# Axial Compressor Mean-Line Analysis: Choking Modelling and Fully-Coupled Integration in Engine Performance Simulations

^{*}

## Abstract

**:**

## 1. Introduction

- De-coupled: the higher fidelity code is used to generate compressor characteristics which are then used as data in the conventional 0D cycle code (e.g., [3]).
- Semi-coupled: An iterative scheme is implemented, where the 0D cycle analysis provides boundary conditions to the 1D code and then the 0D component performance is updated according to the higher fidelity results. This loop is repeated until 0D and 1D performance is matched within a user-defined tolerance (e.g., [17]).
- Fully-coupled: The higher fidelity component is fully integrated in the cycle analysis (e.g., as an external object) directly replacing the corresponding 0D component [18].

## 2. Mean-Line Code (MLC)

## 3. Choke Modelling

#### 3.1. Indices for Annulus and Throat Passage Choke Modelling

#### 3.2. Estimating Choke Point

#### 3.3. Choked Part of a Speed-Line

## 4. 1D Component Model for Integration in 0D Cycle Analysis

_{choke}) below which choking occurs. The independent variables for reading this table are compressor corrected speed NcRdes and ${T}_{t,in}$. So, when the component operates in the 1D mode, the current value of BETA is compared with BETA

_{choke}to determine if operation is in the choked region or not.

_{choke}, the logic presented in Figure 2 is implemented, where for $\dot{m}={\dot{m}}_{MLC}^{*}$, the required $PR$ is the one obtained from the map, but corrected for the current compressor operating conditions, as illustrated in Figure 4. The algorithm establishes the choked row losses that will result to the required compressor exit pressure. The complete algorithmic logic of the new 0D/1D component is depicted in the flowchart of Figure 5.

## 5. Example Compressor Map Results

^{3}HPC [33]. NASA’s ISG-74A is a 3-stage transonic, high-speed compressor developed and tested during the 80s, while NASA/GE’s E

^{3}HPC is a 10-stage transonic, high-speed, high-aerodynamic loading compressor developed and tested during the late 70s-early 80s. Basic geometry and performance details for both compressors are summarized in Table 1.

#### 5.1. Choke Modelling

#### 5.2. Example of MLC Integration in Engine Model

^{3}HPC as far as the necessary stage, blade, and geometric input data required [33]. Following the pre-processing step, a map (Figure 13) is generated for four compressor inlet temperatures, covering the expected range for the considered application (${T}_{t,in}=$ 200, 288, 400, and 500 K). In fact the map is a set of 3D tables describing ${\dot{m}}_{c}$, $\eta $ and $PR$ in terms of BETA, NcRdes, and ${T}_{t,in}$. In addition, the table of BETA for each corrected speed below which choking occurs is produced (0.20 < BETA

_{choke}< 0.27).

- 1.
- 0D mode using single MLC-generated map at standard temperature of ${T}_{std}=288$ K.
- 2.
- 0D mode using MLC-generated maps according to actual HPC inlet temperature ${T}_{t,in}$
- 3.
- 1D mode with constant $\gamma $
- 4.
- 1D mode with variable gas properties

^{®}CoreTM2 Duo CPU at 3GHz with 8GB RAM) required less than 5 s for a complete engine simulation to converge to a tolerance of less than 10

^{−6}. In comparison, the 0D mode executes in less than 0.05 s.

_{choke}= 0.25. The setup of the simulation case was identical for both modes, meaning that choking was handled automatically in 1D mode as in 0D mode. Figure 17 shows the operating points on the map for each of the four cases (0D and 1D modes for the two NcRdes values). For the NcRdes value present in the map data (95%), the results of both 0D and 1D modes overlap on the map speed line and demonstrate that the 1D mode is capable to transition from the unchoked to the choked region and work in the choked region. This is also shown for the other NcRdes value (96.3%), although map interpolation causes visible differences between the two modes, thus highlighting one of the 0D approach limitations.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

