# Comparison of Infrared Thermography and Heat Flux Method for Dynamic Thermal Transmittance Determination

^{*}

## Abstract

**:**

## 1. Introduction

^{2}K)), which is the initial parameter for determining the heating and cooling energy demands [3].

_{k}is the thickness (m) and λ

_{k}is the thermal conductivity (W/(m K)) of each layer, and h

_{si}and h

_{se}are surface heat transfer coefficients (W/(m

^{2}K)), which quantify heat transfer from indoor and outdoor air to element surface.

^{2}K) [7]. If significant differences occur in measured heat flow and temperature rates, then the DYNM must be used. In practice, however, the HFM can, with the above disadvantages, be too expensive and very time-consuming. If conditions are ideal (boundary conditions are stationary, the temperature difference between the indoor and outdoor air is more than 10 °C, no wind on the exterior surface, and no direct solar radiation), the measurement period should be 72 hours; otherwise, it should be more than seven days. For both methods, the HFM and the IRT, results are profoundly affected by environmental conditions: direct solar radiation should be avoided, wind speed should not exceed 1 m/s (best results are expected for wind speed < 0.2 m/s), and minimal temperature difference between the indoor and outdoor air should be 10–15 °C.

## 2. Nondestructive Methods for Building Inspections

#### 2.1. Heat Flux Method (HFM)

- (1)
- Heat flux sensor should not be installed close to the parts with high thermal conductivity (i.e., thermal bridges), cracks, or other causes of error.
- (2)
- The surface under examination has to be shielded from rain, snow, and solar radiation.
- (3)
- Data acquisition intervals should be less than 30 minutes. This interval depends on the thermal inertia of the element, indoor an and outdoor air temperature difference, and method used for data analysis (AVGM or DYNM).
- (4)
- The minimal measurement period should be 72 hours for stable boundary temperatures; otherwise, it should be more than seven days.
- (5)
- The minimal indoor and outdoor air temperature difference should be 10 °C.
- (6)
- It is necessary to achieve quasi-stationary conditions three to four hours before the measurement.
- (7)
- The U-value should be approximated for the first 24 hours and 2/3 of the overall measurement period. These values should not vary more than 5% from the U-value calculated at the end of the measurement period.

#### 2.1.1. Determining the U-Value According to ISO 9869

#### Average Method (AVGM)

_{j}is the heat flux, T

_{i}and T

_{e}are the indoor and outdoor air temperatures, respectively, while index j counts the individual measurement.

_{rad}and q

_{conv}are approximated by the measured surface, reflected, and air temperatures as described in Section 3.

#### Dynamic Method (DYNM)

_{1}, K

_{2}, P

_{n}, and Q

_{n}, and they have no real meaning. In Equation (4), p represents a subset of data points used for numerical integration corresponding to sum over j. The variables β

_{n}are exponential functions of the time constant τ

_{n}. Derivatives of indoor and outdoor air temperatures are calculated using Equations (5) and (6) [7].

_{1}, τ

_{2}, …, τ

_{m}), then 2 × m + 3 unknown parameters are needed. These parameters are given by Equation (7) [7]:

_{1}= r × τ

_{2}= r

^{2}× τ

_{3}) to accurately represent the relation between the measured values (q, T

_{i}and T

_{e}) where r is the ratio between time constants [7].

^{H}is a complex conjugate transpose matrix and if its components are real numbers, which is true in this case, Equation (9) is then transformed into:

^{*}of the vector {Z} is calculated, and for every {Z}

^{*}the approximated vector {q}

^{*}is determined—Equation (11):

_{n}through the procedure described in Section 2.2.

#### 2.2. Procedure for Determining the U-Value Using the DYNM

- (1)
- Choosing m numbers of the time constants (1 ≤ m ≤ 3).
- (2)
- Choosing the constant ratio between the time constants (3 ≤ r ≤ 10) in such way that τ
_{1}= r × τ_{2}= r^{2}× τ_{3}. - (3)
- Choosing the number of equations M (2 × m + 3 ≤ M ≤ N).
- (4)
- Calculating the minimum and the maximum values of the time constants (Δt ≤ τ
_{1}≤ p × Δt/2). - (5)
- Calculating the estimated vector {Z}
^{*}for the series of time constants using Equation (11),$${\left\{\mathrm{q}\right\}}^{*}=\left[\mathrm{X}\right]\times {\left\{\mathrm{X}\right\}}^{*}$$ - (6)
- and estimate vector {q}
^{*}using Equation (12). - (7)
- Calculating the total square deviation between vectors {q} and {q}
^{*}using Equation (13).$${\mathrm{S}}^{2}={\left(\left\{\mathrm{q}\right\}-{\left\{\mathrm{q}\right\}}^{*}\right)}^{2}={\displaystyle {\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{q}}_{\mathrm{i}}-{\mathrm{q}}_{\mathrm{i}}^{*}\right)}^{2}}$$ - (8)
- Repeating the steps (5) and (6) until the smallest square deviation S
_{min}is obtained. - (9)
- A good approximation {Z}
^{*}of the vector {Z} is obtained this way. The first component Z_{1}of vector {Z}^{*}is the approximation of the U-value.

