Experimental Study on Heat Transfer Characteristics of Hollowing Defect Areas on Building Facade

Abstract

Infrared detection is more and more widely used in the field of non-destructive testing of buildings to detect whether there is a defect on the surface of the building facade. In many cases, it is necessary to obtain more information about the defect, such as the depth of the defect, so as to evaluate the severity of the defect and repair. The theoretical formula of hollowing defect depths was derived in this paper based on the heat transfer characteristics of the intact and defective areas on the building facade, and the influence of defects with different shapes, sizes and cavity thicknesses on the temperature distribution of the building facade was summarized quantitatively. Firstly, the theoretical formula of the hollowing defect depth and the factors affecting the distribution of the temperature gradient on the building facade excited by external thermal source was derived and restricted by the boundary condition. Secondly, three sets of physical building facade models that contained hollowing defects with different shapes, sizes and cavity thicknesses were fabricated and designed, and the experimental platform was built. The infrared thermograms and the temperature characteristic curves of the hollowing defect in a natural light environment were obtained and fitted according to the temperature differences of the defective area, while analyzing the influence of the size, shape and cavity thicknesses on surface temperature distribution. Finally, the theoretical formula of the defect depth that is applicable to the building façade was validated through the experimental simulation of 14 forms of hollowing. The experimental results demonstrated that the revised formula of defect depth is consistent with the actual defect depth, and the three-dimensional positioning of the hollow drum defect of the building facade can be effectively carried out and combined with the defect size taken from the obtained infrared thermal image.

1. Introduction

During the construction and service of the building facade, the bonding quality of mortar is reduced because of internal factors such as nonstandard construction [1] and unqualified mortar [2], as well as external factors such as gravity, temperature and humidity, resulting in hollowing defects. When the adhesive force of a building facade with hollowing decreases to a certain extent, the finishing layer will peel off the basal layer, causing high-altitude falling object accident [3]. The existence of hollowing defects on building facades not only brings serious accidents by threatening the safety of pedestrians and the society but also reduces the thermal insulation performance of the building, resulting in serious energy loss problems [4].

In the field of architecture and civil engineering, infrared thermography is widely used for the defect detection of building facades due to its non-destructive testing, large detection area, strong penetration and rapidity. Nonetheless, the existing infrared thermal images are mainly based on the temperature characteristics of the target and the background to detect and identify hollowing defects, and it is difficult to quantitatively analyze the severity of defects.

The research on infrared thermal imaging non-destructive testing technology has been applied many years ago. For example, the thermal insulation qualitative detection of irregularities in building envelopes infrared method (ISO6781-1983) was first developed by the International Organization for Standardization (ISO) in 1983, and the revised version were published in 1999 [5,6]. Gary J. Weil. et al. [7,8,9] used infrared thermal imaging technology to carry out non-destructive tests on large concrete components and found that it has a good effect on detecting the internal defects. Oral Büyüköztürk [10] detected the hollowing defects of building exterior walls and achieved good results using infrared thermal imaging technology successfully. However, the above-mentioned research focused on the qualitative observation of concrete internal defects. To analyze the defects correctly, quantitative infrared detection technology has attracted more attention.

D. J. Titman [11] discussed the application of infrared non-destructive detection, especially on the application of concrete structures with internal defects in different cases, and the best detection time, condition and visual angle of infrared detection methods under different working conditions, as well as the limitation of detection, was proposed. Takahide Sakagami, Shiro Kubo, et al. [12] worked on the quantitative testing of concrete internal damage by infrared non-destructive testing technology, and the relationship curves between the depth and the geometric size of embedded hollowing defects were obtained. Maierhofer et al. [13] verified that the infrared thermogram is more effective in hollowing defect detection for cement mortar shallow layers. N. Avdelidis et al. [14] externally heated four types of building materials and continuously recorded the heat spectra during the transient phase, and the thermal behavior of building materials with defects under heat excitation was simulated by thermal calculation software, which showed that the infrared thermogram can be used to detect and quantitatively analyze the defects of building structures. Patricia Cotič et al. [15] studied 51 man-made defects in concrete specimens with active thermography and detected defects at various depths, and the final measured sizes are equal to or less than that obtained using the thermal contrast method. Maria Teresa Gomes Barbosa [16] quantified the amount of moisture existing on building facades using infrared thermography and made it possible to prioritize interventions while considering the causes of the damages.

