# A Model Calibration Approach to U-Value Measurements with Thermography

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Envelope Model

#### 2.2. Model Calibration

#### 2.3. Experimental Setup

## 3. Experimental Results

## 4. Uncertainty Analysis

#### 4.1. Uncertainty Analysis of the HFM Method

- Approx. $5\%$ error due to the accuracy of the calibration of the HFM and temperature sensors if the sensors are well calibrated.
- Approx. $5\%$ variation due to slight difference in thermal contact between HF sensor and the wall surface.
- $2\%$ to $3\%$ uncertainty due to operational error of the HFM.
- Approx. $10\%$ error caused by the variations over time of the temperatures and heat flow.
- Approx. $5\%$ error in thermal transmittance measurement due to temperature variations within the space and difference between air and radiant temperatures.

#### 4.2. Uncertainty Analysis of the Model Calibration Approach

## 5. Discussion

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

C | capacitor |

EU | European Union |

HFM | Heat flux meter |

LW | long wave |

HVAC | Heating, ventilation, air-conditioning, and cooling |

IR | Infrared |

R | resistance |

SW | short wave |

TM | Thermal mass |

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**Figure 1.**Sketch of processes that determine the heat flow through a wall element. Heat is transferred through the wall by thermal conduction. It exchanges heat with the wall by convective heat transfer to the air and long-wave (LW) radiation exchange with objects in its surroundings. In this figure these objects are represented by the sketch of a sofa on the internal side and a sketch of a building and plants on the external side of the wall. On the external side, additional thermal energy is supplied to the wall by the absorption of short-wave (SW) radiation from the sun.

**Figure 2.**2TM wall model without the boundary conditions. The visualization was generated using the Openmodelica Connection Editor [28].

**Figure 4.**The setup of sensors for validation purposes on the measured wall and two blackbodies in front of it. The blackbodies are used to have a better control of the IR camera uncertainty. The patches of aluminium foil serve as a diffuse reflectors for thermal radiation from surrounding objects.

**Figure 5.**An exemplary image from the IR time series. For the model calibration the temperature of a small area that is marked with a yellow square and denoted A1 was used.

**Figure 6.**Time series of corrected IR camera readings and simulated surface temperatures for a period of five consecutive days for the experiment at the test walls after optimization of wall resistance and capacitance values.

**Figure 7.**Mean value (

**top**) and standard deviation (

**bottom**) of the thermal resistance for sample sizes of 10 to 1000.

**Table 1.**Results of the model calibration for the thermal resistance and the thermal transmittance using different time periods.

Days | ${\mathit{R}}_{\mathbf{tot}}\phantom{\rule{4pt}{0ex}}\left. [\mathbf{K}/\mathbf{W}\right]$ | $\mathit{U}\phantom{\rule{4pt}{0ex}}\left. [\mathbf{W}/\mathbf{K}{\mathbf{m}}^{2}\right]$ | Deviation from ${\mathit{R}}_{\mathbf{ref}}\phantom{\rule{4pt}{0ex}}\left[\%\right]$ | ${\mathit{\chi}}_{\mathit{r}}^{2}$ | $\mathit{C}\phantom{\rule{4pt}{0ex}}\left. [\mathbf{J}/\mathbf{K}\right]$ |
---|---|---|---|---|---|

1–5 | 0.085 | 1.47 | 13.2 | 0.04 | 1.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

1–3 | 0.083 | 1.51 | 15.4 | 0.04 | 1.39 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

2–4 | 0.086 | 1.46 | 12.6 | 0.03 | 1.83 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

3–5 | 0.087 | 1.44 | 11.7 | 0.02 | 1.65 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

1 | 0.081 | 1.55 | 17.8 | 0.05 | 1.22 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

2 | 0.085 | 1.47 | 13.3 | 0.01 | 1.88 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

3 | 0.086 | 1.46 | 12.8 | 0.01 | 1.11 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

4 | 0.088 | 1.41 | 9.8 | <0.01 | 1.9 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

**Table 2.**The errors of the input quantities for the model calibration. The second column shows the source of the signal.

Quantity | Source | Error |
---|---|---|

Outside ${T}_{\mathrm{air}}$ | NTC Sensor | $\pm 0.22\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

Outside ${T}_{\mathrm{wall}}$ | IR camera | $\pm 1.50\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

Outside ${T}_{\mathrm{surr}}$ | IR camera on aliminium foil | $\pm 1.50\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

External h | Nusselt number correlation | $\pm 0.50\phantom{\rule{0.166667em}{0ex}}\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ |

$\epsilon $ | Literature [35,36,37] | $\pm 0.03$ |

Inside ${T}_{\mathrm{air}}$ | NTC Sensor | $\pm 0.22\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

Inside ${T}_{\mathrm{surr}}$ | NTC Sensor | $\pm 0.22\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

Internal h | Literature [38] | $\pm 3.00\phantom{\rule{0.166667em}{0ex}}\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ |

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**MDPI and ACS Style**

Patel, D.; Estevam Schmiedt, J.; Röger, M.; Hoffschmidt, B.
A Model Calibration Approach to U-Value Measurements with Thermography. *Buildings* **2023**, *13*, 2253.
https://doi.org/10.3390/buildings13092253

**AMA Style**

Patel D, Estevam Schmiedt J, Röger M, Hoffschmidt B.
A Model Calibration Approach to U-Value Measurements with Thermography. *Buildings*. 2023; 13(9):2253.
https://doi.org/10.3390/buildings13092253

**Chicago/Turabian Style**

Patel, Dhruvkumar, Jacob Estevam Schmiedt, Marc Röger, and Bernhard Hoffschmidt.
2023. "A Model Calibration Approach to U-Value Measurements with Thermography" *Buildings* 13, no. 9: 2253.
https://doi.org/10.3390/buildings13092253