# Prediction of Heat Transfer in a Jet Cooled Aircraft Engine Compressor Cone Based on Statistical Methods

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedure

#### 2.1. Design Parameters

#### 2.2. Design Parameter Space

## 3. Results and Discussion

#### 3.1. Measurement Uncertainty

#### 3.2. Sensitivity Study

#### 3.3. Derived Correlation

#### 3.4. Correlation Quality

## 4. Summary and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Latin Symbols | |

A | Area (m^{2}) |

d | Diameter (m) |

C | Coefficient of a model (variable) |

H | Structural thickness (m) |

L | Characteristic length (m) |

$\dot{Q}$ | Heat flux (W) |

T | Temperature (K) |

$\dot{V}$ | Volumetric flow (m^{3}/s) |

W | Mechanical power (W) |

c_{p} | Specific heat (J/(kg K)) |

e | Exponent of a model (–) |

h | Heat transfer coefficient (W/m^{2}/K) |

k | Thermal conductivity (W/(m K)) |

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

n | Number of inlet holes (–) |

p | Pressure (Pa) |

$\dot{q}$ | Heat flux density (W/m^{2}) |

r | Radial coordinate (m) |

s | Gap width (m) |

u | Radial velocity component u = u_{r} (m/s) |

z | Axial coordinate (m) |

Greek Symbols | |

Δ | Absolute uncertainty (variable) |

γ | Incidence angle (rad) |

μ | Dynamic viscosity (Pa s) |

ν | Kinematic viscosity (m/s^{2}) |

ω | Angular velocity (rad/s) |

ϕ | Angular coordinate (rad) |

ρ | Density (kg/m^{3}) |

σ | Normal stress (N/m^{2}) |

θ | Cone angle (rad) |

Non-dimensional Groups | |

Bi | Biot number Bi = h · H/k_{S} |

Non-dimensional Groups | |

C_{w} | Non-dim. mass flow C_{w} = $\dot{m}$/(μ · r_{o}) = $\dot{V}$/(v · r_{o}) |

D | Non-dim. jet hole diameter D = d/(2 · r_{o}) |

E_{c} | Eckert number E_{c} = u^{2}/(c_{p} · ΔT) |

G | Gap ratio G = s/r_{o} |

K_{1} | Core rotation factor at gap inlet K_{1} = (cos γ · u_{jet})/(r_{o} · ϕ) |

Nu | Nusselt number Nu = h · r_{o}/k_{f} |

$\overline{\mathit{Nu}}$ | Global Nusselt number $\overline{\mathit{Nu}}$ = ʃ Nu · ΔT dA/ʃ ΔT dA |

Pr | Prandtl number Pr = μ · c_{p}/k |

Re | Reynolds number Re = u · r_{o}/v |

X | Radius ratio X = r_{i}/r_{o} |

R | Recovery factor R = Pr^{1/3} |

Indices | |

1 | Gap inlet property |

ad | Adiabatic |

c | Cooled |

f | Fluid |

h | Heated |

i | Inner |

jet | Jet |

o | Outer |

ϕ | Circumferential |

S | Solid |

W | Wall |

L | Leakage |

Abbreviations | |

RMSE | Root Mean Square Error |

## Appendix A. Finite-Element Model

**Figure A1.**Boundary conditions of finite-element model and discretization of the rotor-stator system. (

**a**) Finite-element model including boundary conditions: interpolated temperature and measurement point ( ), extrapolated temperature (―). (