In China, the infrared thermogram for hollowing defect detection on building facades was also developed steadily. Yuan X et al. [17] analyzed the influence of the types, sizes, thicknesses and the embedded depth of hollowing defects, with the sun and the surrounding scenery, on the surface temperature distribution of the decorative layer. Fang XM [18] proposed a detection method by using the characteristic of the temperature difference between the defect-free and defect areas. The detection method works as the basis of detecting the hollowing defects of building facades, which can be used to effectively calculate the relative area of the defect, resulting in the quantitative detection of defects. Chen J et al. [19] measured the target temperature parameters and obtained the changed threshold value of cement mortar surface temperature with the infrared digital image analysis method to identify hollowing defects. Xu XH et al. [20] analyzed the influence of defect size, different time, space and shooting angle on the infrared detection through experiments. Hong M et al. [21] used passive infrared thermography and digital image correlation techniques to study the temperature and deformation fields containing void and crack defects and proposed a significant power function relationship between thickness × depth and temperature difference.

The daily inspection of hollowing defects on building facades mainly includes three aspects: the defect size, location and depth. The size and location of defects in infrared thermal images can be extracted effectively by image processing technology, and the theoretical depth of the hollowing defects can be calculated by acquiring the surface temperature difference characteristics of the defect-free and defect areas on building facades. Huang WH [22] studied the influence of geometric sizes (the size and thickness of the defective cavity) and the embedding depth on the surface temperature of decorative bricks by means of Fluent numerical simulation and experimentation, which showed that the influence of the defects on the surface temperature increases with the size and depth of the hollowing defects. Huang HM et al. [23] proposed the method of using finite element numerical simulation to solve the defect depth according to the defect surface temperature, but did not carry out specific experimental verification. Li QX et al. [24] proposed a two-stage network with an integrated attention mechanism for the quantitative identification of debonding defects in building facades. The target detection and semantic segmentation network were combined to realize the location and description of defects with higher detection accuracy.

From the above research developments, it can be concluded that it is reasonable and effective to use infrared thermal imaging to detect the void defect of the building facade, and qualitative analysis can be carried out according to the obtained temperature distribution of the building facade. In quantitative detection, based on the simulation software or the single-shape model of the defect areas, the theoretical research will affect the authenticity and reliability of the data. However, it is difficult to obtain quantitative size and depth information of regional defects by using real walls as research objects for experimental exploration. In this paper, the equivalent calculation model of a building facade was constructed, and the formula for calculating the depth of a defect was derived based on the theory of heat conduction. This paper carried out several experiments on building facades containing hollowing defects, where three block models of building facades containing hollowing defects of different shapes, sizes and thicknesses were tested experimentally using a thermal infrared imager. The influence of physical parameters such as the size, shape and thickness of the hollowing defects on surface temperature distribution is analyzed through infrared thermal images collected in the experiment. On this basis, the formula for calculating the depth of a defect is verified and improved so as to accurately locate the hollowing defect.

2. Theoretical Derivation of Heat Transfer of Hollowing Defects on Building Facades

2.1. Heat Transfer Process of Hollowing Defect on Building Facades

When the building facade is irradiated by the sun, the heat is transmitted from the outer surface to the inner. For the heat transfer analysis, it is assumed that the materials and physical properties of each medium layer in the intact wall are homogeneous [25], and the heat transmits vertically and uniformly from the outer surface to the interior.

