolis forces in Equation (14) are accounted for by $\rho \left(2\text{}\overrightarrow{\mathsf{\Omega}}\times {\overrightarrow{U}}_{r}\right)$ and $\rho \left(\text{}\overrightarrow{\mathsf{\Omega}}\times \text{}\overrightarrow{\mathsf{\Omega}}\times \overrightarrow{r}\right)$ terms, respectively. The term $\nabla p$ represents the pressure gradient, and the term ${\tau}_{r}$ describes the viscous stress tensor, which is calculated as

#### 3.2.3. Model Setup

#### 3.3. Grid Independent Investigation

_{M}) was documented. The rotor was operated at a flow speed of 1.5 m/s and a rotational speed of 376 RPM. The residuals were decreased below $5\times {10}^{-5}$ if the moment and thrust did not converge. The initial wall spacing of the blade surfaces ($\mathsf{\Delta}{y}_{w}$) was maintained at $\mathsf{\Delta}{y}_{w}^{+}\approx 1$ to resolve the boundary layer (BL). The optimum rotor mesh size was approved at around 4.06 million elements (The total model element number, including outer domain, was about 6.171 million). At this mesh size, the resulting change in C

_{M}was about 0.019% when the rotor domain grid was uniformly increased from 4.06 million to 7.322 million elements. Increasing the mesh size larger than 4.06 million elements did not yield a noticeable change in C

_{M}. For this optimum mesh size, the number of grids along the hydrofoil was 110.

#### 3.4. Experimental Validation

_{P}versus TSR curve did not experience a decline in its magnitude as TSR decreased. The stall delay caused a longer span of the blade to operate at angles of attack below the stall angle [46]. Consequently, as the applied load on the turbine increased and reached a value more significant than the turbine’s generated torque, the rotor came to a halt. The validation results in Figure 4a show that the C

_{P}satisfactorily agreed with the water tunnel measurements. The C

_{T}was slightly underestimated and then likely overestimated as TSR decreased from 4.54 to values less than 3.75 (Figure 4b). The close match of the steady RANS results to the experimental measurements provided confidence in the steady approach. Therefore, it was used in the study of the blockage effects on the turbine performance.

## 4. Evaluating the Operational Rotational Speed

_{l}and C

_{d}) for all used chord lengths required by the BEM model were obtained from 2D CFD models utilizing the SST k-ω turbulence model. The angle of attack, α (α = θ in 2D flow), was altered by rotating the meshes of the hydrofoil and the surrounded flow domain from −10° to 30° with an increment step of 1°. The boundary conditions for this 2D CFD model are illustrated in Figure 5. The flow speed at the inlet was set at 1.5 m/s, which was the same flow speed considered in the blockage investigation. To account for the rotational and 3D effects, the C

_{l}and C

_{d}were modified using models from [53] and [54], respectively. These rotational and 3D effects were made based on a hydrofoil located at 80% of the considered blades spans as this was the recommended design span [41]. The modified C

_{l}and C

_{d}were then extrapolated over a broad range of α (±180°) using the Viterna model [55]. This broad range of α is needed by the BEM model while searching for the solution parameters iteratively.

_{P}for the different considered rotors is shown in Figure 6a. The BEM prediction of a rotor with a blade chord length of 6.35 cm (2.5 in.) was verified against 3D CFD unconfined model ($\epsilon =$ 4.168%), and the results are shown in Figure 6b. The blockage ratio of $\epsilon \le 5\%$ is considered effectively unblocked [21]. The BEM slightly underestimated the C

_{P}peak and then started to overestimate C

_{P}as TSR increased. This was likely attributed to the used 3D effects correction models, which were considered at one representative radial location $\left(\mathrm{radial}\text{}\mathrm{location}\left(r\right)/\mathrm{rotor}\text{}\mathrm{radius}\left(R\right)=0.8\right)$ and using one representative rotational speed (Ω = 200 RPM). Additionally, the 3D correction model is designed for larger blades where the aspect ratio (r/c) is less than one over most of the blade span.

## 5. Experimental Design

_{P}and C

_{T}(i.e., $\Delta {C}_{P}$ and $\Delta {C}_{T}$) as a result of the blockage effects were the response variables of the DOE and are defined in Equation (16).

