# Turbine Passage Design Methodology to Minimize Entropy Production—A Two-Step Optimization Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Strategy Overview

#### 2.2. 1D Optimization Methodology

_{01}) and total temperature (T

_{01}) at the inlet, rotational speed (RPM), total-to-static pressure ratio (P

_{01}/P

_{s3}), inlet Mach number (M

_{1}), and degree of reaction (r

_{p}). Geometrical constraints include inlet height (H

_{1}), and ratio of channel heights (i.e., H

_{2}/H

_{1}and H

_{3}/H

_{2}). The inlet and outlet blade metal angles (α

_{2}and β

_{3}) were iterated upon to balance the mass-flow through the stage. An axial inflow (α

_{1}= 0), moderate stator and rotor efficiencies (respectively 92.5% and 87%), and constant gas properties were used. Constraints included an upstream total pressure of 5 bar and the rotational speed was fixed to 7500 RPM to maintain the structural integrity of the future rotating assembly.

_{2}/H

_{1}and H

_{3}/H

_{2,}varied from 1 to 2 and 0.5 to 1.5, respectively. The mean radius was constrained by the dimensions of the tunnel and was held at a constant of 389mm.

_{2}and β

_{3,}were limited to 80 deg., the relative rotor inlet angle β

_{2}to 40 deg., and a maximum allowed turning of 120 deg. in the rotor passage was imposed. To avoid high supersonic designs, the outlet Mach numbers, M

_{2}and M

_{3r,}were constrained to 1.3 and 1.05, respectively, while M

_{2r}was kept below 0.42. Furthermore, mass-flow was limited to a maximum of 30 kg/s based on the capacity of the upstream high-pressure vessels and downstream dump tank. The target was to simultaneously maximize the stage loading (Equation (1)) and stage efficiency (Equation (2)).

#### 2.3. Three-Dimensional Optimization Strategy

#### 2.4. 2D Airfoil Parameterization

_{1}, β

_{2}) and outlet (α

_{2}, β

_{3}) metal angles. The stagger angle (γ) and axial chord (C

_{ax}) determines the position of point 1. Throughout the optimization, the stator inlet flow angle, α

_{1}, was fixed to zero while the other blade metal angles, as well as the chord and stagger angle, could vary.

#### 2.5. 3D Blade Parameterization

_{1}, H

_{2}and H

_{3}. To control the local curvature, points 4, 5 and 6 could move axially from the profiles mid-chord all the way up to the vicinity of the stator-rotor interface.

#### 2.6. Computational Domain and Grid Sensitivity

#### 2.7. Optimization Setup

#### 2.8. Blade Count Selection

## 3. Results

#### 3.1. 1D Optimization Results

_{2}) and 1.04 (M

_{3r}), are in the high subsonic-transonic regime while both the stator and rotor passage heights are increasing, with 16% and 32% respectively. The pressure ratio, inlet total temperature will be used for the boundary conditions and the metal angles will be used to define a range for the following 3D optimization.

#### 3.2. 3D Optimization Results

#### 3.2.1. Pareto Front

_{p}). On the right of Figure 9, the same design space is colored with the rotor turning angle (Δβ). Individuals with higher efficiency contain higher degrees of reaction and have lower turning. However, designs with higher stage loading, feature lower degrees of reaction and more turning. Degrees of reaction in the 0.3-0.35 range with turning up to 120 degrees represent the top right of the pareto front.

#### 3.2.2. Trends in Stator Loss Generation

_{2}). The upper regions are characterized by low degree of reactions which results in more turning of the stator and have higher exit Mach numbers. Losses can be kept as low as 0.6% if one limits the stator turning to 76 deg. and a degree of reaction above 0.4.

#### 3.2.3. Trends in Rotor Loss Generation

#### 3.2.4. Horlock Efficiency

_{3}and v

_{3ss}are equal, which however may result in errors for high speed turbines.

_{3s}-h

_{3ss}is related to h

_{2}-h

_{2s}by the temperature ratio.

_{2,}extracted at the exit of the stator (Equation (9)). Similarly, for the blade losses, the mass-flow average relative velocity, W

_{3,}was taken at one half the axial chord downstream of the rotor (Equation (10)).