In the process of solar radiation excitation and heat transfer, the temperature of the outer surface of the intact wall gradually rises and the temperature distribution is uniform. When the hollowing defects are present in the finishing layer, a very thin air layer will be formed in that position. Due to the air layer having good thermal insulation performance, the thermal conductivity of the wall is reduced, and the heat cannot be evenly conducted to the interior and accumulates at the defect, forming a high temperature abnormal area, which is called a “heat spot” in the project. However, when the intensity of solar radiation received by the external wall gradually decreases, the heat is transmitted from the inside to the outside, and a “cold spot” is formed at the defect. Figure 1 is a schematic diagram of the heat transfer process of the wall with a hollowing defect under solar radiation heating.

2.2. The Derivation of Heat Transfer Theory of Hollowing Defect

The hollowing defect in large civil structures, such as a high-rise building, can be approximated as a one-dimensional heat conduction problem, because the thickness of the wall is less than the height and length [26]. Thus, the thermal conductivity is usually thought of as the main factor to evaluate the heat leakage in the thermal analysis of multi-story building envelopes [27], and the effect of thermal conductivity is the result of the joint action of the materials. In order to acquire an accurate value of the comprehensive thermal conductivity, the equivalent thermal conductivity of the model is calculated by the concept of equivalent thermal resistance [28].

Based on the above considerations, the detected wall is divided into area 1 without defects and area 2 with defects in this paper, as shown in Figure 2. According to the resistance series-parallel connection calculation formula, the multi-story building structure is simplified, and the equivalent total thermal resistance is calculated. The multi-layer medium structure composed of four layers of materials (finishing layer, bonding layer, leveling layer and basal layer) is regarded as a series structure.

Figure 2 is the equivalent calculation model of a building specimen. According to the relationship between thermal conductivity and thermal resistance, the process can be deduced as follows.

The building wall specimen is in the ambient temperature with temperature $T_f(°\mathrm{C})$, the thickness of the wall is $L\left(\mathrm{m}\right)$, the depth and thickness of hollowing defect are $d\left(\mathrm{m}\right)$ and $l_d\left(\mathrm{m}\right)$ and the heat flux is $q(\mathrm{J}/{\mathrm{m}}^2)$. It is assumed that the thermal conductivity, thermal diffusivity, density and thickness of the basal layer, leveling layer and finishing layer are ${\lambda }_1,{\lambda }_2,{\lambda }_3$; ${\alpha }_1,{\alpha }_2,{\alpha }_3$; ${\rho }_1,{\rho }_2,{\rho }_3$ and $l_1,l_2,l_3$, respectively.

Thermal resistance of the basal layer: $R_1={\displaystyle \frac{l_1}{{\lambda }_1}}$; thermal resistance of the leveling layer: $R_2={\displaystyle \frac{l_2}{{\lambda }_2}}$; thermal resistance of the bonding layer and the finishing layer: $R_3={\displaystyle \frac{l_3}{{\lambda }_3}}$.

The total thermal resistance is as follows:

$$R=R_1+R_2+R_3={\displaystyle \frac{l_1}{{\lambda }_1}}+{\displaystyle \frac{l_2}{{\lambda }_2}}+{\displaystyle \frac{l_3}{{\lambda }_3}}$$

(1)

The expression for the equivalent thermal conductivity is as follows:

$$\lambda ={\displaystyle \frac{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}}={\displaystyle \frac{l_1+l_2+l_3}{R}}={\displaystyle \frac{{\lambda }_1{\lambda }_2{\lambda }_3(l_1+l_2+l_3)}{l_1{\lambda }_2{\lambda }_3+l_2{\lambda }_1{\lambda }_3+l_3{\lambda }_1{\lambda }_2}}$$

(2)

The equivalent density is given in the following equation:

$$\rho ={\displaystyle \frac{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}}={\displaystyle \frac{{\rho }_{1}s{l}_1+{\rho }_{2}s{l}_2+{\rho }_{3}s{l}_3}{s{l}_1+s{l}_2+s{l}_3}}={\displaystyle \frac{{\rho }_1{l}_1+{\rho }_2{l}_2+{\rho }_3{l}_3}{l_1+l_2+l_3}}$$