0,1,2,3D | 0,1,2,3-Dimensional |

AMB | Ambient |

BRM | Blade Row Module |

BRN | Burner |

CFD | Computational Fluid Dynamics |

D13/25/30/45/50 | Interconnecting ducts |

GE | General Electric |

HP | High-Pressure |

HPC | High-Pressure Compressor |

HPS | High-Pressure Shaft |

HPT | High-Pressure Turbine |

IGV | Inlet Guide Vane |

INL | Engine Inlet |

ISA | International Standard Atmosphere |

IVM | Inter-Volume Module |

LE | Leading Edge |

LP | Low-Pressure |

LPC | Low-Pressure Compressor |

LPS | Low-Pressure Shaft |

LPT | Low-Pressure Turbine |

MLC | Mean-Line Code |

NASA | National Aeronautics and Space Administration |

NBP | Bypass Nozzle |

NCO | Core Nozzle |

OTAC | Object-oriented Turbomachinery Analysis Code |

PROOSIS | Propulsion Object Oriented SImulation Software |

R/S | Rotor/Stator |

SLS | Sea-Level Static |

TE | Trailing Edge |

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

$\mathit{A}\mathit{L}\mathit{T}$ | Altitude (m) |

$\mathit{\beta}$ | Flow angle relative to frame of reference (^{o}) |

$\mathit{\gamma}$ | Specific heats ratio (–); stagger angle (^{o}) (for loss and deviation correlations) |

$\mathit{\delta}$ | Deviation angle (^{o}) |

${\mathit{\delta}}_{\mathit{c}}$ | Radial clearance (m) |

$\mathit{\epsilon}$ | User-defined tolerance ($\mathit{\epsilon}>0$) |

$\mathit{\eta}$ | Compressor overall isentropic efficiency (–) |

${\mathit{\eta}}_{\mathit{p}}$ | Compressor overall polytropic efficiency (–) |

$\mathit{\theta}$ | Blade camber angle (^{o}) |

$\mathit{\nu}$ | Prandtl-Meyer function (^{o}) |

$\mathit{\rho}$ | Flow density (kg/m^{3}) |

$\mathit{\kappa}$ | Blade metal angle w.r.t. axial direction (^{o}) |

$\mathit{\sigma}$ | Blade row solidity (–) |

$\mathit{\omega}$ | Total pressure loss coefficient (–) |

$\mathit{a}$ | Position of blade maximum camber (m) |

$\mathit{A}\mathit{V}\mathit{D}\mathit{R}$ | Axial Velocity Density Ratio (–) |

$\mathit{b}$ | Blade row throat width for a single passage (m) |

${\mathit{c}}_{\mathit{p}}$ | Heat capacity at constant pressure (J/kgK) |

$\mathit{c}$ | Blade chord length (m) |

${\mathit{C}}_{\mathit{L}}$ | Blade lift coefficient (–) |

BETA | Auxiliary map parameter (–) |

$\mathit{D}{\mathit{F}}_{\mathit{e}\mathit{q}}$ | Equivalent diffusion factor (–) |

$\mathit{E}\mathit{G}\mathit{T}$ | Exhaust Gas Temperature (K) |

$\mathit{F},\text{}\mathit{G}$ | Function/ Functional |

$\mathit{F}\mathit{N}$ | Net Thrust (N) |

$\mathit{h}$ | Blade height (m) |

$\mathit{i}$ | Stage number (–); incidence angle (^{o}) |

$\mathit{g}$ | Distance between two blades measured normal to chord line (m) |

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

$\mathit{M}$ | Flow Mach number (–) |

$\mathit{N}$ | Compressor mechanical rotational speed (rpm) |

NcRdes | Corrected speed relative to design (–) |

$\mathit{p}$ | Pressure (Pa) |

$\mathit{P}\mathit{R}$ | Total Pressure Ratio (–) |

$\mathit{R}$ | Gas constant (J/kgK) |

$\mathit{s}$ | Blade row pitch length (m) |

$\mathit{S}\mathit{F}\mathit{C}$ | Specific Fuel Consumption (g/kN·s) |

$\mathit{t}$ | Blade thickness (m) |

$\mathit{T}$ | Temperature (K) |

$\mathit{W}$ | Flow velocity relative to frame of reference (m/s) |

${\mathit{{\rm Z}}}_{\mathit{s}\mathit{t}\mathit{g}}$ | Number of stages (–) |

1,2 | Blade row inlet (1) and outlet (2) |

$\mathit{b}\mathit{r}$ | Blade row |

$\mathit{c}$ | Corrected |

$\mathit{d}\mathit{e}\mathit{s}$ | Design conditions |

$\mathit{e}\mathit{x}$ | Compressor exit |

$\mathit{i}\mathit{n}$ | Compressor inlet |

$\mathit{i}\mathit{n}\mathit{a}\mathit{n}$ | Blade row inlet flow annulus |

$\mathit{L}\mathit{E}$ | Blade Leading Edge |

$\mathit{m}\mathit{a}\mathit{p}$ | Map quantity |

$\mathit{m}\mathit{i}\mathit{n}$ | Minimum |

$\mathit{s}\mathit{t}\mathit{d}$ | Standard-day conditions |

$\mathit{t}$ | Total conditions |

$\mathit{t}\mathit{h}$ | Blade row throat |

$\mathit{o}\mathit{u}\mathit{t}\mathit{a}\mathit{n}$ | Blade row outlet flow annulus |