_{1}is larger than the maximum value τ

_{max}= p × Δt/2, then the selected number of data points for the system of equations is not sufficient and results obtained in this way are not reliable (for a subset of data points M and ratio r).

^{2}, Y(1, 1) is the first element of the [Y] = ([X]

^{T}× [X]

^{−1}) matrix and F is a significance limit of Student’s t-distribution; where P = 0.95 is the probability and M −2 × m − 5 are the degrees of freedom.

#### 2.3. Infrared Thermography (IRT)

_{tot}(J) is total energy that reaches the camera lens when the surface is focused and W

_{obj}(J) is radiated energy from the surface, and it is the only term of importance for measuring the surface temperature. The total reflected energy that reaches the examined surface is W

_{refl}(J). It is a function of the average temperature of all the objects surrounding the surface under analysis, and the physical properties of the surface itself (mostly its reflectiveness). W

_{atm}(J) is the total energy emitted by the atmosphere. It is the function of the air temperature, relative humidity (RH) and the distance between the IR camera and the examined surface. ε is surface emissivity, τ is the atmosphere’s transmittance, and σ is Stefan–Boltzmann constant [20,21].

## 3. The Surface Heat Transfer Coefficient

_{cond}, q

_{conv}, and q

_{rad}are the heat transfer rates by conduction, convection and radiation (W/m

^{2}), h

_{c}and h

_{r}are the convective and radiative heat transfer coefficients (W/(m

^{2}K)), T

_{i}is the indoor air temperature, and T

_{si}is the surface temperature (K).

_{r}can also be calculated using the reflected surface temperature T

_{ref}instead of T

_{si}.

_{f}(W/(m K)) are the Nusselt number and thermal conductivity of the fluid (air in this case), respectively, and x is the distance (m) from the edge of the object that first contacts the air (i.e., leading edge).

_{c}can be expressed using Equation (19) for all surfaces [5].

_{i}− T

_{si}.

_{c}in Table 1 is derived from experimental data sets using Equation (19).

^{2}K) for vertical surfaces that are not directly heated or near windows where the surface is not plain.

#### Procedure for Calculating the Convection Heat Transfer Coefficient of Vertical Isothermal Planes

_{c}can be calculated, Nusellt, Grashof, and Rayleigh numbers are needed. The analytical procedure for calculating the average convective heat transfer coefficient $\overline{\mathrm{h}}$ is described below.

_{f}—Equation (22) [42]:

^{2}), L is the height of the vertical surface (m), ν is cinematic viscosity (m

^{2}/s), and β represents the temperature coefficient of thermal conductivity (K

^{−1}). β is equal to the 1/T

_{f}.

_{f}. The average Nusselt number is given by Churchill and Chu [39]—Equation (25):

## 4. Experimental Setup

#### Measuring Procedure

## 5. Results

_{avg}) and the DYNM (U

_{dyn}) were determined. One time constant was adopted because it gave the best confidence interval (from 2.13 to 3.87%). For two and three time constants, τ it was not possible to get satisfactory confidence interval I, and because of the large number of data points, for three time constants analysis time was around 30 minutes.

## 6. Conclusions and Further Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Section through a typical heat flowmeter showing the various parts (the vertical scale is enlarged).

**Figure 2.**Flowchart of the dynamic method (DYNM) used in Microsoft Excel VBA code based on the least square adjustment.