(3)

The equivalent specific heat capacity can be calculated as follows:

$$c={\alpha }_1\%c_1+{\alpha }_2\%c_2+{\alpha }_3\%c_3={\displaystyle \frac{{\rho }_1{l}_1{\alpha }_1+{\rho }_2{l}_2{\alpha }_2+{\rho }_3{l}_3{\alpha }_3}{{\rho }_1{l}_1+{\rho }_2{l}_2+{\rho }_3{l}_3}}$$

(4)

$$\alpha ={\displaystyle \frac{\lambda }{\rho c}}={\displaystyle \frac{{\lambda }_1{\lambda }_2{\lambda }_3{(l_1+l_2+l_3)}^2}{(l_1{\lambda }_2{\lambda }_3+l_2{\lambda }_1{\lambda }_3+l_3{\lambda }_1{\lambda }_2)·({\rho }_1{l}_1{\alpha }_1+{\rho }_2{l}_2{\alpha }_2+{\rho }_3{l}_3{\alpha }_3)}}$$

(5)

The thermodynamic parameters of each media layer on the building facade are shown in

Table 1. Physical properties of experimental materials.

Substance	Thermal Conductivity $/\mathbf{W}/(\mathbf{m}·\mathbf{K})$	Density $/\mathbf{K}\mathbf{g}/{\mathbf{m}}^3$	Specific Heat $/\mathbf{J}/\left(\mathbf{K}\mathbf{g}·\mathbf{K}\right)$
Basal layer	1.51	2300	920
Leveling layer	0.93	1800	1800
Bonding layer	1.74	1800	1050
Finishing layer	1.74	1800	1050
Air	0.0251	1.205	1005

[29]. The equivalent specific heat capacity of the building facade can be calculated by Equation (5) and Table 1.

The equivalent heat conduction coefficient $\lambda (\mathrm{W}/(\mathrm{m}·\mathrm{K}\left)\right)$, the equivalent density $\rho (\mathrm{K}\mathrm{g}/{\mathrm{m}}^3)$, the equivalent specific heat capacity $c$ and the equivalent heat diffusion coefficient $\alpha ({\mathrm{m}}^2/\mathrm{s})$ are obtained.

According to the heat transfer analysis of the equivalent calculation model of building wall specimens, the following conclusions can be drawn [30]:

$${\displaystyle \frac{{\partial T}_s\left(0,t\right)}{\partial t}}={\displaystyle \frac{q\alpha }{\lambda d}}$$

(6)

Equation (6) shows that the wall size $L$ and the thickness $l_d$ of the cavity defect do not affect the values of ${\displaystyle \frac{{\partial T}_s\left(0,t\right)}{\partial t}}$, and the ascending speed of the surface temperature is not affected by the corresponding parameters.

The defect area of the hollowing is in the direction of heat conduction [31]:

$${\displaystyle \frac{{\partial }^{2}T}{{\partial }^{2}x}}={\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{\partial T\left(x, t\right)}{\partial t}}(x>0, t>0)$$

(7)

$$-\lambda {\displaystyle \frac{\partial T}{\partial x}}=\epsilon \sigma \left(T_f^4-T_s^4\right), x=0, t>0$$

(8)

$$T=0, t=0, x>0$$

(9)

where $\epsilon$ is the emissivity of the external surface of the building, $\sigma$ is the Stefan–Boltzmann constant, $T_f$ is the ambient temperature and $T_s$ is the temperature at the external surface of the building ($x=0$).

Definition by boundary conditions:

$$f\left(t\right)-\epsilon \sigma T_s^4=C$$

(10)

$$\varnothing \left(T_s\right)=\epsilon \sigma T_s^4$$

(11)

where $f\left(t\right)$ is a given function and $\mathrm{\varnothing }\left(T_s\right)$ is a function of the temperature $T_s$ at the outer surface of the building ($x=0)$.

