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Fan Stage Design and Performance Optimization for Low Specific Thrust Turbofans^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Streamline Curvature Model Setup

#### 2.2. Streamline Curvature Loss Correlations

#### 2.3. Blade Loading Parameter

- Calculate the loading parameter for each streamline and the associated annulus area fraction.
- Sort the streamlines by loading parameter.
- Calculate the cumulative distribution of the area with respect to the loading parameter.
- Calculate the loading parameter at a cumulative area fraction of $75\%$.

#### 2.4. Optimization Setup

#### 2.5. Surrogate Modeling

#### 2.6. Engine Systems Modeling

#### 2.6.1. Engine Performance

#### 2.6.2. Engine Dry Weight

## 3. Results

#### 3.1. Part: 1—Fan-Stage Parameter Interdependencies

#### 3.2. Part: 2—Engine Performance and Weight

#### 3.2.1. Engine Model Parameter Sweep

#### 3.2.2. Engine Model Trade-Off Study

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

AR | Aspect ratio |

DCA | Double circular arc |

HTR | Hub-tip-ratio |

PR | Pressure ratio |

LHS | Latin hyper-cube sampling |

SLC | Streamline curvature |

TSFC | Thrust specific fuel consumption |

Symbols | |

A | Area, (m${}^{2}$) |

c | Chord, (m) |

${d}_{fan}$ | Fan diameter, (m) |

D | Lieblein’s diffusion factor, (-) |

${D}^{*}$ | Normalized blade loading, (-) |

${D}_{eq}$ | Equivalent diffusion, (-) |

h | Specific enthalpy, ($\frac{\mathrm{J}}{\mathrm{kg}}$) |

i | Incidence angle, ($\xb0$) |

$\dot{m}$ | Mass flow, ($\xb0$) |

P | Pressure, (Pa) |

$r,z$ | Radial and axial coordinate, (m) |

${r}_{le},{r}_{te}$ | Leading and trailing edge radius, (m) |

s | Specific entropy, ($\frac{\mathrm{J}}{\mathrm{kgK}}$) |

T | Temperature, (K) |

t | Max profile thickness, (m) |

V | Velocity, (m/s) |

${V}_{w}$ | Whirl velocity, (m/s) |

U | Rotor blade velocity, (m/s) |

${W}_{RE}$ | Equivalent velocity ratio, (-) |

W | Weight (mass), (kg) |

Z | Blade count |

$\alpha $ | Flow angle, ($\xb0$) |

$\delta $ | Tip clearance, (m) |

${\eta}_{p}$ | Polytropic efficiency, (-) |

$\sigma $ | Solidity, (-) |

$\psi $ | Stage loading, (-) |

$\varphi $ | Flow coefficient, (-) |

$\mathit{\xi}$ | Design variable vector |

$\mathbf{\Phi}$ | Optimization parameter vector |

$\zeta $ | Entropy loss coefficient, (-) |

${\theta}_{w}$ | Wake momentum thickness (m) |

$\theta $ | Camber angle (rad) |

$\gamma $ | Stagger angle (rad) |

$\omega $ | Pressure loss coefficient |

Subscripts | |

1, 2, 3 … | Station number |

$hub$ | Parameter at the hub |

$i,j$ | Meridional and normal grid index |

R | Rotor |

S | Stator |

$tip$ | Parameter at the blade tip (or casing) |

t | Total/stagnation property |

$tgt$ | Parameter target value |

$r,z$ | Radial and axial component |

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**Figure 1.**Flow chart showing the modeling setup comprising three processes; the flow-geometry modeler, the curve parameterizer, and the streamline curvature program. Explicit inputs to the model are visualized as white parallelograms and are either explicitly specified as parametric attributes or used as design variables during the optimization. In order to be able to specify the average stage loading the normalized stage pressure ratio distribution ${\widehat{PR}}_{j}$ is scaled with a parameter ${k}^{PR}$ which is iterated upon until the output stage loading $\overline{\psi}$ matches the target value ${\overline{\psi}}_{tgt}$ within a given tolerance.

**Figure 2.**Meridional view of the computational domain for an example design: (

**a**) flow path geometry with station numbering 1–6; (

**b**) flow path with streamlines and quasi-orthogonal planes overlaid; and (

**c**) the meridional projection of the rotor and the stator blade outline.

**Figure 3.**Illustration of the parameterization of a spanwise distributed variable y in terms of a cubic Bezier curve.

**Figure 4.**Prediction profiler of fan-stage polytropic efficiency and rotor solidity in design. Solid black curves shows the surrogate model prediction when a given parameter is varied around a reference value and other parameters are fixed. The reference values are shown by vertical dashed lines. The gray scatter plots display the results upon which the surrogate model is trained.

**Figure 5.**Contours of polytropic efficiency against stage loading and flow coefficient for two different fan-face Mach numbers—predicted by the surrogate model. Rotor aspect ratio and blade loading parameter held fixed at their respective reference values.

**Figure 6.**Results from the parameter sweep of the engine systems model. Each column (

**a**–

**c**) shows responses when varying one design variable keeping the others constant at their baseline value. All y axis values are further normalized with their respective baseline.

