# Detailed Simulations of a Three-Stage Supercritical Carbon Dioxide Axial Compressor with a Focus on the Shrouded Stator Cavities Flow

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## Abstract

**:**

## 1. Introduction

## 2. Three-Stage Design

## 3. Numerical Setup

_{2}compressor and the sub-scaled three-stage design. Along with the results, files containing the parametric definitions of the 3D blade are also generated which is then used by T-Blade3, an in-house developed general parametric 3D blade geometry builder [16], to generate the 3D geometry of all blade rows for the compressor. The tool can create geometries based on a few basic parameters. Autogrid was used for generating the structured mesh and Cadence’s Fine/Turbo was used to perform high-fidelity CFD simulations. Turbulent Navier–Stokes was used as a flow model and the Spalart–Allmaras model was used for turbulence modeling. The boundary condition at inlet was defined by the total pressure profile generated from the estimation of the duct boundary layer at the University of Notre Dame’s testing facility and the total temperature was set to be 371.15 K. A turbulent viscosity ratio of 50 was defined at the inlet as a turbulent quantity. The mass imposed boundary condition was used at the outlet with a design mass flow rate of 127 kg/s and initial pressure of 6.28 MPa. The rotation speed of 19,800 rpm was set for rotors. While the primary flow converged within 10,000 iterations. It required 30,000 iterations on an AMD Ryzen Threadripper 3970X 32-Core Processor model utilizing 18 cores for full convergence of flow inside the cavities, taking approximately 200 h of wall clock time (about 8 days).

## 4. R/S Interfaces

## 5. Stage Performance

## 6. Near Hub Performance

## 7. Leakage Flow Characteristics

## 8. Windage Heating

## 9. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\dot{m}$ | Massflow rate |

A | Area |

${h}_{0}$ | Total Enthalpy |

$\overline{)V}$ | Volume |

$\mathit{\mu}$ | Viscosity |

$\mathit{\rho}$ | Density |

$\mathit{\tau}$ | Shear Stress |

p | Pressure |

v | Fluid Velocity |

V | Surface Velocity |

C_{p} | Specific heat at constant pressure |

f | Body force |

q | Heat flux |

TT | Total Temperature |

C_{f} | Coefficient of friction |

Subscripts | |

in | at inlet |

out | at outlet |

m | mean |

w | wall |

tot | total |

stg | stage |

Abbreviations | |

TPR | Total Pressure Ratio |

TTR | Total Temperature Ratio |

NLH | Non-linear Harmonic |

CFD | Computational Fluid Dynamics |

## References

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**Figure 1.**Three-stage design (