_{x}is the response variable, ${C}_{{x}_{confined}}$ and ${C}_{{x}_{unconfined}}$ are the coefficients when a turbine operates in confined and unconfined flows, respectively. The independent parameters (factors) considered in the DOE study are blockage ratio ($\epsilon $), solidity ($\sigma ,\text{}$varied by changing chord length), tip speed ratio (TSR), and the pitch angle ($\theta $). These parameters affect the C

_{P}and C

_{T}and are likely to affect the blockage behavior. A full factorial DOE investigation was considered with four parameters and three levels per each. This required 81 3D CFD runs (${3}^{4}=81$). The considered parameters and their levels are listed in Table 2.

_{P}and ∆C

_{T}) due to blockage effects.

## 6. Results and Discussion

#### 6.1. Influence of the DOE Parameters on the Blockage Behavior

_{P}peak) for the rotors with c = 6.35 cm ($\sigma $ = 0.191) and c = 3.81 ($\sigma $ = 0.115) was insensitive to the alteration of θ. Thus, for these rotors, when comparing performances within the selected range of TSR, the possibility of the change in performance due to the shifting of the C

_{P}vs. TSR curve is eliminated. For this reason, this work started the analyses of the DOE parameters using the rotor with the highest solidity ($\sigma $ = 0.191).

_{P}, of the highest solidity rotor with varying tip speed ratio, TSR, and blockage ratio, $\epsilon ,$ are represented in Figure 7. Each response surface was generated at a fixed pitch angle, θ. Different edge colors were used to differentiate between the surfaces. The intersections between surfaces were indicated by parallel curves that were given the same colors as the edges of the intersecting surfaces. This was also illustrated in the legend of Figure 7. This figure shows that the blockage effects on the turbine performance increase with increasing TSR for all pitch angles. Figure 7 also shows that the surface that represents θ = 5° was the most affected by the blockage as indicated by the surface with the red edges. This surface changed from the lowest C

_{P}to the highest C

_{P}as the TSR and $\epsilon $ increased. Additionally, the performance of this high solidity rotor when θ = 5° and 10° was larger than unity (i.e., C

_{P}> 1) at the extreme level of confinement and highest rotational speed. This was caused by the large increase in the kinetic flux passed through and around the highly confined rotor. Meanwhile, the power coefficient was calculated via normalizing the extracted power from the augmented confined kinetic flux by the kinetic flux at the upstream undistorted region (i.e., unconfined environment).

_{P}with TSR and θ is presented for each blockage ratio separately in Figure 8. The increase and then decrease in C

_{P}as TSR increase in Figure 8a,b indicates that the C

_{P}peak was located within this range of TSR. On the other hand, this pattern of increase and decrease in the C

_{P}curve is not observed or less pronounced in Figure 8c,d, which indicates that the C

_{P}peak was not within this range of TSR at these relatively high levels of $\epsilon $. The increase in the blockage caused a slight shift in the C

_{P}peak to higher TSRs. This was attributed to the fact that increasing the blockage (i.e., decreasing the tunnel’s cross-sectional area) augmented the flow around the rotor. Because the TSR range was fixed, the angle of attack increased (see Equations (7) and (8)), and thus larger portions of the blades’ spans were stalled. The rotors need to rotate faster to bring the angles of attack along the blades at or below the stall angle. This means the rotors need to operate at higher TSR to reach the C

_{P}peak. Additionally, in Figure 8a–d, it is clear that at the highest TSR, the smaller the θ, the faster the C

_{P}increased with increasing the ε.

_{P}, the relationship between C

_{P}vs. ε was plotted for all pitch angles at fixed TSRs. The results are shown in Figure 9. The ∆C

_{P}in the legend represents the change in performance due to changing the blockage from ε = 4.168 % to ε = 62.533%. The similar curves slope in Figure 9a,b indicates that varying θ has insignificant effects on how the blockage alters the turbine performance at this relatively low range of TSRs. When TSR = 1.563, the change in ∆C

_{P}as a result of varying θ was 0.08 (e.g., $\Delta {C}_{P}{}_{\left(\theta ={15}^{\xb0}\right)}-\Delta {C}_{P}{}_{\left(\theta ={5}^{\xb0}\right)}=0.08$) as seen in Figure 9a. The change in ∆C

_{P}caused by changing θ was 0.07 when TSR = 2.865 (Figure 9b).