**b**) Discretization of the rotor-stator gap to calculate the adiabatic wall temperature.

## Appendix B. Adiabatic Wall Temperature

## References

- Von der Bank, R.; Donnerhack, S.; Rae, A.; Cazalens, M.; Lundbladh, A.; Dietz, M. LEMCOTEC—Improving the Core-Engine Thermal Efficiency: Paper-Nr. GT2014-25040. In Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, Düsseldorf, Germany, 16–20 June 2014. [Google Scholar]
- Kervistin, R. Cooling System for a Gas Turbine Engine Compressor. U.S. Patent No. 5297386, 29 March 1994. [Google Scholar]
- Bleier, F.; Hufnagel, M.; Pychynski, T.; Eichler, C.; Bauer, H.J. Numerical Conjugate Heat Transfer Study of a Cooled Compressor Rear Cone: Paper-Nr. GT2014-26289. In Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, Düsseldorf, Germany, 16–20 June 2014. [Google Scholar]
- Bleier, F.; Höfler, C.; Pychynski, T.; Bauer, H.J. Design and Validation of a Test Rig for Heat Transfer Measurements on a Rotating Cone: Paper-Nr. ISABE-2015-20273. In Proceedings of the 22nd International Symposium on Air Breathing Engines, Phoenix, AZ, USA, 25–30 October 2015. [Google Scholar]
- Fénot, M.; Bertin, Y.; Dorignac, E.; Lalizel, G. A review of heat transfer between concentric rotating cylinders with or without axial flow. Int. J. Therm. Sci.
**2011**, 50, 1138–1155. [Google Scholar] [CrossRef] - Harmand, S.; Pellé, J.; Poncet, S.; Shevchuk, I.V. Review of fluid flow and convective heat transfer within rotating disk cavities with impinging jet. Int. J. Therm. Sci.
**2013**, 67, 1–30. [Google Scholar] [CrossRef] - Owen, J.M.; Rogers, R.H. Flow and Heat Transfer in Rotating-Disc Systems: Rotating Cavities; Research Studies Press: Taunton, UK; John Wiley & Sons: New York, NY, USA, 1995; Volume 2. [Google Scholar]
- Owen, J.M.; Rogers, R.H. Flow and Heat Transfer in rotating-Disc Systems: Rotor-Stator Systems; Research Studies Press: Taunton, UK; John Wiley & Sons: New York, NY, USA, 1989; Volume 1. [Google Scholar]
- Alexiou, A.; Hills, N.J.; Long, C.A. Heat Transfer in High-Pressure Compressor Gas Turbine Internal Air Systems: A Rotating Disc-Cone Cavity with Axial Throughflow. Exp. Heat Transf.
**2000**, 13, 299–328. [Google Scholar] [CrossRef] - Wimmer, M.; Zierep, J. Transition from Taylor vortices to cross-flow instabilities. Acta Mech.
**2000**, 140, 17–30. [Google Scholar] [CrossRef] - Long, C.A.; Turner, A.B.; Kais, G.; Tham, K.M.; Verdicchio, J.A. Measurement and CFD Prediction of the Flow within an HP Compressor Drive Cone. J. Turbomach.
**2003**, 125, 165–172. [Google Scholar] [CrossRef] - Dorfman, L.A. Hydrodynamic Resistance and the Heat Loss of Rotating Solids; Oliver & Boyd: Edinburgh, Scotland, UK, 1963. [Google Scholar]
- Daily, J.W.; Nece, R.E. Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks. J. Basic Eng.
**1960**, 82, 217–230. [Google Scholar] [CrossRef] - Pellé, J.; Harmand, S. Heat transfer measurements in an opened rotor-stator system air-gap. Exp. Therm. Fluid Sci.
**2007**, 31, 165–180. [Google Scholar] [CrossRef] - Mitchell, J.W.; Metzger, D.E. Heat Transfer From a Shrouded Rotating Disk to a Single Fluid Stream. J. Heat Transf.
**1965**, 87, 485. [Google Scholar] [CrossRef] - Djaoui, M.; Dyment, A.; Debuchy, R. Heat transfer in a rotor-stator system with a radial inflow. Eur. J. Mecha.-B/Fluids
**2001**, 20, 371–398. [Google Scholar] [CrossRef] - Tang, H.; Shardlow, T.; Michael Owen, J. Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks. J. Turbomach.
**2015**, 137, 121003. [Google Scholar] [CrossRef] - Geis, T. Strömung und Reibungsinduzierte Leistungs- und Wirkungsgradverluste in Komplexen Rotor-Stator Zwischenräumen. Ph.D. Thesis, Department of Mechanical Engineering, University of Karlsruhe, Baden-Württemberg, Germany, 2002. [Google Scholar]
- Seber, G.A.F.; Wild, C.J. Nonlinear Regression; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1989. [Google Scholar]

**Figure 1.**Heat transfer regimes for an open discoidal rotor-stator system (Reprinted from [14] with permission from Elsevier): regime I (laminar, merged boundary layers), regime II (laminar, separated boundary layers), regime III (turbulent, merged boundary layers), and regime IV (turbulent, separated boundary layers).

**Figure 2.**Scheme of the physical problem with variable input parameters and test rig setup with schematic illustration of the temperature measurements ( ) [4].

**Figure 4.**Comparison between nominal test plan and realized test plan for Reynolds number ${\mathit{Re}}_{\phi}$ and mass flow ${\mathit{C}}_{\mathit{w}}$.

**Figure 5.**Comparison between nominal test plan and realized test plan for Reynolds number ${\mathit{Re}}_{\phi}$ and mass flow ${\mathit{C}}_{\mathit{w}}$.

**Figure 6.**Detailed uncertainty analysis including linear propagation of uncertainties and repeatability of the experiments. (

**a**) Nusselt number uncertainty for all measurements. (

**b**) Deviation between repeated measurements.

**Figure 7.**Main effects of all variable parameters on the averaged relative Nusselt number change $\overline{\delta \overline{\mathit{Nu}}}$.

**Figure 8.**Illustration of the interactions between the variable parameters and their effect on the averaged global Nusselt number $\overline{\Delta \overline{\mathit{Nu}}}$.

**Figure 9.**Absolute and relative distribution of the residuals between predicted Nusselt numbers ${\overline{\mathit{Nu}}}^{*}$ and measured global Nusselt numbers $\overline{\mathit{Nu}}$.

**Figure 10.**Comparison between measured data (symbols) and the derived correlation (lines) as a function of the mass flow rate ${\mathit{C}}_{\mathit{w}}$ for each incidence angle $\gamma $ and the varations: ${\mathit{Re}}_{\phi 1}$: ${\mathit{G}}_{1}$ ( ), ${\mathit{G}}_{2}$ ( ), ${\mathit{G}}_{3}$ ( ); ${\mathit{Re}}_{\phi 2}$: ${\mathit{G}}_{1}$ ( ), ${\mathit{G}}_{2}$ ( ), ${\mathit{G}}_{3}$ ( ); ${\mathit{Re}}_{\phi 3}$: ${\mathit{G}}_{1}$ ( ), ${\mathit{G}}_{2}$ ( ), ${\mathit{G}}_{3}$ ( ).

