# Optimizing the Geometric Parameters of a Stepped Labyrinth Seal to Minimize the Discharge Coefficient

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## Abstract

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## 1. Introduction

## 2. Numerical Method and Verification

## 3. Results and Discussion

## 4. Conclusions

- In the stepped labyrinth seal, when CW, SH, and SP were varied, the discharge coefficient of SH was the most sensitive at a factor of approximately 30%;
- In the stepped labyrinth seal, when a small recirculation flow appeared at the clearance inlet, the clearance inlet pressure was reduced, and the axial velocity gently increased, which reduced the size of the vena contracta. The shapes with these characteristics exhibited a small discharge coefficient;
- The wall shear stress and discharge coefficient trends were very similar for the SH changes of the stepped labyrinth seal. However, no correlations existed between the wall shear stress and the discharge coefficient for CW and SP changes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\mathrm{A}}_{\mathrm{c}}$ | Cross-sectional area of labyrinth seal (${m}^{2}$) |

AVn | Normalized axial velocity $\mathrm{AVN}=\frac{\mathrm{local}\text{}\mathrm{axial}\text{}\mathrm{velocity}}{\mathrm{average}\text{}\mathrm{axial}\text{}\mathrm{velocity}}$ |

${\mathrm{C}}_{\mathrm{d}}$ | Discharge coefficient |

${\mathrm{C}}_{\mathrm{n}}$ | Normalized clearance ${\mathrm{C}}_{\mathrm{n}}=\frac{\mathsf{\xi}}{\mathrm{clearance}}$ |

CW | Cavity width |

$\dot{m}$ | Mass flow rate (kg/s) |

P | Pressure (Pa) |

R | Specific gas constant |

SH | Step height |

SP | Step position |

T | Temperature (K) |

W | Width of tooth |

Greek Symbols | |

$\gamma $ | Isentropic coefficient |

ξ | Coordinate from tooth tip |

Subscripts | |

id | Ideal |

in | Inlet |

out | Outlet |

Superscript | |

* | Normalized |

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**Figure 1.**Labyrinth seal geometrical sketch and parameters. (

**a**) Straight-through seal (

**b**) Stepped seal.

**Figure 2.**Computational domain of the straight-through seal. (

**a**) Whole shape of the grids. (

**b**) Enlarged part of the grids. (

**c**) Computational domain of the stepped labyrinth seal.

**Figure 4.**Comparison of the 2D and 3D discharge coefficients of the straight-through labyrinth seal.

**Figure 6.**Prediction of the discharge coefficient according to the changes in the turbulence model discharge coefficient using geometrical parameters. (

**a**) CW*. (

**b**) SH*. (

**c**) SP*.

**Figure 7.**Contours of the labyrinth-seal static pressure. (

**a**) Straight-through seal. (

**b**) Stepped seal at SH*1.4.

**Figure 8.**Change in the cavity flow structure according to CW*. (

**a**) High-discharge-coefficient group. (

**b**) Low-discharge-coefficient group.

**Figure 9.**Change in the cavity flow structure according to SH. (

**a**) High-discharge-coefficient group. (

**b**) Low-discharge-coefficient group.

**Figure 10.**Change in the cavity flow structure according to SP. (

**a**) High-discharge-coefficient group. (

**b**) Low-discharge-coefficient group.

**Figure 12.**Normalized axial-velocity changes by geometrical parameter at the second tooth. (

**a**) CW. (

**b**) SH. (

**c**) SP.

**Figure 13.**Changes in the wall shear stress of the geometrical parameters. (

**a**) CW*. (

**b**) SH*. (

**c**) SP*.

Geometry Parameter | Straight-Through Seal [16] | Stepped Seal |
---|---|---|

Number of teeth (N) | 6 | 5 |

Clearance (C) | 0.5 mm | 0.5 mm |

Tooth width (T) | 2.5 mm | 2.5 mm |

Tooth height (H) | 10.5 mm | 10.5 mm |

Cavity width (CW) | 9.5 mm | 3.5–33.5 mm |

Step height (SH) | - | 0–5.6 mm |

Step position (SP) | - | 0.15–10.35 mm |

Surface | Boundary Condition |
---|---|

Inlet | Pressure inlet, 300 K, 202,650 (Pa, absolute pressure) |

Outlet | Pressure outlet, 300 K, 101,325 (Pa, absolute pressure) |

Shaft | 0 RPM, adiabatic wall |

Casing | 0 RPM, adiabatic wall |

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**MDPI and ACS Style**

Chun, Y.H.; Ahn, J. Optimizing the Geometric Parameters of a Stepped Labyrinth Seal to Minimize the Discharge Coefficient. *Processes* **2022**, *10*, 2019.
https://doi.org/10.3390/pr10102019

**AMA Style**

Chun YH, Ahn J. Optimizing the Geometric Parameters of a Stepped Labyrinth Seal to Minimize the Discharge Coefficient. *Processes*. 2022; 10(10):2019.
https://doi.org/10.3390/pr10102019

**Chicago/Turabian Style**

Chun, Ye Hwan, and Joon Ahn. 2022. "Optimizing the Geometric Parameters of a Stepped Labyrinth Seal to Minimize the Discharge Coefficient" *Processes* 10, no. 10: 2019.
https://doi.org/10.3390/pr10102019