_{P}peak (Figure 6b). The separation also existed at this TSR but with less intensity and covered a relatively smaller portion of the blade span for this unconfined rotor [46]. To examine the effects of the blockage ratio and angle of attack on flow separation and thus analyze the results in Figure 9b, Figure 10 was generated. Figure 10 utilizes surface streamlines to detect the flow behavior. The fully attached flow at the blade outboard is denoted by a red dot at the leading edge (LE) and an orange dot at the trailing edge (TE). Only Figure 10c does not have the red dot because the flow is not fully attached to all LE span. The percentage numbers represent the ratio of the radial distance to the rotor diameter (r/R). Increasing the pitch angle resulted in expanding the area of the fully attached flow, as observed when comparing Figure 10a to Figure 10b or Figure 10c to Figure 10d. However, the C

_{P}was still insensitive to θ alteration (Figure 9b), which is similar to the previous lower TSR. This was likely due to the insignificant effect of the change in lift and drag coefficients at the regions of fully and partially attached flows. However, ∆C

_{P}at this TSR was more significant than in the previous case (TSR = 1.563). The increase in blockage augmented both the flow and the angle of attack (see Equations (7) and (8)). Therefore, as ε increased, the fully attached flow covered a smaller outboard region of the blade; this can be observed when comparing Figure 10a to Figure 10c or Figure 10b to Figure 10d. However, the rise in the lift due to the increase in blockage and thus flow speed (e.g., increase in Reynolds number) outperformed the decrease in the lift due to the increase in flow separation. Therefore, the ∆C

_{P}when TSR = 2.865 (Figure 9b) improved compared to that when TSR = 1.563. The increase in lift coefficients along the blade due to the increase in blockage when TSR = 2.865 and θ = 10° was verified by monitoring the change in moment coefficient (${C}_{M}$). The ${C}_{M}$ increased from 0.111 to 0.31 as ε increased from 4.168% to 62.533%.

_{P}was more pronounced when TSR was 4.167, as indicated by the slopes of the lines and the legend in Figure 9c. This TSR is located at the right side of the C

_{P}peak (see Figure 6b), where the fully attached flow covered a larger portion of the blade than the previous TSRs. Therefore, the increase in blockage augmented the angles of attack along the blade to levels close to the optimum value, and thus the ∆C

_{P}was larger compared to the lower TSRs. It was also observed that varying the pitch angle affects the blockage behavior at this TSR of 4.167. The pitch angle of 15° had the lowest ∆C

_{P}as blockage increased. That was because θ = 15° caused the angles of attack along most of the blade span to decrease to values below the optimum angle of attack, which lowered the lift and, thus, the moment.

_{P}as solidity decreased.

_{P}with respect to ε increased with increasing the TSR (i.e., ∆C

_{P}is proportional to TSR). The proportionality between blockage effects and TSR (i.e., rotational speed) was partly attributed to the augmented wake turbulence and elevated bypass flow [5]. The blockage that influences the performance of a confined turbine comprises both the rotor blockage and wake blockage [56], where the latter is mainly affected by rotor rotational speed. The increase in rotational speed results in a stronger wake boundary and stronger wake blockage [5], which further enhances the kinetic flux and thus the turbine performance.

_{P}enhancement (i.e., ∆C

_{P}was very small) as the blockage increased. As the flow speed increased with increasing the blockage and as the turbine started to decrease its rotational speed under loading, the stall started at the inboard region of the untwisted blades and then propagated towards the tip under Coriolis and centrifugal force effects [40,57]. The large stall-dominated portion of the blade span at this low TSR caused a drop in the lift, which unfavorably affected the turbine performance. As rotational speed increased, the range of angles of attack along the blade span was lowered, where angles of attack were smaller towards the tip. That means as TSR increased, the separation became less intense and covered a smaller portion of the blade, mostly at the inboard segment. Thus, at the elevated TSRs, as the flow speed increased due to increased blockage, an increasing portion of the outboard of the blade had angles of attack that increased to levels below or near the optimum angle, which enhanced the performance (i.e., ∆C

_{P}increased).

_{P}vs. $\epsilon $ decreased. Because a larger portion of the blade span was stalled at TSR = 2.865, the line slope was affected less by increasing the θ compared to TSR = 4.167 (this was explained in Figure 9 discussion). The slope of the TSR = 4.167 line was decreased the most and approached the slope of the TSR = 2.865 line as the pitch angle increased to 15°. This was again due to the decrease in the angles of attacks below the optimum value.