#### 3.2.5. Analysis of Optimal Profiles

_{3}also increases from 66 to 74 deg at tip. However, β

_{2}decreases from 40 deg at hub to 38 at the tip. Higher loading designs typically feature fatter profiles at the tip with more turning. More efficient designs on the other hand, have higher turning at the hub and less at the tip. In design E, the inlet metal angle β

_{2}varies along the span from 38 to 43 to 38 degrees at the tip. β

_{3}shows a different trend. It decreases from the hub to midspan, 73 to 65 degrees and slightly increases at the tip to 66 deg. In contrast, the suction side wedge angle fluctuates from 3 to 5 deg, then back to 3 deg. at the tip.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c | Chord [mm] |

C_{p} | Specific heat [J/(kgK)] |

H | Blade height [mm] |

h | Specific enthalpy [J/kg] |

M | Mach number |

P | ressure [bar] |

r_{p} | Degree of reaction |

RPM | Rotational speed |

RTZ | Radius, Tangential, Axial |

s | Specific Entropy [J/(kgK)] |

T | Temperature [K] |

Subscripts | |

0 | Total flow quantity |

1 | Inlet |

2 | Rotor-stator interface |

3 | Outlet |

ax | Axial component |

θ | Tangential component |

r | Radial component |

s | Isentropic change across one row |

ss | Isentropic change across two rows |

Greek symbols | |

α | Absolute flow angle |

β | Relative flow angle |

γ | Ratio of specific heats |

η | Efficiency |

ψ | Stage loading |

ω | Angular velocity |

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**Figure 4.**(

**a**,

**b**) Three-dimensional stacking of the blade profiles through a lean distribution. (

**c**) Channel parameterization.

**Figure 6.**Stator and rotor pressure losses for four meshes of the grid sensitivity analysis. (

**a**) Stator pressure loss; (

**b**) Rotor pressure loss.

**Figure 10.**Relation of the stator losses with the aerodynamic quantities (

**left**) and geometrical characteristics (

**right**).

**Figure 12.**The stage efficiency in function of the aerodynamic losses (

**left**) and the compensation strategy for the Horlock efficiency equation (

**right**).

Stator/Rotor | Mesh Size | Spanwise Divisions | Suction Side | Pressure Side |
---|---|---|---|---|

Coarser | 1.5/1.5M | 81/81 | 133/177 | 65/65 |

Coarse | 2.4/2.2M | 97/97 | 161/213 | 81/81 |

Fine | 3.4/4.4M | 117/141 | 193/257 | 97/97 |

Finer | 5.6/7.0M | 141/169 | 233/309 | 113/117 |

Symbol | Value | |
---|---|---|

Mutation scale factor | F | 0.6 |

Mutation rate | C | 0.8 |

1D | 3D | |
---|---|---|

# Parameters | 7 | 75 |

Design of Experiments | 80 | 256 |

Population size | 40 | 30 |

Populations Evaluated | 35 | 15 |

Flow Angles | Performance | Mach Number | ||||
---|---|---|---|---|---|---|

α_{2} | 73 | Power | MW | 3.64 | M_{2} | 0.89 |

α_{3} | −33 | Massflow | kg/s | 22.8 | M_{2r} | 0.34 |

β_{2} | 39.3 | Degree of Reaction | - | 0.39 | M_{3} | 0.48 |

β_{3} | −67.7 | Stage Loading | - | 1.74 | M_{3r} | 1.04 |

- | - | T_{01} | K | 676 | - | - |

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**MDPI and ACS Style**

Juangphanich, P.; De Maesschalck, C.; Paniagua, G.
Turbine Passage Design Methodology to Minimize Entropy Production—A Two-Step Optimization Strategy. *Entropy* **2019**, *21*, 604.
https://doi.org/10.3390/e21060604

**AMA Style**

Juangphanich P, De Maesschalck C, Paniagua G.
Turbine Passage Design Methodology to Minimize Entropy Production—A Two-Step Optimization Strategy. *Entropy*. 2019; 21(6):604.
https://doi.org/10.3390/e21060604

**Chicago/Turabian Style**

Juangphanich, Paht, Cis De Maesschalck, and Guillermo Paniagua.
2019. "Turbine Passage Design Methodology to Minimize Entropy Production—A Two-Step Optimization Strategy" *Entropy* 21, no. 6: 604.
https://doi.org/10.3390/e21060604