**Figure 5.**An illustration of the general infrared thermography (IRT) measurement.

**1**: surroundings;

**2**: surface;

**3**: environment;

**4**: IR camera.

**Figure 6.**(

**a**) Measurement setup for the IRT and the heat flux method (HFM); (

**b**) measurement location at Walls 2 and 3 showing shading of walls from direct solar radiation.

**Figure 8.**Surface temperatures (thermograms) during the measurement period: (

**a**) Wall 1, (

**b**) Wall 2 and (

**c**) Wall 3.

Authors | h_{c} (W/(m^{2} K)) |
---|---|

Awbi et al. [30] | 1.49 × ΔT^{0.345} |

Khalifa et al. [31] | 2.07 × ΔT^{0.230} |

Michejev [32] | 1.55 × ΔT^{0.330} |

King [33] | 1.51 × ΔT^{0.330} |

Nusselt [34] | 2.56 × ΔT^{0.250} |

Heilman [35] | 1.67 × ΔT^{0.250} |

Wilkers et al. [36] | 3.04 × ΔT^{0.120} |

ASHRAE [37] | 1.31 × ΔT^{0.330} |

Bejan [38] and Chu [39] | Equation (20) |

Almadari and Hammond [40] | Equation (21) |

ISO 9869 [7] | 3.00 |

Layer | λ | D | R |
---|---|---|---|

-- | W/m K | mm | m^{2} K/W |

R_{si} | -- | -- | 0.13 |

Lime cement mortar | 1 | 30 | 0.03 |

Concrete | 2.00–2.60 | 250 | 0.1–0.13 |

Thermal insulating plaster | 0.11 | 30 | 0.27 |

R_{se} | -- | -- | 0.04 |

ΣR (m^{2} K/W) | 0.57–0.60 | ||

U (W/m^{2} K) | 1.67–1.76 |

Layer | λ | D | R |
---|---|---|---|

-- | W/m K | mm | m^{2} K/W |

R_{si} | -- | -- | 0.13 |

Gypsum cardboards | 0.25 | 12.5 | 0.05 |

Plaster | 1 | 20 | 0.02 |

Brick | 0.81 | 380 | 0.469 |

Cement mortar | 0.90 | 5.0 | 0.006 |

Thermal insulating plaster | 0.11 | 40 | 0.364 |

Mineral plaster | 0.70 | 15 | 0.021 |

R_{se} | -- | -- | 0.04 |

ΣR (m^{2} K/W) | 1.10 | ||

U (W/m^{2} K) | 0.91 |

Model | ThermaCAM P640 |

Serial code | 3009001177 |

Lens | FOV 24° (38 mm) |

IR resolution | 640 × 480 |

Temperature sensitivity | ±0.05 K |

Temperature range | −40 °C/2000 °C |

Model | gSKIN® Heat Flux Sensor |

Sensitivity | 10.93 μV/(W/m^{2}) |

Correction factor | 0.0137 (μV/(W/m^{2}))/°C |

Dimensions | 30.0 × 30.0 mm |

Thickness | 2.0 mm |

Electrical resistance at 22.5 °C | < 100 Ω |

Relative error | ±3% |

Temperature range | −50 °C/+150 °C |

Wall 1 | Wall 2 | Wall 3 | ||||||
---|---|---|---|---|---|---|---|---|

Authors | Method | Period | U_{avg} ± σ_{95%} | U_{dyn} ± I_{95%} | U_{avg} ± σ_{95%} | U_{dyn} ± I_{95%} | U_{avg} ± σ_{95%} | U_{dyn} ± I_{95%} |

days | W/(m^{2} K) | W/(m^{2} K) | W/(m^{2} K) | W/(m^{2} K) | W/(m^{2} K) | W/(m^{2} K) | ||

ISO 9869 [7] | HFM | 1 | 2.095 ± 0.517 | 2.172 ± 0.082 | 0.675 ± 0.682 | 2.153 ± 0.399 | 1.447 ± 0.661 | 1.147 ± 0.084 |

2 | 1.991 ± 0.446 | 2.123 ± 0.054 | 0.599 ± 0.625 | 1.153 ± 0.115 | 0.998 ± 0.620 | 1.190 ± 0.052 | ||

3 | 2.159 ± 0.964 | 2.134 ± 0.063 | 0.623 ± 0.656 | 1.166 ± 0.211 | 0.877 ± 0.532 | 1.186 ± 0.045 | ||

Awbi et al. [30] | IRT | 1 | 1.496 ± 0.367 | 1.568 ± 0.054 | 0.811 ± 0.815 | 2.641 ± 0.423 | 1.232 ± 0.573 | 1.050 ± 0.080 |

2 | 1.427 ± 0.312 | 1.526 ± 0.037 | 0.684 ± 0.709 | 1.151 ± 0.133 | 0.861 ± 0.563 | 1.064 ± 0.062 | ||

3 | 1.563 ± 0.766 | 1.513 ± 0.048 | 0.657 ± 0.677 | 1.184 ± 0.272 | 0.744 ± 0.467 | 1.083 ± 0.056 | ||

Khalifa et al. [31] | IRT | 1 | 1.584 ± 0.390 | 1.656 ± 0.059 | 0.819 ± 0.814 | 2.732 ± 0.445 | 1.215 ± 0.587 | 0.995 ± 0.076 |