_{P}was proportional to σ, and this proportionality increased with increasing TSR. For TSR of 2.865 (Figure 12a–c), the highest solidity rotor (σ = 0.191) showed the highest values of ∆C

_{P}for all pitch angles while the lower solidities (σ = 0.0504 and 0.115) showed minimal values of ∆C

_{P}at all pitch angles. The ∆C

_{P}with a negative sign in Figure 12a indicates a slight decrease in performance due to increasing the blockage for this lowest solidity rotor with a pitch angle of 5°. This was due to the increase in stall effects as a result of increasing the U/ωr ratio.

#### 6.2. Analysis of Variance of the DOE Parameters

_{P}and ∆C

_{T}). If ANOVA output, p-value, is less than a predetermined significance level (0.05 used in the current study), the null hypothesis is rejected, concluding that: a change in at least one of the treatment combinations (i.e., a set of factors’ levels) results in a statistically significant change in ∆C

_{P}or ∆C

_{T}. This helps to identify which of the selected factors and factors’ interactions affect the blockage behavior.

_{P}were tested, and results are also listed in Table 3. The p-values for all linear terms are less than 0.05, indicating that their effects on ∆C

_{P}are statistically significant. For the 2-way interactions, only the p-value from the interaction between blockage and pitch angle (ε*θ) is greater than 0.05; therefore, this interaction does not significantly affect the response variable, ∆C

_{P}. All other 2-way interactions have p-values less than 0.05; therefore, their effects on ∆C

_{P}are statistically significant. For the 3-way interactions, both $\sigma $*ε*θ and ε*θ*TSR interactions do not have a significant effect on the response variable, ∆C

_{P}, while $\sigma $*ε*TSR and $\sigma $*θ*TSR interactions do.

_{T}) was conducted and presented in Table 4. The p-values for all model, linear, and 2-way interactions terms are less than 0.05, indicating their statistically significant effects on ∆C

_{T}. The 3-way interactions terms also have p-values less than 0.05, and their effects are statistically significant, except the term ε*θ*TSR, which has p-values of 0.143.

_{P}and ∆C

_{T}. This means the 4-way interactions do not have significant effects on the response variable.

_{P}and ∆C

_{T}. However, the pitch angle is inversely proportional to ∆C

_{P}and ∆C

_{T}. The order of the main effect intensity of the factors on both response variables is such that solidity has the highest effect, followed by blockage ratio, tip speed ratio, and then pitch angle, which has the least effect on ∆C

_{P}and ∆C

_{T}.

## 7. Practical Implications

- When designing a confined hydrokinetic turbine, designers should be aware of the sensitivity of the design variables to the confinement effects.
- After optimizing the turbine for an open environment, further considerations should be given toward design parameters. Priority should be given in the following order: the solidity, blockage ratio, rotational speed, and pitch angle.
- Interaction effects should also be considered; for example, the performance of the highest solidity rotor changed from the lowest in an open flow (due to the increased flow impedance) to the highest in a confined flow (due to the increased kinetic flux).

## 8. Conclusions

- C
_{P}was insensitive to θ alteration at relatively low TSRs as ε increased. The effect of varying θ was noticeable at the high TSR. The ∆C_{P}was inversely proportional to θ at the TSR of 4.167. That was due to the decrease in the angle of attack to levels below the optimum values. - For all pitch angles, the rate of increment of C
_{P}with respect to ε was proportional to the TSR. This proportionality was attributed to the augmented wake turbulence and the improvement in the angles of attack along the blade span due to the increase in rotational speed. - The blockage effects were proportional to the solidity level, and this proportionality increased with increasing TSR. Additionally, the performance of the highest solidity rotor changed from the poorest to the highest as $\epsilon $ increased.

- The ε*θ,$\text{}\sigma $*ε*θ, and ε*θ*TSR interactions do not have significant effects on ∆C
_{P}. All other linear, 2-way, and 3-way interactions are statistically significant in affecting the ∆C_{P}. - All model, linear, 2-way, and 3-way (excluding ε*θ*TSR) interactions are statistically significant in affecting the ∆C
_{T}. - All 4-way interactions do not have significant effects on both response variable

_{P}and ∆C

_{T}, followed by blockage ratio, tip speed ratio, and then pitch angle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Pitch angle, angle of attack, and incoming flow angle at a section located at r radial distance from the rotor center.

