# Optimizing the Geometric Parameters of a Straight-Through Labyrinth Seal to Minimize the Leakage Flow Rate and the Discharge Coefficient

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometry and Boundary Conditions

## 3. Validation of Numerical Methods

_{t}is assumed to be the same in all directions, and the assumption shows good results for a lot of simple turbulent flows, even though the assumption is not true [22]. Equations (1)–(7) could not be solved analytically, thus CFD simulations were the only way to obtain solutions for the equations, which required a turbulence model to close the equations mathematically.

^{−6}were used as the solution convergence criteria. Figure 4 shows the 3D baseline mesh of the straight-through labyrinth seal composed of around 8 million hexahedron cells; the clearance was 0.5 mm, the tooth inclination angle was 0°, the tooth height was 10.5 mm, the cavity width was 9.5 mm, the tooth width was 2.5 mm, and the number of cavities was 5, as described in Wittig et al. [15]. Y

^{+}values around the walls were set to less than 3 so as to capture the normal velocity gradient of the wall in the viscous sublayer accurately.

## 4. Results

#### 4.1. The Effect of Varying the Clearance

#### 4.2. The Effect of Varying the Tooth Inclination Angle

#### 4.3. The Effect of Varying the Tooth Height

#### 4.4. The Effect of Varying the Cavity Width

#### 4.5. The Effect of Varying the Tooth Width

#### 4.6. Sensitivity Analysis of the Geometrical Parameters

#### 4.7. An Example of Optimized Geometry for a Straight-Through Labyrinth Seal

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\mathit{A}}_{\mathit{c}}$ | Cross-sectional area of the seal |

C_{d}C _{p}k | Discharge coefficient = $\dot{m}/\dot{{m}_{id}}$ = Real mass flow rate / Ideal mass flow rate Specific heat of fluid Turbulence |

k ${\mathit{T}}_{\mathit{o}\mathit{,}\mathit{i}\mathit{n}}$ | Isentropic coefficient Temperature at the inlet |

u u’ ${\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}$ ${\mathit{P}}_{\mathit{o}\mathit{,}\mathit{i}\mathit{n}}$ R | Flow velocity [m/s] Root-mean-square of the turbulent velocity fluctuations (m/s) Pressure at the outlet Pressure at the inlet Specific gas constant |

PR | Pressure ratio |

$\dot{{\mathit{m}}_{\mathit{i}\mathit{d}}}$ | ${\dot{m}}_{id}=\frac{{P}_{o,in}{A}_{c}}{\sqrt{{T}_{o,in}}}\sqrt{\frac{2k}{R\left(k-1\right)}\left[{\left(\frac{{P}_{out}}{{P}_{o,in}}\right)}^{\frac{2}{k}}-{\left(\frac{{P}_{out}}{{P}_{o,in}}\right)}^{\frac{k+1}{k}}\right]}$ |

x | X-direction coordinate |

y | Y-direction coordinate |

z | Z-direction coordinate |

Greek symbols | |

β | Jet divergence angle |

$\mathit{\rho}$ ν | Density (kg/m^{3})Local kinematic viscosity (m ^{2}/s) |

µ | Local dynamic viscosity (m^{2}/s) |

ε ω | Dissipation rate of turbulent kinetic energy Specific dissipation rate of turbulent kinetic energy |

Subscripts | |

id | Ideal |

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**Figure 4.**Baseline mesh of the labyrinth seal (

**a**) whole shape of the mesh, (

**b**) enlarged part of the mesh, and (

**c**) cross-sectional view of the mesh.

**Figure 5.**Grid sensitivity test of the Reynolds-Averaged Navier–Stokes (RANS) calculation for the effect of the pressure ratio on the discharge coefficient of the baseline labyrinth seal.

**Figure 6.**Effect of pressure ratio on the discharge coefficient of the baseline labyrinth seal. C

_{d}—discharge coefficient.

**Figure 7.**Pressure distribution in the labyrinth seal for a pressure ratio of 1.38. Fin number refers to the number assigned to six fins in the baseline labyrinth seal.

**Figure 8.**Effect of changing the clearance on the discharge coefficient of the labyrinth seal. C

_{d}—discharge coefficient.

**Figure 9.**(

**a**–

**c**) The effect of changing the clearance (C) on the streamlines of the labyrinth seal under a pressure ratio of 2.

**Figure 10.**The effect of varying the tooth inclination angle on the discharge coefficient of the labyrinth seal under a pressure ratio of 2. The y-axis units on the right-hand side plot have been expanded. C

_{d}—discharge coefficient.

**Figure 11.**(

**a**–

**f**) The effect of varying the tooth inclination angle (A) on the streamlines of the labyrinth seal under a pressure ratio of 2.