$\mathit{w}$ | Quantity relative to frame of reference |

* | Choke conditions; design conditions (for loss and deviation correlations) |

## Appendix A. MLC Loss and Deviation Models

References | Correlation | Formula |
---|---|---|

[21] | Design incidence | ${\mathit{i}}^{*}=\mathit{i}\left(\mathit{\theta},\mathit{\sigma},{\mathit{\beta}}_{1},\mathit{t}/\mathit{c},\mathit{a}/\mathit{c}\right)$ |

[21] | Design deviation | ${\mathit{\delta}}^{*}=\mathit{\delta}\left(\mathit{\theta},\mathit{\sigma},{\mathit{\beta}}_{1},\mathit{t}/\mathit{c}\right)$ |

[22] | Off-design deviation | $\mathit{\delta}=\mathit{\delta}\left({\mathit{M}}_{\mathit{w}1},\mathit{D}{\mathit{F}}_{\mathit{e}\mathit{q}}\right)$ |

[26] | IGVs off-design deviation | $\mathit{\delta}=\mathit{\delta}\left({\mathit{M}}_{1},\mathit{\theta},\mathit{\sigma},\mathit{\gamma},\mathit{A}\mathit{V}\mathit{D}\mathit{R},\mathit{t}/\mathit{c}\right)$ |

[21,25] | Design profile loss corrected for Mach $({\mathit{s}}_{\mathit{M}\mathit{a}},{\mathit{a}}_{\mathit{M}\mathit{a}})$ and Reynolds $({\mathit{s}}_{\mathit{R}\mathit{e}}$$,{\mathit{a}}_{\mathit{R}\mathit{e}}$) number effects | ${\mathit{\omega}}_{\mathit{p}\mathit{r}}^{*}={\mathit{a}}_{\mathit{M}\mathit{a}}+{\mathit{s}}_{\mathit{M}\mathit{a}}{\mathit{s}}_{\mathit{R}\mathit{e}}\left[{\mathit{a}}_{\mathit{R}\mathit{e}}+\mathit{\omega}\left(\mathit{D}{\mathit{F}}_{\mathit{e}\mathit{q}}\right)\right]$ |

[21] | Off-design profile loss | ${\mathit{\omega}}_{\mathit{p}\mathit{r}}={\mathit{\omega}}_{\mathit{p}\mathit{r}}^{*}\mathit{f}\left(\mathit{i}-{\mathit{i}}^{*}\right)$ |

[24] | Shock loss | ${\mathit{\omega}}_{\mathit{s}\mathit{h}}=\mathit{\omega}\left({\mathit{M}}_{\mathit{w}1},\mathit{\sigma},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2}\right)$ |

[21] | Secondary loss | ${\mathit{\omega}}_{\mathit{s}\mathit{c}}=\mathit{\omega}\left({\mathit{C}}_{\mathit{L}},\mathit{\sigma},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2}\right)$ |

[21] | Endwall loss | ${\mathit{\omega}}_{\mathit{e}\mathit{w}}=\mathit{\omega}\left(\mathit{s}/\mathit{h},\mathit{\sigma},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2}\right)$ |

[23] | Clearance loss | ${\mathit{\omega}}_{\mathit{c}\mathit{l}}=\mathit{\omega}\left({\mathit{C}}_{\mathit{L}},\mathit{s},\mathit{h}/\mathit{c},{\mathit{\delta}}_{\mathit{c}},\mathit{\sigma},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2}\right)$ |

[21] | Overall loss | $\mathit{\omega}={\mathit{\omega}}_{\mathit{p}\mathit{r}}+{\mathit{\omega}}_{\mathit{s}\mathit{h}}+{\mathit{\omega}}_{\mathit{s}\mathit{c}}+{\mathit{\omega}}_{\mathit{e}\mathit{w}}+{\mathit{\omega}}_{\mathit{c}\mathit{l}}$ |

[26] | IGVs overall loss | $\mathit{\omega}=\mathit{\omega}\left({\mathit{M}}_{1},\mathit{\theta},\mathit{\sigma},\mathit{\gamma},\mathit{t}/\mathit{c}\right)$ |

## Appendix B. Blade Row Throat Passage Choke Modelling

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**Figure 5.**Algorithm flow chart to establish the auxiliary map parameter (BETA) for mass flow compatibility.