2 | 1.511 ± 0.331 | 1.615 ± 0.040 | 0.690 ± 0.710 | 1.197 ± 0.131 | 0.861 ± 0.576 | 0.990 ± 0.062 | ||

3 | 1.653 ± 0.800 | 1.604 ± 0.050 | 0.661 ± 0.678 | 1.202 ± 0.281 | 0.738 ± 0.474 | 1.028 ± 0.056 | ||

Michejev [32] | IRT | 1 | 1.505 ± 0.370 | 1.577 ± 0.055 | 0.820 ± 0.815 | 2.739 ± 0.447 | 1.230 ± 0.574 | 1.044 ± 0.080 |

2 | 1.436 ± 0.314 | 1.535 ± 0.037 | 0.690 ± 0.710 | 1.201 ± 0.131 | 0.861 ± 0.564 | 1.056 ± 0.062 | ||

3 | 1.572 ± 0.769 | 1.523 ± 0.048 | 0.662 ± 0.679 | 1.203 ± 0.281 | 0.743 ± 0.468 | 1.077 ± 0.056 | ||

King [33] | IRT | 1 | 1.496 ± 0.366 | 1.569 ± 0.054 | 0.801 ± 0.813 | 2.557 ± 0.394 | 1.232 ± 0.573 | 1.049 ± 0.080 |

2 | 1.428 ± 0.311 | 1.526 ± 0.037 | 0.678 ± 0.706 | 1.112 ± 0.136 | 0.862 ± 0.563 | 1.062 ± 0.062 | ||

3 | 1.563 ± 0.764 | 1.513 ± 0.048 | 0.652 ± 0.675 | 1.175 ± 0.264 | 0.744 ± 0.468 | 1.082 ± 0.056 | ||

Nusselt [34] | IRT | 1 | 1.693 ± 0.436 | 1.757 ± 0.068 | 0.818 ± 0.814 | 2.714 ± 0.445 | 1.193 ± 0.612 | 0.937 ± 0.075 |

2 | 1.611 ± 0.369 | 1.724 ± 0.045 | 0.689 ± 0.709 | 1.185 ± 0.131 | 0.857 ± 0.592 | 0.912 ± 0.063 | ||

3 | 1.765 ± 0.861 | 1.717 ± 0.054 | 0.661 ± 0.678 | 1.194 ± 0.279 | 0.732 ± 0.490 | 0.968 ± 0.057 | ||

Heilman [35] | IRT | 1 | 1.508 ± 0.363 | 1.583 ± 0.053 | 0.796 ± 0.808 | 2.470 ± 0.385 | 1.230 ± 0.578 | 1.038 ± 0.078 |

2 | 1.440 ± 0.309 | 1.539 ± 0.037 | 0.674 ± 0.702 | 1.064 ± 0.140 | 0.863 ± 0.567 | 1.047 ± 0.062 | ||

3 | 1.575 ± 0.762 | 1.525 ± 0.047 | 0.650 ± 0.672 | 1.147 ± 0.258 | 0.744 ± 0.470 | 1.071 ± 0.055 | ||

Wilkers et al. [36] | IRT | 1 | 1.728 ± 0.427 | 1.799 ± 0.066 | 0.824 ± 0.819 | 2.774 ± 0.460 | 1.193 ± 0.629 | 0.903 ± 0.071 |

2 | 1.647 ± 0.363 | 1.762 ± 0.044 | 0.693 ± 0.713 | 1.217 ± 0.130 | 0.866 ± 0.606 | 0.861 ± 0.062 | ||

3 | 1.800 ± 0.857 | 1.752 ± 0.053 | 0.664 ± 0.681 | 1.207 ± 0.285 | 0.737 ± 0.505 | 0.928 ± 0.057 | ||

ASHRAE [37] | IRT | 1 | 1.451 ± 0.348 | 1.527 ± 0.050 | 0.825 ± 0.820 | 2.791 ± 0.464 | 1.241 ± 0.569 | 1.073 ± 0.081 |

2 | 1.386 ± 0.296 | 1.482 ± 0.035 | 0.694 ± 0.714 | 1.225 ± 0.130 | 0.864 ± 0.560 | 1.093 ± 0.062 | ||

3 | 1.518 ± 0.740 | 1.467 ± 0.046 | 0.664 ± 0.682 | 1.210 ± 0.287 | 0.748 ± 0.466 | 1.104 ± 0.056 | ||