**Figure 12.**Effect of changing the ratio of the tooth height to the clearance (H/C) on the discharge coefficient of the labyrinth seal under a pressure ratio of 2. The x-axis units on the right-hand side plot have been expanded. C

_{d}—diffusion coefficient.

**Figure 13.**(

**a**–

**h**) The effect of changing the ratio of the tooth height to the clearance (H/C) on the streamlines of a straight-through labyrinth seal.

**Figure 14.**The effect of the cavity width of the clearance (S/C) ratio on the discharge coefficient of a straight-through labyrinth seal under a pressure ratio of 2 when the whole axial length is fixed.

**Figure 15.**(

**a**–

**e**) The effect of the cavity width to the clearance (S/C) ratio on the discharge coefficient of a straight-through labyrinth seal under a pressure ratio of 2 when the whole axial length is fixed.

**Figure 16.**The effect of varying the tooth width to clearance (W/C) ratio on the streamlines of a straight-through labyrinth seal under a pressure ratio of 2 when the whole axial length was fixed.

**Figure 17.**(

**a**–

**f**) The effect of changing the tooth width to clearance (W/C) ratio on the streamlines of a straight-through labyrinth seal.

**Figure 18.**Sensitivity analysis of the geometric parameters for decreasing the discharge coefficient.

**Figure 19.**Streamlines of the labyrinth seal under a pressure ratio of 2 for (

**a**) baseline and the (

**b**) suggested design.

**Figure 20.**Contours of the static pressure in the labyrinth seal under a pressure ratio of 2 for (

**a**) baseline and the (

**b**) suggested design.

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

Inlet | Pressure inlet |

Outlet | Pressure outlet |

Stator | Wall |

Rotor | Wall (no rotation) |

Grid | Number of Cells in the r Direction | Number of Cells in the θ Direction | Number of Cells in the z Direction | Total Cells Number |
---|---|---|---|---|

First | 70 | 100 | 590 | 3,245,000 |

Second | 90 | 120 | 660 | 5,456,400 |

Third | 110 | 130 | 730 | 7,967,700 |

Fourth | 115 | 135 | 765 | 9,134,100 |

Fifth | 120 | 140 | 800 | 10,413,200 |

Clearance (mm) | Jet Divergence Angle (°) |
---|---|

0.5 | 1.66689 |

1.0 | 1.47669 |

2.5 | 0.90164 |

Tooth Inclination Angle (°) | Jet Divergence Angle (°) |
---|---|

0 | 1.66689 |

4 | 2.05151 |

8 | 2.16896 |

12 | 2.62472 |

16 | 2.80864 |

20 | 2.77528 |

Clearance (mm) | Tooth Inclination Angle (°) | Tooth Height (mm) | Cavity Width (mm) | Tooth Width (mm) | Number of Cavities | |
---|---|---|---|---|---|---|

Baseline | 0.5 | 0 | 10.5 | 9.5 | 2.5 | 5 |

Suggested geometry | 0.5 | 16 | 2.1 | 3.5 | 0.5 | 15 |

**Table 6.**Jet divergence angle and discharge coefficients for the baseline and suggested optimal geometry for a labyrinth seal under pressure ratios of 1.1, 1.5, and 2.

Pressure Ratio | Jet Divergence Angle (°) | Discharge Coefficient | ||
---|---|---|---|---|

Baseline | Suggested Geometry | Baseline | Suggested Geometry | |

1.1 | 2.14659 | 3.08846 | 0.3773 | 0.2904 |

1.5 | 1.82968 | 2.39175 | 0.4166 | 0.3593 |

2 | 1.66689 | 2.03137 | 0.454 | 0.39978 |

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**MDPI and ACS Style**

Baek, S.I.; Ahn, J.
Optimizing the Geometric Parameters of a Straight-Through Labyrinth Seal to Minimize the Leakage Flow Rate and the Discharge Coefficient. *Energies* **2021**, *14*, 705.
https://doi.org/10.3390/en14030705

**AMA Style**

Baek SI, Ahn J.
Optimizing the Geometric Parameters of a Straight-Through Labyrinth Seal to Minimize the Leakage Flow Rate and the Discharge Coefficient. *Energies*. 2021; 14(3):705.
https://doi.org/10.3390/en14030705

**Chicago/Turabian Style**

Baek, Seung Il, and Joon Ahn.
2021. "Optimizing the Geometric Parameters of a Straight-Through Labyrinth Seal to Minimize the Leakage Flow Rate and the Discharge Coefficient" *Energies* 14, no. 3: 705.
https://doi.org/10.3390/en14030705