**Figure 2.**Structured mesh of (

**a**) outer flow domain near the rotor location and (

**b**) one-third of the rotor domain (blue planes were established to show the mesh characteristics).

**Figure 6.**(

**a**) BEM results show the range of operational TSR in an unconfined flow and (

**b**) BEM prediction of the rotor with a chord length of 6.35 cm verified against 3D CFD unconfined model.

**Figure 7.**Response surfaces show the effect of ε and TSR on the performance of the highest solidity rotor (σ = 0.191) configured with different pitch angles.

**Figure 8.**Change in C

_{P}with varying the TSR and θ plotted at different ε of (

**a**) 4.168%, (

**b**) 20.842%, (

**c**) 41.684%, and (

**d**) 62.526% for rotor with the highest solidity (σ = 0.191).

**Figure 9.**Effect of θ on how ε alters C

_{P}of the highest solidity rotor (σ = 0.191) operating at different TSRs of (

**a**) 1.563, (

**b**) 2.865, and (

**c**) 4.167.

**Figure 10.**Surface streamlines on the suction side of the highest solidity rotor (σ = 0.191) operating at TSR = 2.865: (

**a**) ε = 20.842% & θ = 5°, (

**b**) ε = 20.842% & θ = 15°, (

**c**) ε = 62.533% & θ = 5°, and (

**d**) ε = 62.533 & θ = 15°.

**Figure 11.**Effect of TSR on how ε alters C

_{P}of the highest solidity rotor (σ = 0.191) that pitched to different θ values of (

**a**) 5°, (

**b**) 10°, and (

**c**) 15°.

**Figure 12.**Effect of σ on how ε alters C

_{P}at: (

**a**) TSR = 2.865 & θ = 5°, (

**b**) TSR = 2.865 & θ = 10°, (

**c**) TSR = 2.865 & θ =15°, (

**d**) TSR = 4.167 & θ = 5°, (

**e**) TSR = 4.167 & θ = 10°, and (

**f**) TSR = 4.167 & θ = 15°.

**Figure 13.**The main effects of factors’ levels on the response variables (

**a**) ∆C

_{P}, and (

**b**) ∆C

_{T}.

Geometry/Operation Type | Specification |
---|---|

Hydrofoil | Eppler 395 |

Number of blades (N) | 3 blades |

Rotor radius (R) | 15.557 cm (6.125 in.) |

Chord length (c) | 1.676, 3.81, 6.35 cm (0.66, 1.5, 2.5 in.) |

Rotor solidity ($\sigma $) | 0.0504, 0.115, 0.191 |

Pitch angle (θ) | 5°, 10°, 15°, |

Flow domain circular cross-section radius | 0.197, 0.241, 0.341, 0.762 m (7.746, 9.487, 13.416, 30 in.) |

Flow velocity (U) | 1.5 m/s |

Levels | Parameters | |||
---|---|---|---|---|

ε (%) | $\mathit{\sigma}$ | $\mathit{\theta}(\xb0)$ | TSR | |

Low | 20.842 | 0.0504 | 5 | 1.563 |

Medium | 41.684 | 0.115 | 10 | 2.865 |

High | 62.533 | 0.191 | 15 | 4.167 |

Source | Degrees of Freedom | Adjusted Sum of Squares | Adjusted Mean Squares | F-Value | p-Value |
---|---|---|---|---|---|