**a**) Cantilevered configuration (without cavities) and (

**b**) Shrouded configuration (with cavities).

**Figure 3.**Grid dependency study for fine and medium meshes. (

**a**) Isentropic efficiency and (

**b**) Total pressure ratio.

**Figure 4.**The y+ value comparison (

**a**) Cantilevered configuration (without cavities) and (

**b**) Shrouded configuration (with cavities).

**Figure 5.**Rotor–Stator Configurations. (

**a**) RS-1 (Cavities in the rotating frame). (

**b**) RS-2 (Cavities in the stationary frame). (

**c**) RS-3 (Cavities in the stationary/rotating frames). (

**d**) RS-4 (Cavities in the rotating/stationary frames).

**Figure 6.**Tangentially mass-averaged entropy comparison at the rotor inlets. (

**a**) at Rotor1 inlet. (

**b**) at Rotor2 inlet. (

**c**) at Rotor3 inlet.

**Figure 7.**Stream tubes drawn along S1 (Stator 1). (

**a**) for Cantilevered Case (

**b**) For Shrouded (NLH) case.

**Figure 8.**Stream tubes drawn along S2 (Stator 2). (

**a**) for Cantilevered Case (

**b**) For Shrouded (NLH) case.

**Figure 9.**Axial component of vorticity plotted at 1% span. (

**a**) at Stator 1 (cantilevered) and (

**b**) at Stator 1 (shrouded) (

**c**) at Stator 2 (cantilevered) (

**d**) at Stator2 (shrouded).

**Figure 10.**Cross-stream streamwise vorticity plots just upstream of stator leading edge (

**a**) for cantilevered configuration (without cavities) and (

**b**) for shrouded configuration (with cavities).

**Figure 11.**Tangentially mass-averaged entropy comparison at the rotor outlets. (

**a**) at Rotor1 outlet. (

**b**) at Rotor2 outlet. (

**c**) at Rotor3 outlet.

**Figure 12.**Vortical flow structures in Cavities (

**a**) Cavity-1. (

**b**) Cavity-2. (

**c**) Cavity-3. (

**d**) Cavity-4.

**Figure 13.**Axial velocity plotted at 1% span (

**a**) at Stator1 (cantilevered) and (

**b**) at Stator1 (shrouded) (

**c**) at Stator2 (cantilevered) (

**d**) at Stator2 (shrouded).

**Figure 16.**Tangentially mass-averaged span-wise axial velocity plots. (

**a**) at Rotor1 inlet. (

**b**) at Stator1 inlet. (

**c**) at Rotor2 inlet. (

**d**) at Stator2 inlet. (

**e**) at Rotor3 inlet. (

**f**) at Stator3 inlet.

**Figure 17.**Tangentially mass-averaged span-wise radial velocity plots. (

**a**) at Rotor1 inlet. (

**b**) at Stator1 inlet. (

**c**) at Rotor2 inlet. (

**d**) at Stator2 inlet. (

**e**) at Rotor3 inlet. (

**f**) at Stator3 inlet.

**Figure 18.**Total temperature contour in Cavity flow. (

**a**) Cavity-1, (

**b**) Cavity-2, (

**c**) Cavity-3, (

**d**) Cavity-4.

**Figure 19.**Tangentially mass-averaged span-wise total temperature plots. (

**a**) at Stator1 inlet. (

**b**) at Stator2 inlet. (

**c**) at Stator3 inlet. (

**d**) at Rotor1 inlet. (

**e**) at Rotor2 inlet. (

**f**) at Rotor3 inlet.

Row | Span-Wise Points | Grid Level | Hub Fillet (mm) | Tip Gap (mm) | Tip Fillet (mm) | Blade Count |
---|---|---|---|---|---|---|