The heat conduction equation with a nonlinear boundary condition is transformed into a nonlinear integral equation by the Duhamel integral transformation.

$$T\left(x, t\right)={\displaystyle \frac{1}{\sqrt{\lambda \rho c\pi }}}{\int }_{\tau =0}^L{\displaystyle \frac{f\left(t\right)-\varnothing \left(T_s\right)}{{\left(t-\tau \right)}^{\frac{1}{2}}}}\exp \left(-{\displaystyle \frac{x^2}{4\alpha \left(t-\tau \right)}}\right)d\tau$$

(12)

Into the boundary condition, the solution is:

$$\begin{gathered} T\left(x, t\right)={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}{\int }_{\tau =0}^L{\displaystyle \frac{\left(T_f^4-T_s^4\right)}{{\left(t-\tau \right)}^{\frac{1}{2}}}}\exp \left(-{\displaystyle \frac{x^2}{4\alpha \left(t-\tau \right)}}\right)d\tau \\ ={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}\left(T_f^4-T_s^4\right){\int }_{\tau =0}^L{\displaystyle \frac{1}{{\left(t-\tau \right)}^{\frac{1}{2}}}}\exp \left(-{\displaystyle \frac{x^2}{4\alpha \left(t-\tau \right)}}\right)d\tau \end{gathered}$$

(13)

Newton–Cotes is used to integrate the following: ${\int }_{\tau =0}^L{\displaystyle \frac{1}{{\left(t-\tau \right)}^{\frac{1}{2}}}}\exp \left(-{\displaystyle \frac{x^2}{4\alpha \left(t-\tau \right)}}\right)d\tau$, and the result is shown as follows:

$${\int }_{\tau =0}^L{\displaystyle \frac{1}{{\left(t-\tau \right)}^{\frac{1}{2}}}}\exp \left(-{\displaystyle \frac{x^2}{4\alpha \left(t-\tau \right)}}\right)d\tau ={\displaystyle \frac{\sqrt{t}}{2}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{x^2}{4\alpha t}})$$

(14)

The surface temperature can be calculated as follows:

$$T\left(x,t\right)={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}·{\displaystyle \frac{\sqrt{t}}{2}}·\left(T_f^4-T_s^4\right)\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{x^2}{4\alpha t}})$$

(15)

where $T\left(x,t\right)=T_S$. The numerical solution is obtained by means of successive approximation. The initial temperature $T_{s\left(0\right)}\left(t\right)=0$. The iteration expression is as follows:

$$T_{s\left(n+1\right)}\left(x,t\right)={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}·{\displaystyle \frac{\sqrt{t}}{2}}·\left(T_f^4-T_{s\left(n\right)}^4\left(t\right)\right)\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{x^2}{4\alpha t}}), n=\mathrm{1,2},3\cdots \cdots$$

(16)

When $x$ = 0, the surface temperature of the defect-free area 1 after radiation heating can be obtained:

$$T_{s\left(n+1\right)}\left(0,t\right)={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}·{\displaystyle \frac{\sqrt{t}}{2}}·\left(T_f^4-T_{s\left(n\right)}^4\left(t\right)\right), n=\mathrm{1,2},3\cdots \cdots$$

(17)

In the presence of a hollowing, the heat will be hindered as it moves inward and propagates back, with the distance $x=2m$.

$$T_{m\left(n+1\right)}\left(t\right)={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}·{\displaystyle \frac{\sqrt{t}}{2}}·\left(T_f^4-T_{s\left(n\right)}^4\left(t\right)\right)\left[1+\exp \left(-{\displaystyle \frac{d^2}{\alpha t}}\right)\right],n=\mathrm{1,2},3\cdots \cdots$$

(18)

