# Development of an Analytic Convection Model for a Heated Multi-Hole Probe for Aircraft Applications

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## Abstract

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## 1. Introduction and Motivation

## 2. Theory and Analytical Model

#### 2.1. Heat Convection Theory

#### 2.2. Heat Convection Model for the Heated Probe

#### 2.2.1. The Front Hemisphere

#### 2.2.2. The Curved Surface of a Cylinder in Axial Flow

#### 2.2.3. Resulting Analytical Model

## 3. Experimental Setup and Probe Assembly

## 4. Results

#### 4.1. Temperature Measurements

#### 4.2. Comparison to the Analytical Model

#### 4.3. Evaluation of the Heating System Anti-Icing Capability

## 5. Discussion and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AM | Additive Manufacturing |

IPS | Icing Protection System |

PBF | Powder Bed Fusion |

UAV | Unmanned Aerial Vehicles |

## Appendix A. Additional Information of the Front Hemisphere Heat Convection

#### Appendix A.1. Experimental Explanation

The heat transfer from this part of the sphere is dominating $Re\sim 4000$ but, as the contribution from the rear part increases faster with increasing Reynolds number, both become equally important at $Re\sim $ 70,000.

**Figure A1.**Average Nusselt number estimations for the whole sphere and the front hemisphere in the range $3.0\times {10}^{3}<R{e}_{D}<1.0\times {10}^{5}$.

#### Appendix A.2. Mathematical Explanation

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**Figure 4.**Stagnation point position and expected flow streamlines for the considered geometry under an axial flow.

**Figure 5.**Stagnation point position and flow streamlines for the disc and cylinder under an axial flow [19].

**Figure 8.**Additive manufactured probe after printing and the side opening to the axial heater cavity.

**Figure 14.**Comparison between the analytical model output ${T}_{analytical}$ and the corresponding experimental test temperature reading ${T}_{test}$. The data labels indicate the test configuration, respectively.

**Figure 15.**Anti-icing evaluation graph containing test data from configurations 4, 5 and 6 translated to the considered Standard Atmosphere cases in ${}^{\xb0}$C.

**Figure 16.**Anti-icing evaluation graph for the probe showing the accordance between the test translated data and the predicted temperature values by the regression model.

Configuration | V [m/s] | q [W] |
---|---|---|

1 | 25 (${V}_{I}$) | 10 (${q}_{I}$) |

2 | 25 (${V}_{I}$) | 20 (${q}_{II}$) |

3 | 25 (${V}_{I}$) | 30 (${q}_{III}$) |

4 | 25 (${V}_{I}$) | 40 (${q}_{IV}$) |

4 | 25 (${V}_{I}$) | 40 (${q}_{IV}$) |

5 | 35 (${V}_{II}$) | 40 (${q}_{IV}$) |

6 | 45 (${V}_{III}$) | 40 (${q}_{IV}$) |

T_{1} | T_{2} | T_{3} | T_{4} | T_{5} | T_{6} | |
---|---|---|---|---|---|---|

Distance from probe tip [mm] | 0.5 | 6.5 | 22.9 | 42.8 | 66.0 | 72.5 |

Config. | T_{∞} | T_{1} | T_{2} | T_{3} | T_{4} | T_{5} | T_{6} |
---|---|---|---|---|---|---|---|

1 | 22.7 | 36.5 | 46.5 | 88.6 | 58.4 | 37.9 | 34.8 |

2 | 23.1 | 51.1 | 71.6 | 145.3 | 92.1 | 52.2 | 46.2 |

3 | 23.4 | 66.8 | 97.6 | 200.5 | 124.1 | 65.5 | 57.0 |

4 | 23.8 | 81.3 | 113.5 | 250.7 | 150.9 | 77.0 | 66.3 |

5 | 24.9 | 59.1 | 77.8 | 219.8 | 123.1 | 62.4 | 55.8 |

6 | 26.3 | 49.7 | 62.7 | 196.8 | 106.8 | 56.4 | 51.2 |

Config. | T_{analytical} [${}^{\xb0}$C] | T_{test} [${}^{\xb0}$C] | $\Delta \mathit{T}$ [${}^{\xb0}$C] | $\mathit{\delta}\mathit{T}$ [%] |
---|---|---|---|---|

1 | 61.0 | 64.5 | −3.5 | −4.4% |

2 | 99.6 | 103.0 | −3.4 | −2.9% |

3 | 140.1 | 140.7 | −0.6 | −0.4% |

4 | 176.3 | 171.7 | 4.6 | 2.5% |

5 | 144.6 | 140.2 | 4.4 | 2.8% |

6 | 126.2 | 122.1 | 4.1 | 3.0% |

**Table 5.**Anti-icing evaluation initial values for the probe head and the static ring containing test data from configurations 4, 5 and 6 in [${}^{\xb0}$C].

Config. | T_{1} | T_{5} |
---|---|---|

Head | Static Ring | |

4 | 81.3 | 77.0 |

5 | 59.1 | 62.4 |

6 | 49.7 | 56.4 |

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**MDPI and ACS Style**

Nieto Muro, P.; Heckmeier, F.M.; Jenkins, S.; Breitsamter, C.
Development of an Analytic Convection Model for a Heated Multi-Hole Probe for Aircraft Applications. *Sensors* **2021**, *21*, 6218.
https://doi.org/10.3390/s21186218

**AMA Style**

Nieto Muro P, Heckmeier FM, Jenkins S, Breitsamter C.
Development of an Analytic Convection Model for a Heated Multi-Hole Probe for Aircraft Applications. *Sensors*. 2021; 21(18):6218.
https://doi.org/10.3390/s21186218

**Chicago/Turabian Style**

Nieto Muro, Pablo, Florian M. Heckmeier, Sean Jenkins, and Christian Breitsamter.
2021. "Development of an Analytic Convection Model for a Heated Multi-Hole Probe for Aircraft Applications" *Sensors* 21, no. 18: 6218.
https://doi.org/10.3390/s21186218