Bejan [38] and Chu [39] | IRT | 1 | 1.420 ± 0.339 | 1.497 ± 0.049 | 0.822 ± 0.817 | 2.761 ± 0.456 | 1.244 ± 0.567 | 1.082 ± 0.082 |

2 | 1.357 ± 0.289 | 1.450 ± 0.034 | 0.692 ± 0.712 | 1.210 ± 0.130 | 0.865 ± 0.558 | 1.105 ± 0.062 | ||

3 | 1.486 ± 0.727 | 1.435 ± 0.045 | 0.663 ± 0.680 | 1.204 ± 0.284 | 0.750 ± 0.465 | 1.113 ± 0.056 | ||

Almadari and Hammond [40] | IRT | 1 | 1.707 ± 0.394 | 1.770 ± 0.060 | 0.840 ± 0.817 | 2.886 ± 0.513 | 1.239 ± 0.571 | 1.065 ± 0.081 |

2 | 1.635 ± 0.334 | 1.737 ± 0.040 | 0.703 ± 0.714 | 1.265 ± 0.127 | 0.864 ± 0.561 | 1.083 ± 0.062 | ||

3 | 1.775 ± 0.788 | 1.726 ± 0.049 | 0.672 ± 0.686 | 1.208 ± 0.299 | 0.747 ± 0.467 | 1.097 ± 0.056 | ||

Theoretical | IRT | 1 | 1.422 ± 0.313 | 1.512 ± 0.043 | 0.820 ± 0.815 | 2.743 ± 0.447 | 1.283 ± 0.559 | 1.160 ± 0.087 |

2 | 1.363 ± 0.270 | 1.456 ± 0.031 | 0.690 ± 0.710 | 1.203 ± 0.130 | 0.880 ± 0.545 | 1.202 ± 0.062 | ||

3 | 1.487 ± 0.700 | 1.435 ± 0.043 | 0.662 ± 0.679 | 1.204 ± 0.282 | 0.769 ± 0.460 | 1.184 ± 0.056 |

Days | AVGM | DYNM | |||||
---|---|---|---|---|---|---|---|

U_{IRT,min} | U_{IRT,max} | ΔU | U_{IRT,min} | U_{IRT,max} | ΔU | ||

W/(m^{2} K) | W/(m^{2} K) | % | W/(m^{2} K) | W/(m^{2} K) | % | ||

Wall 1 | 1 | 1.42 | 2.095 | 32.22% | 1.497 | 2.172 | 31.08% |

2 | 1.357 | 1.991 | 31.84% | 1.45 | 2.123 | 31.70% | |

3 | 1.486 | 2.159 | 31.17% | 1.435 | 2.134 | 32.76% | |

Wall 2 | 1 | 0.675 | 0.84 | 19.64% | 2.153 | 2.886 | 25.40% |

2 | 0.599 | 0.703 | 14.79% | 1.064 | 1.265 | 15.89% | |

3 | 0.623 | 0.672 | 7.29% | 1.147 | 1.21 | 5.21% | |

Wall 3 | 1 | 1.193 | 1.447 | 17.55% | 0.903 | 1.16 | 22.16% |

2 | 0.857 | 0.998 | 14.13% | 0.861 | 1.202 | 28.37% | |

3 | 0.732 | 0.877 | 16.53% | 0.928 | 1.186 | 21.75% | |

$\mathsf{\Delta}\mathrm{U}=\left({\mathrm{U}}_{\mathrm{I}\mathrm{R}\mathrm{T},\mathrm{max}}-{\mathrm{U}}_{\mathrm{I}\mathrm{R}\mathrm{T},\mathrm{min}}\right)/{\mathrm{U}}_{\mathrm{I}\mathrm{R}\mathrm{T},\mathrm{max}}$ |

© 2019 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**

Gaši, M.; Milovanović, B.; Gumbarević, S. Comparison of Infrared Thermography and Heat Flux Method for Dynamic Thermal Transmittance Determination. *Buildings* **2019**, *9*, 132.
https://doi.org/10.3390/buildings9050132

**AMA Style**

Gaši M, Milovanović B, Gumbarević S. Comparison of Infrared Thermography and Heat Flux Method for Dynamic Thermal Transmittance Determination. *Buildings*. 2019; 9(5):132.
https://doi.org/10.3390/buildings9050132

**Chicago/Turabian Style**

Gaši, Mergim, Bojan Milovanović, and Sanjin Gumbarević. 2019. "Comparison of Infrared Thermography and Heat Flux Method for Dynamic Thermal Transmittance Determination" *Buildings* 9, no. 5: 132.
https://doi.org/10.3390/buildings9050132