# Impeller Optimization in Crossflow Hydraulic Turbines

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. PRS Turbine Design

_{V}= 0.98 and ξ = 2.1 are constant coefficients [13], ΔH is the head drop between inlet and outlet PRS sections, ω is the impeller rotational velocity, D is the outer impeller diameter, and g is the gravity acceleration. The momentum Equation (1) can be coupled with the relative velocity optimality condition, which is represented by Equation (2):

_{r}is the optimal velocity ratio, for PRS turbines equal to 1.7 [14] and α is the velocity inlet angle, with respect to the tangent direction, approximately equal to 15°. For a given value of the impeller rotational velocity ω, usually chosen among a finite number of possible speeds of the electric generator coupled to the turbine, Equations (1) and (2) can be solved in the V and D unknowns. Note that the rotational velocity ω is function only of the frequency f of the alternating current (AC) grid and of the number p of the polar couples of the electrical generator coupled to the turbine, according to the following equation:

_{max}is the maximum inlet angle, equal to 110° as shown in Figure 1. A more extended discussion of the turbine design and management criteria can be found in [13,14,16].

## 3. Fluid Dynamic Investigation

^{−4}s. The root mean square residual was used for the convergence criterion with a residual target equal to 1.0 × 10

^{−5}. The boundary conditions selected in the simulation according to the design data are the following: (a) the total pressure per unit weight at the nozzle inlet, corresponding to the piezometric level plus the kinetic energy per unit weight, (b) the flow rate at the outlet section of the casing. The initial condition for unsteady state simulation was the fluid field output computed, according to the steady state flow assumption.

## 4. Design of Blade Shape and Number

_{2}is equal to 90°. The inlet angle β

_{1}guarantees to the inlet water particles a relative velocity tangent to the inlet blade surface. Note that, because the velocity norm V in Equations (1), (2) and (4) represents the mean value along all the inlet impeller surface, the radial velocity component in P is smaller than the mean value, due to the φ angle existing between the directions tangent to the internal and the external surface. This also implies that the velocity ratio in Equation (2) is smaller than the ratio ${\mathit{V}}_{\mathit{r}}^{\mathit{P}}$ computed using the local velocity in P instead of the mean one. Previous experiments suggest a ${\mathit{V}}_{\mathit{r}}^{\mathit{P}}$ value equal to 2 [20]. According to this hypothesis, the radius ρb, the angle β

_{1}and the central angle θ of the blade can be computed as a solution of Equation (5), setting the optimal value of the impeller inner diameter ratio D

_{i}/D equal to 0.75 [16].

_{max}, scaled by the best efficiency obtained for an angle φ = β

_{1}(see Figure 5). The efficiencies are computed with 2D CFD simulations, setting a fixed pressure value in the inlet and outlet boundaries. The number of blades is optimized for each couple of φ and t

_{max}values. Sharp edges are well known to provide a local stress concentration along the same edges, which can be avoided by rounding off the edges with a circular profile, as shown in Figure 7a. A value t

_{min}= 0.1 t

_{max}corresponds to a maximum stress located outside the edge, without a significant reduction of the hydraulic efficiency with respect to the value obtained with t

_{min}= 0. We note in Figure 6 that the maximum efficiency is computed for both PRS1 and PRS2 with φ = β

_{1}, corresponding to an external surface tangent to the inlet impeller surface. The reason is likely to be that the increment of the φ angle provides a reduction of the α attack angle of the water particles along a large part of the channel inlet, and a corresponding efficiency increment. For φ values larger than β

_{1}, the attack angle becomes negative, with a sharp efficiency reduction.

_{max,opt}the maximum thickness corresponding to the condition φ = β

_{1}. Note that, if the maximum thickness does not guarantee the blade structural resistance and the computed maximum von Mises stress is higher than the admissible limit, we need to give up the maximum efficiency condition in favor of a more robust design.

_{min}) = tan(β

_{1}− δ

_{min})

_{min}) = t

_{0}

_{min}

_{max}

_{max}is the maximum thickness, t

_{min}is the minimum thickness at the outlet extremity, t’ is the derivative with respect to δ in the interval (δ

_{min}; θ), t

_{min}is empirically set equal to 0.1 t

_{max}and δ* is a fifth auxiliary unknown.