Model | 64 | 4.501 | 0.070 | 52.55 | 1.363 × 10^{−11} |

Linear | 8 | 2.974 | 0.372 | 277.74 | 1.100 × 10^{−15} |

$\sigma $ | 2 | 1.454 | 0.727 | 543.22 | 2.000 × 10^{−15} |

ε (%) | 2 | 0.360 | 0.180 | 134.43 | 9.905 × 10^{−11} |

θ (°) | 2 | 0.015 | 0.007 | 5.58 | 0.015 |

TSR | 2 | 1.145 | 0.572 | 427.73 | 1.290 × 10^{−14} |

2-Way Interactions | 24 | 1.253 | 0.052 | 39.02 | 4.059 × 10^{−10} |

$\sigma $*ε (%) | 4 | 0.283 | 0.071 | 52.95 | 5.001 × 10^{−09} |

$\sigma $*θ (°) | 4 | 0.050 | 0.013 | 9.40 | 4.163 × 10^{−04} |

$\sigma $*TSR | 4 | 0.593 | 0.148 | 110.78 | 1.898 × 10^{−11} |

ε (%)*θ (°) | 4 | 0.002 | 0.000 | 0.35 | 0.838 ** |

ε (%)*TSR | 4 | 0.266 | 0.067 | 49.77 | 7.879 × 10^{−09} |

θ (°)*TSR | 4 | 0.058 | 0.015 | 10.89 | 1.858 × 10^{−04} |

3-Way Interactions | 32 | 0.274 | 0.009 | 6.40 | 1.379 × 10^{−04} |

$\sigma $*ε (%)*θ (°) | 8 | 0.008 | 0.001 | 0.79 | 0.623 ** |

$\sigma $*ε (%)*TSR | 8 | 0.178 | 0.022 | 16.61 | 2.178 × 10^{−06} |

$\sigma $*θ (°)*TSR | 8 | 0.074 | 0.009 | 6.88 | 5.595 × 10^{−04} |

ε (%)*θ (°)*TSR | 8 | 0.014 | 0.002 | 1.34 | 0.295 ** |

Error | 16 | 0.021 | 0.001 | ||

Total | 80 | 4.523 |

Source | Degrees of Freedom | Adjusted Sum of Squares | Adjusted Mean Squares | F-Value | p-Value |
---|---|---|---|---|---|

Model | 64 | 10.661 | 0.167 | 58.46 | 5.916 × 10^{−12} |

Linear | 8 | 7.858 | 0.982 | 344.72 | 2.000 × 10^{−16} |

$\sigma $ | 2 | 5.230 | 2.615 | 917.72 | 0.000 |

ε (%) | 2 | 1.158 | 0.579 | 203.11 | 4.253 × 10^{−12} |

θ (°) | 2 | 0.333 | 0.167 | 58.44 | 4.420 × 10^{−08} |

TSR | 2 | 1.138 | 0.569 | 199.60 | 4.863 × 10^{−12} |

2-Way Interactions | 24 | 2.325 | 0.097 | 34.00 | 1.169 × 10^{−09} |

$\sigma $*ε (%) | 4 | 0.892 | 0.223 | 78.26 | 2.691 × 10^{−10} |

$\sigma $*θ (°) | 4 | 0.384 | 0.096 | 33.72 | 1.303 × 10^{−07} |

$\sigma $*TSR | 4 | 0.533 | 0.133 | 46.76 | 1.245 × 10^{−08} |

ε (%)*θ (°) | 4 | 0.057 | 0.014 | 4.96 | 0.009 |

ε (%)*TSR | 4 | 0.263 | 0.066 | 23.07 | 1.776 × 10^{−06} |

θ (°)*TSR | 4 | 0.197 | 0.049 | 17.24 | 1.187 × 10^{−05} |

3-Way Interactions | 32 | 0.477 | 0.015 | 5.23 | 4.933 × 10^{−04} |

$\sigma $*ε (%)*θ (°) | 8 | 0.066 | 0.008 | 2.87 | 0.034 |

$\sigma $*ε (%)*TSR | 8 | 0.174 | 0.022 | 7.65 | 3.065 × 10^{−04} |

$\sigma $*θ (°)*TSR | 8 | 0.196 | 0.024 | 8.58 | 1.553 × 10^{−04} |

ε (%)*θ (°)*TSR | 8 | 0.042 | 0.005 | 1.84 | 0.143 ** |

Error | 16 | 0.046 | 0.003 | ||

Total | 80 | 10.707 |

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**MDPI and ACS Style**

Abutunis, A.; Menta, V.G.
Comprehensive Parametric Study of Blockage Effect on the Performance of Horizontal Axis Hydrokinetic Turbines. *Energies* **2022**, *15*, 2585.
https://doi.org/10.3390/en15072585

**AMA Style**

Abutunis A, Menta VG.
Comprehensive Parametric Study of Blockage Effect on the Performance of Horizontal Axis Hydrokinetic Turbines. *Energies*. 2022; 15(7):2585.
https://doi.org/10.3390/en15072585

**Chicago/Turabian Style**

Abutunis, Abdulaziz, and Venkata Gireesh Menta.
2022. "Comprehensive Parametric Study of Blockage Effect on the Performance of Horizontal Axis Hydrokinetic Turbines" *Energies* 15, no. 7: 2585.
https://doi.org/10.3390/en15072585