IGV | 193 | 12 | 1.6 | - | 1.6 | 43 |

Rotor1 | 289 | 18 | 1.6 | 0.201 | - | 69 |

Stator1 | 193 | 12 | 1.6 | - | 1.6 | 114 |

Rotor2 | 305 | 20 | 1.6 | 0.192 | - | 88 |

Stator2 | 193 | 15 | 1.6 | - | 1.6 | 112 |

Rotor3 | 305 | 20 | 1.6 | 0.186 | - | 83 |

Stator3 | 193 | 15 | 1.6 | - | 1.6 | 101 |

Cantilevered | Shrouded | |
---|---|---|

(without Cavity) | (with Cavity) | |

First Cell Width | $1.2\times {10}^{-7}$ m | $5.22\times {10}^{-8}$ m |

IGV | 3.82 million | 3.5 million |

Rotor1 | 9.9 million | 7 million |

Stator1 | 8.6 million | 6.5 million |

Rotor2 | 11 million | 8.5 million |

Stator2 | 8 million | 6.2 million |

Rotor3 | 10 million | 8.5 million |

Stator3 | 8 million | 7.2 million |

Cavity1 | - | 3 million |

Cavity2 | - | 7.5 million |

Cavity3 | - | 8 million |

Cavity4 | - | 6.5 million |

Total Grid Points | 60 million | 72 million |

Configuration | Efficiency | Pressure Ratio | Massflow (kg/s) |
---|---|---|---|

Cantilevered | 89.46 | 2.538 | 127.10 kg/s |

NLH-Shrouded | 89.60 | 2.549 | 127.22 kg/s |

RS-1 | 88.95 | 2.455 | 127.20 kg/s |

RS-2 | 87.33 | 2.434 | 127.20 kg/s |

RS-3 | 87.62 | 2.425 | 127.21 kg/s |

RS-4 | 85.54 | 2.384 | 127.25 kg/s |

Flow | Mass Flow (kg/s) | Percentage of Main Flow (%) |
---|---|---|

Main Flow | 127.20 | 100 |

Cavity 1 | 0.0014 | 0.0011 |

Cavity 2 | 0.071 | 0.056 |

Cavity 3 | 0.098 | 0.077 |

Cavity 4 | 0.38 | 0.299 |

Configuration | Stage | ${\mathit{TPR}}_{\mathit{stg}}$ | ${\mathit{TTR}}_{\mathit{stg}}$ | ${\mathit{IsenEff}}_{\mathit{stg}}$ | ${\mathit{TPR}}_{\mathit{tot}}$ | ${\mathit{TTR}}_{\mathit{tot}}$ | ${\mathit{IsenEff}}_{\mathit{tot}}$ |
---|---|---|---|---|---|---|---|

Cantilevered | First | 1.380 | 1.079 | 0.904 | 2.5382 | 1.2412 | 0.8946 |

Second | 1.380 | 1.077 | 0.901 | ||||

Third | 1.335 | 1.068 | 0.905 | ||||

Shrouded | First | 1.389 | 1.081 | 0.900 | 2.5491 | 1.2419 | 0.8960 |

Second | 1.382 | 1.078 | 0.903 | ||||

Third | 1.330 | 1.067 | 0.910 |

Cav1_in | Cav2_out | Cav2_in | Cav3_out | Cav3_in | Cav4_in | |
---|---|---|---|---|---|---|

$\dot{m}$ (kg/s) | $1.07\times {10}^{-4}$ | $5.66\times {10}^{-4}$ | $5.66\times {10}^{-4}$ | $8.40\times {10}^{-4}$ | $8.40\times {10}^{-4}$ | $2.08\times {10}^{-4}$ |

${h}_{{0}_{m}}$ (kJ/kg) | 559.54 | 601.36 | 582.49 | 618.993 | 607.38 | 630.98 |

Periodicity | 43 | 114 | 114 | 112 | 112 | 101 |

$\dot{m}{h}_{{0}_{m}}$ (kW) | 2.647 | 38.797 | 37.580 | 58.235 | 57.142 | 132.691 |

$\Delta \left(\dot{m}{h}_{{0}_{m}}\right)$ (kW) | - | 1.217 | 1.093 | - | ||

$TT$ (K) | 373.736 | 418.614 | 401.936 | 440.092 | 431.344 | 457.337 |

$\Delta TT$ (K) | - | 16.678 | 8.748 | - |

Cavity 2 | Cavity 3 | |||||
---|---|---|---|---|---|---|

Analytical | CFD | Difference (%) | Analytical | CFD | Difference (%) | |

$\dot{m}$ (kg/s) | $5.66\times {10}^{-4}$ | $5.66\times {10}^{-4}$ | - | $8.40\times {10}^{-4}$ | $8.40\times {10}^{-4}$ | - |

$\Delta \left(\dot{m}{h}_{{0}_{m}}\right)$ (kW) | 1.343 | 1.217 | 9.4 | 1.198 | 1.093 | 12.6 |

$\Delta TT$ (K) | 20.60 | 16.678 | 19 | 12.60 | 8.748 | 30.5 |

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**MDPI and ACS Style**

Ghimire, S.; Turner, M.
Detailed Simulations of a Three-Stage Supercritical Carbon Dioxide Axial Compressor with a Focus on the Shrouded Stator Cavities Flow. *Processes* **2023**, *11*, 1358.
https://doi.org/10.3390/pr11051358

**AMA Style**

Ghimire S, Turner M.
Detailed Simulations of a Three-Stage Supercritical Carbon Dioxide Axial Compressor with a Focus on the Shrouded Stator Cavities Flow. *Processes*. 2023; 11(5):1358.
https://doi.org/10.3390/pr11051358

**Chicago/Turabian Style**

Ghimire, Saugat, and Mark Turner.
2023. "Detailed Simulations of a Three-Stage Supercritical Carbon Dioxide Axial Compressor with a Focus on the Shrouded Stator Cavities Flow" *Processes* 11, no. 5: 1358.
https://doi.org/10.3390/pr11051358