_{min}and the thickness t

_{0}are the parameters of the tip of the blade and are computed to guarantee the tangent condition of the external blade surface to the impeller inlet surface. The inner blade extremity has a circular profile with radius r

_{f}, tangent in point P

_{1}to the cubic spline profile of the external surface and in P

_{2}to the circular profile of the internal surface (Figure 7b). r

_{f}is empirically set equal to 0.1 t

_{max}.

_{max}thickness value.

## 5. Maximum von Mises Stress Computation

_{max}. Using several CFD 2D test cases the behavior of this parameter was analyzed as a function of the number of blades of the impeller, within the range 22–35 suggested by most of the authors (see Table 3). The results obtained for two different impellers are summarized in Figure 10. We can observe that the maximum value of the ratio τ drops very slowly, along with the increase in the number of blades, in the analyzed range. For this reason, the maximum von Mises stress s is initially assumed independent from the number of blades.

#### Design of the Maximum Admissible von Mises Stress S_{adm}

_{f}) versus nominal stress S

_{f}.

_{f}ranges:

- A first quasi-static resistance or low cycle fatigue range (N
_{f}< 10^{3÷4}), where S_{f}remains constant; - A range with a high number of cycles (10
^{3÷4}< N_{f}< 10^{6}), where the Wöhler curve equation is of the type S_{f}^{µ}N_{f}= K, with µ and K constants relative to the material; - A third range with a very high number of cycles (N
_{f}> 10^{6}) where S_{f}again remains constant, but much smaller than in the first N_{f}range.

_{f}range is called the fatigue limit, S

_{l}, and is the maximum alternating stress value at which no breakage occurs. Experimental tests show that, for steel, the fatigue limit varies between 40 and 60% of the tensile strength S

_{r}, and the average fatigue limit for rotating bending specimens can be obtained with the following relationships:

- S
_{l}= 0.5 Sr for S_{r}< 1400 MPa; - S
_{l}= 700 MPa for S_{r}> 1400 MPa.

## 6. The Proposed Methodology for Impeller Design

- Compute width B and diameter D according to the procedure described in Section 2. Choose a small maximum blade thickness t
_{max}, as the initial tentative value. - Compute the internal and external blade profiles according to the procedure explained in the previous section.
- Solve a first 2D CFD model using an impeller with 35 blades, which is the upper limit of the usual range, and export the pressure distribution on the blade surface.
- Create a 3D CAD model of a single blade, based on the impeller width B and on the previously computed profile. Add to the CAD model a small portion of the two disks at the lateral contours of the blade; compute the fillet radius at the blade-disk connection and, after the first iteration, at the connection with eventual baffles. After the first iteration, use the pressure field on the blade previously computed in point 6.
- Using a 3D FEM code, compute the stress field and the maximum von Mises stress S in the selected blade.
- If the maximum thickness used in point 4 leads to a maximum von Mises stress value above the admissible one, here indicated as S
_{adm}, then a new attempt must be made. To this end, either increase the maximum thickness or introduce a new reinforcing baffle. Using the new geometry, compute again the corresponding blade section and update the number of blades with the optimal one corresponding to the new maximum thickness by iteratively solving a 2D CFD model. Update the pressure distribution on the blade surface and go back to point 5. The trial and error procedure must be repeated until the computed maximum stress is below the admissible one. - Perform a final validation of the impeller geometry using only one 3D CFD simulation coupled with 3D FEM analysis.

_{max}is required. A reasonable choice would be to start with a small reliable value and gradually increase it until the maximum von Mises stress is less than S

_{adm}.

_{f}, must be introduced at the blade-baffle connection. A fillet radius is always present in the real impeller and cannot be neglected to avoid exceptionally high stresses along the surface intersections. A parametric study was carried out to determine an approximate optimal value of the fillet radius as a function of the maximum thickness t for different impellers.

_{f}/t

_{max}ratio higher than 0.833. A ratio R

_{f}/t

_{max}= 0.833 is also a reasonable design choice.

## 7. A Case Study: Fontes Episcopi Power Plant

_{max}= 110° (Figure 1), a new PRS turbine, called PRS2, was designed with the proposed procedure, according to the input parameters listed in Table 1. Trough Equations (1–4), the resulting impeller diameter D and width B were found to, respectively, be 234 and 55 mm.