Parameter | Definition | Test Rig Conditions | Engine Conditions [8,9,18] |
---|---|---|---|

Circ. Reynolds number ${\mathit{Re}}_{\phi}$ | $\left(\right)open="("\; close=")">\omega \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{o}}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\varrho $ | < 5 · 10^{6} | > 10 · 10^{6} |

Non-dim. mass flow rate ${\mathit{C}}_{\mathit{w}}$ | $\dot{m}/\left(\right)open="("\; close=")">\mu \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{o}}$ | 16 · 10^{3} to 25 · 10^{3} | 5 · 10^{3} to 50 · 10^{3} |

Relative gap width $\mathit{G}$ | $s/{r}_{\mathrm{o}}$ | 0.03 to 0.11 | >0.05 |

Incidence angle $\gamma $ | $\frac{\pi}{6}$ to $\frac{\pi}{2}$ | $\frac{\pi}{6}$ to $\frac{\pi}{2}$ | |

Jet diameter ratio $\mathit{D}$ | $d/\left(\right)open="("\; close=")">2\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{o}}$ | 6.82 · 10^{−3} | tbd |

Number of jets n | 48 | tbd | |

Leakage mass flow rate ${\mathit{C}}_{\mathit{w}L}$ | ${\dot{m}}_{L}/\left(\right)open="("\; close=")">\mu \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{o}}$ | 8 · 10^{3} | |

Diameter ratio $\mathit{X}$ | ${r}_{\mathrm{i}}/{r}_{\mathrm{o}}$ | 0.45 | >0.4 |

Cone angle $\theta $ | 35° | 36° | |

Initial swirl ratio ${\mathit{K}}_{1}$ | ${u}_{\phi ,\mathrm{jet}}/\left(\right)open="("\; close=")">\omega \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{o}}$ | <1.4 | <1.5 |

Biot number $\mathit{Bi}$ | $h\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}H/{k}_{\mathrm{S}}$ | 0.2 to 5 | 0.1 to 5 |

Eckert number $\mathit{Ec}$ | $\Delta {u}_{\phi}^{2}/\left(\right)open="("\; close=")">{c}_{p}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\Delta T$ | <2 | <1 |

Prandtl number $\mathit{Pr}$ | ${c}_{p}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mu /{k}_{\mathrm{f}}$ | 0.71 | 0.72 |

Group | ${\mathit{Re}}_{\mathit{\phi}}$/10^{6} | ${\mathit{C}}_{\mathit{w}}$/10^{4} | $\mathit{G}$ | $\mathit{\gamma}$ |
---|---|---|---|---|

1 | 3.08 | 1.61 | 0.03 | $\pi /6$ |

2 | 3.83 | 2.12 | 0.07 | $\pi /4$ |

3 | 4.57 | 2.45 | 0.11 | $\pi /3$ |

4 | – | – | – | $\pi /2$ |

**Table 3.**Summary of the observed main effects (major diagonal) and interactions (minor diagonals) for increasing the variable parameters from the lowest to the highest factor level.

${\mathit{Re}}_{\mathit{\phi}}$ | ${\mathit{C}}_{\mathit{w}}$ | $\mathit{G}$ | $\mathit{\gamma}$ | |
---|---|---|---|---|

${\mathit{Re}}_{\phi}$ | $++$ | − | ∘ | ∘ |

${\mathit{C}}_{\mathit{w}}$ | − | ∘ | ∘ | + |

$\mathit{G}$ | ∘ | ∘ | $--$ | + |

$\gamma $ | ∘ | + | + | $+++$ |

**Table 4.**Quality parameters of the correlation according to (Equation (12)).

${\mathit{R}}^{2}$ in % | RMSE | $\mathit{\mu}$ in % | $\mathit{\sigma}$ in % |
---|---|---|---|

96.3 | 129 | −0.812 | 7.53 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bleier, F.; Schwitzke, C.; Bauer, H.-J.
Prediction of Heat Transfer in a Jet Cooled Aircraft Engine Compressor Cone Based on Statistical Methods. *Aerospace* **2018**, *5*, 51.
https://doi.org/10.3390/aerospace5020051

**AMA Style**

Bleier F, Schwitzke C, Bauer H-J.
Prediction of Heat Transfer in a Jet Cooled Aircraft Engine Compressor Cone Based on Statistical Methods. *Aerospace*. 2018; 5(2):51.
https://doi.org/10.3390/aerospace5020051

**Chicago/Turabian Style**

Bleier, Fabian, Corina Schwitzke, and Hans-Jörg Bauer.
2018. "Prediction of Heat Transfer in a Jet Cooled Aircraft Engine Compressor Cone Based on Statistical Methods" *Aerospace* 5, no. 2: 51.
https://doi.org/10.3390/aerospace5020051