**Figure 6.**Comparison between MLC prediction and experimental map for NASA’s “Stage-35” [34] compressor.

**Figure 7.**MLC pressure ratio map for the example 3-stage compressor geometry [32].

**Figure 8.**Choke indices variation for the example 3-stage compressor geometry [32], for $N/{N}_{des}=85\%$: (

**a**) Minimum choke indices variation and (

**b**) S3 individual choke indices variation against compressor inlet mass flow.

**Figure 9.**Throat passage choking for (

**a**) S3 (at 85% compressor speed) and (

**b**) R1 (at 100% compressor speed), for the example 3-stage compressor geometry [32].

**Figure 10.**“Iso-choke” lines for the 100%, 102.5%, and 105% speed-lines for the example 3-stage compressor geometry [32].

**Figure 11.**MLC pressure-ratio map for the example three-stage compressor geometry [32] showing the “iso-choke” lines.

**Figure 12.**Schematic diagram of turbofan model with 1D compressor component. In the same diagram the components for representing the atmosphere (AMB), engine inlet (INL), fan (FAN), low- (LPC) and high-pressure compressor (HPC), low- (LPT) and high-pressure turbine (HPT), low- (LPS) and high-pressure shaft (HPS), burner (BRN), core (NCO) and bypass nozzle (NBP), and interconnecting ducts (D13/25/30/45/50), are also shown.

**Figure 15.**Results at steady state cruise conditions. (

**a**) Variation of HPC pressure ratio with thrust; (

**b**) 0D and 1D operating points on HPC map.

**Figure 16.**Results at transient cruise conditions. (

**a**) Differences of 0D from 1D mode for selected compressor and engine parameters; (

**b**) 0D and 1D transient operating lines on HPC map.

NASA ISG-74A [32] | NASA/GE E^{3} HPC [33] | |
---|---|---|

Design Pressure Ratio | 4.474 | 25 |

Design Inlet Mass Flow (kg/s) | 29.71 | 54.40 |

Design Rotating Speed (rpm) | 16042.3 | 12416.5 |

No. of Stages | 3 | 10 |

No. of Rotor Rows | 3 | 10 |

No. of Stator Rows | 4 [including inlet guide vanes (IGVs)] | 11 (including IGVs) |

**Table 2.**MLC first and last choke point results for the example 3-stage compressor geometry [32].

$\mathit{N}/{\mathit{N}}_{\mathit{d}\mathit{e}\mathit{s}}$ | First Choke Point Row-Station | Last Choke Point Row-Station |
---|---|---|

50–85% | S3-Throat | Compressor exit |

90–95% | R3-Throat | Compressor exit |

100–105% | R1-Throat | Compressor exit |

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

Engine Net Thrust (kN) | 59.2 |

Engine Overall Pressure Ratio (–) | 56.1 |

Engine Bypass Ratio | 8.85 |

Engine Specific Fuel Consumption (g/kN·s) | 15.52 |

Engine Inlet Mass Flow Rate (kg/s) | 463.5 |

Fuel Flow Rate (kg/s) | 0.918 |

High-Pressure Compressor (HPC) Exit Mass Flow Rate (kg/s) | 45.17 |

HPC Discharge Temperature (K) | 896.6 |

HPC Isentropic Efficiency (-) | 0.812 |

HPC Overall Pressure Ratio (-) | 23.9 |

High-Pressure Spool Rotational Speed (rpm) | 13,265.5 |

Combustor Exit Temperature (K) | 1715 |

Exhaust Gas Temperature (K) | 709.2 |

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**MDPI and ACS Style**

Kolias, I.; Alexiou, A.; Aretakis, N.; Mathioudakis, K. Axial Compressor Mean-Line Analysis: Choking Modelling and Fully-Coupled Integration in Engine Performance Simulations. *Int. J. Turbomach. Propuls. Power* **2021**, *6*, 4.
https://doi.org/10.3390/ijtpp6010004

**AMA Style**

Kolias I, Alexiou A, Aretakis N, Mathioudakis K. Axial Compressor Mean-Line Analysis: Choking Modelling and Fully-Coupled Integration in Engine Performance Simulations. *International Journal of Turbomachinery, Propulsion and Power*. 2021; 6(1):4.
https://doi.org/10.3390/ijtpp6010004

**Chicago/Turabian Style**

Kolias, Ioannis, Alexios Alexiou, Nikolaos Aretakis, and Konstantinos Mathioudakis. 2021. "Axial Compressor Mean-Line Analysis: Choking Modelling and Fully-Coupled Integration in Engine Performance Simulations" *International Journal of Turbomachinery, Propulsion and Power* 6, no. 1: 4.
https://doi.org/10.3390/ijtpp6010004