**Figure 7.**Engine optimization results with (

**a**) showing the trade-off between the two objectives for Pareto-optimal designs and (

**b**) showing the associated response in terms of the design variables along the Pareto-front. Dashed lines showing results from optimizations with the only jet velocity ratio as the design variable. All values are normalized with the corresponding baseline value. The baseline design in terms of thrust specific fuel consumption and engine dry weight is shown in figure (

**a**) as the star.

Parameter | Value Range | Comment |
---|---|---|

Inlet Mach nr. | 0.55–0.65 | Maximized to reduce fan frontal area. |

Exit Mach nr. | 0.3–0.5 | High values will increase losses and may cause choking in downstream components. |

Rotor rel. tip Mach nr. | 1.2–1.5 | High values associated with excessive shock losses. |

Exit swirl angle | 0.0 | Residual swirl downstream of the OGV will reduce the thrust. |

Rotor tip speed | <500 m/s | Limited from above to preserve mechanical integrity. Impacts disc stress levels. |

Hub-to-tip ratio | 0.3–0.4 | At high values, tip clearance will be a larger percentage of blade height. At low values, higher disc and blade stress. |

Stage loading | 0.35–0.55 | Higher values are usually assumed to be associated with reduced polytropic efficiency. |

Rotor axial aspect ratio | 2.0–2.5 | High aspect ratio is beneficial for weight but at the expense of surge margin and efficiency. |

Solidity | - | Specified to avoid excessive diffusion. Related to surge margin, cost and weight. |

**Table 2.**Variable and parameter values and ranges for the parametric optimization. For each combination of optimization parameter values, a Pareto frontier is generated to establish a trade-off between efficiency and the blade loading parameter. Design variables in bold refer to vectors with a size shown in parenthesis. The total number of scalar design variables is 18.

Variable | Comment | Value | Type |
---|---|---|---|

${r}_{1,tip}$ | Inlet casing radius | $1.0$ m | Fixed parameter |

$HT{R}_{1}$ | Inlet hub-tip-ratio | $0.3$ | Fixed parameter |

${\alpha}_{1}$ | Inlet swirl angle | $0.0\xb0$ | Fixed parameter |

${\alpha}_{5}$ | Stator outlet swirl angle | $0.0\xb0$ | Fixed parameter |

${t}_{R}$ | Rotor maximum profile thickness | Distribution | Fixed parameter |

${t}_{S}$ | Stator maximum profile thickness | Distribution | Fixed parameter |

${\overline{\psi}}_{tgt}$ | Stage loading coefficient | $[0.35,0.85]$ | Opti. parameter |

$\overline{\varphi}$ | Stage flow coefficient | $[0.45,1.05]$ | Opti. parameter |

${M}_{2}$ | Fan-face Mach number | $[0.45,0.75]$ | Opti. parameter |

$A{R}_{R}$ | Rotor section aspect ratio | $[1.2,2.4]$ | Opti. parameter |

$A{R}_{S}$ | Stator section aspect ratio | $[1.0,4.5]$ | Design variable |

${A}_{3}/{A}_{2}$ | Rotor section area ratio | $[0.75,1.0]$ | Design variable |

${A}_{5}/{A}_{4}$ | Stator section area ratio | $[0.85,1.0]$ | Design variable |

${Z}_{R}$ | Number of rotor blades | $[10,35]$ | Design variable |

${Z}_{S}$ | Number of stator blades | $[15,65]$ | Design variable |

${\mathit{\xi}}^{PR}$ | Stage pressure ratio dist. parameters | - | Design variable (3) |

${\mathit{\xi}}_{R}^{c}$ | Rotor chord dist. parameters | - | Design variable (1) |

${\mathit{\xi}}_{S}^{c}$ | Stator chord dist. parameters | - | Design variable (1) |

${\mathit{\xi}}_{R}^{i}$ | Rotor incidence dist. parameters | - | Design variable (4) |

${\mathit{\xi}}_{S}^{i}$ | Stator incidence dist. parameters | - | Design variable (4) |

**Table 3.**Design point operating condition and baseline fan-stage parameters. Assumed representative of a 2010 geared turbofan engine during initial cruising at a max-cruise rating for the design mission.

Parameter | Value | |
---|---|---|

Operating conditions | Altitude | 33.0 kft |

Flight Mach nr. | 0.78 | |

Req. net thrust (one engine) | 26.0 kN | |

Fan-stage baseline | Polytropic efficiency | 0.9365 |

Rotor solidity | 1.4546 | |

Stage loading | 0.5742 | |

Flow coefficient | 0.8999 | |

Fan-face Mach nr. | 0.6488 |

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## Share and Cite

**MDPI and ACS Style**

Sjögren, O.; Grönstedt, T.; Lundbladh, A.; Xisto, C.
Fan Stage Design and Performance Optimization for Low Specific Thrust Turbofans. *Int. J. Turbomach. Propuls. Power* **2023**, *8*, 53.
https://doi.org/10.3390/ijtpp8040053

**AMA Style**

Sjögren O, Grönstedt T, Lundbladh A, Xisto C.
Fan Stage Design and Performance Optimization for Low Specific Thrust Turbofans. *International Journal of Turbomachinery, Propulsion and Power*. 2023; 8(4):53.
https://doi.org/10.3390/ijtpp8040053

**Chicago/Turabian Style**

Sjögren, Oliver, Tomas Grönstedt, Anders Lundbladh, and Carlos Xisto.
2023. "Fan Stage Design and Performance Optimization for Low Specific Thrust Turbofans" *International Journal of Turbomachinery, Propulsion and Power* 8, no. 4: 53.
https://doi.org/10.3390/ijtpp8040053