_{r}, was set equal to 500 MPa, with a corresponding fatigue limit S

_{l}equal to 250 MPa. Application of a reasonable safety factor 3 [31,32] provided S

_{adm}= 250/3 MPa = 83.3 MPa. The external surface of the blades of impeller 1 had a circular profile. The corresponding maximum efficiency was η = 79.3%, attained for φ = β

_{1}, t

_{max}= 5.12 mm and a number of blades equal to 34. The maximum von Mises stress S

_{max}= 117.76 MPa was not lower than the admissible limit S

_{adm}(see Figure 16, where the stress is the result of 3D FEM analysis).

_{1}is relaxed, it is possible in impeller 2 to increase t

_{max}up to 7 mm, corresponding to an optimal number of blades equal to 27 (computed by CFD 2D analyses, and not reported here for brevity). In this case, the maximum von Mises stress is less than S

_{adm}(see Figure 17) but the efficiency decreases up to η = 78.2%.

_{1}, t

_{max}= 7 mm and n

_{b}= 27. The resulting efficiency value was η = 79.2% and the maximum von Mises stress was below the admissible limit S

_{adm}(see Figure 18).

## 8. Conclusions

^{®}Xeon(R) E5-2650 v3 processors. The same problem, solved as the search of a 3D coupled structural and hydrodynamic optimization of the whole impeller, subject to the admissible stress constraint, would require a computational time of 16 days per simulation. Even with only two optimization parameters (number of blades and maximum thickness), the required computational time would have been larger than the actual one of several orders of magnitude.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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PRS Parameter | PRS1 | PRS2 |
---|---|---|

ΔH | 40 m | 100 m |

Q | 210 l/s | 100 l/s |

D | 297 mm | 234 mm |

B | 144 mm | 55 mm |

ω | 755 rpm | 1500 rpm |

α | 15° | 15° |

β | 28.2° | 28.2° |

λ_{max} | 110° | 110° |

PRS1 Configuration | 2D Efficiencies | 3D Efficiencies |
---|---|---|

Rotor with 33 blades | 0.855 | 0.779 |

Rotor with 35 blades | 0.856 | 0.780 |

Rotor with 37 blades | 0.854 | 0.777 |

Authors | Optimum Number of Blades | Reference |
---|---|---|

Ceballos Y.C. et al., | 28 | [18] |

Sammartano V., et al. | 35 | [20] |

Choi Y. D., et al. | 30 | [21] |

Aziz N.M., Totapally H.G.S | 30 | [22] |

Olgun H., Ulkun A. | 28 | [23] |

Aziz N. M, Desai V. R. | 25 | [24] |

Mani S., et al. | 22 | [25] |

Acharya N., et al. | 22 | [26] |

Impeller | Impeller 1 | Impeller 2 | Proposed Impeller |
---|---|---|---|

External profile | Circular | Circular | Cubic |

t_{max} | 5.12 mm | 7 mm | 7 mm |

n_{b} | 34 | 27 | 27 |

η | 79.3% | 78.2% | 79.2% |

S_{max} | 117.76 MPa | 47.16 MPa | 45.14 MPa |

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**MDPI and ACS Style**

Sinagra, M.; Picone, C.; Aricò, C.; Pantano, A.; Tucciarelli, T.; Hannachi, M.; Driss, Z.
Impeller Optimization in Crossflow Hydraulic Turbines. *Water* **2021**, *13*, 313.
https://doi.org/10.3390/w13030313

**AMA Style**

Sinagra M, Picone C, Aricò C, Pantano A, Tucciarelli T, Hannachi M, Driss Z.
Impeller Optimization in Crossflow Hydraulic Turbines. *Water*. 2021; 13(3):313.
https://doi.org/10.3390/w13030313

**Chicago/Turabian Style**

Sinagra, Marco, Calogero Picone, Costanza Aricò, Antonio Pantano, Tullio Tucciarelli, Marwa Hannachi, and Zied Driss.
2021. "Impeller Optimization in Crossflow Hydraulic Turbines" *Water* 13, no. 3: 313.
https://doi.org/10.3390/w13030313