The difference in surface temperatures between the defect and defect-free area at time $t$ can be expressed as:

$$\begin{array}{cc}\hfill ∆T_{\left(n+1\right)}& =T_{m\left(n+1\right)}\left(t\right)-T_{s\left(n+1\right)}\left(t\right)\hfill \\ & ={\displaystyle \frac{\epsilon \sigma }{\sqrt{\lambda \rho c\pi }}}·{\displaystyle \frac{\sqrt{t}}{2}}·\left(T_f^4-T_{s\left(n\right)}^4\left(t\right)\right)\exp \left(-{\displaystyle \frac{d^2}{\alpha t}}\right),(n=\mathrm{1,2},3\cdots )\hfill \end{array}$$

(19)

$${\displaystyle \frac{∆T_{\left(n+1\right)}}{T_{s\left(n+1\right)}}}=\exp \left(-{\displaystyle \frac{d^2}{\alpha t}}\right),(n=\mathrm{1,2},3\cdots )$$

(20)

The calculation of the depth of the defect can be simplified by the following equation:

$$d=\sqrt{\alpha t\left[\ln \left(T_{s\left(n+1\right)}\right)-\mathrm{l}\mathrm{n}(∆T_{\left(n+1\right)})\right]}$$

(21)

where $d$ is the depth of the defect($m$), $\alpha$ is the equivalent thermal conductivity ($W/(m·K)$), $t$ is the heating time($s$), $T_{s\left(n+1\right)}$ is the surface temperature at the moment when the sample contains no defect area, $T_{m\left(n+1\right)}$ is the surface temperature of the defective area at time $t$ and $∆T_{\left(n+1\right)}=(T_{m\left(n+1\right)}\left(t\right)-T_{s\left(n+1\right)}\left(t\right))°\mathrm{C}$ is the temperature difference of the surface of the defective area at time $t$.

According to Formula (21), the temperature of the defect area on the external surface has a direct relationship with the depth of the defect, and the surface temperature change rate is independent of the thickness of the defect or the wall. In addition, in engineering practice, the temperature of the hollowing defect area on the external surface of the building facade is affected by the shape, size and thickness of the hollowing defect. In order to quantitatively analyze the effects of the above physical parameters on the external surface temperature of a building facade with hollowing defects, three prefabricated wall models with hollowing defects are experimentally studied by the infrared imager to carry out the experimental research. The collected infrared thermal image was analyzed to extract the temperature change of the thermal image. The influence of different shapes, sizes and cavity thicknesses on the surface temperature field was analyzed, and the correctness of the defect depth formula was verified by using three wall models.

3. Experiment Design and Implementation

3.1. Experimental Model Design

Through the observation, recording and analysis of the surface temperature field of the physical model of the embedded defects, the influence of hollowing defects on surface temperature field was summarized. The thickness, size and shape of the hollowing defects are the control variables of the hollowing defect. Since the thermal conductivity is similar to that of air, polystyrene foam boards were used to simulate realistic hollowing defects. Three sets of models were designed, as shown in Figure 3, and the defect parameters of the prefabricated model are shown in

Table 2. Prefabricated model defect parameters.

Model	Size of Experimental Wall/mm	Defect Parameters
		Quantity	Shape	Size/mm	Thickness/mm	Depth/mm
Model 1	500 × 500 × 70	4	circle	100	10	5
			rectangle	150 × 100	10
			triangle	125	10
			square	100 × 100	10
Model 2	480 × 350 × 70	4	square	50 × 50	10	5
				75 × 75	10
				100 × 100	10
				125 × 125	10
Model 3	480 × 350 × 70	6	square	70 × 70	3	5
				70 × 70	5
				70 × 70	7
				70 × 70	9
				70 × 70	11
				70 × 70	13

.

3.2. Model Creation and Testing

As mentioned in the previous section, experimental wall prefabrication requires setting parameters such as shape, size and cavity thickness. Referring to the industry standard “Technical specification for inspecting the defects of exterior walls cement coating of building using infrared thermography method (CECS 204-2006)” and “Technical specification for inspecting the defects of exterior walls cement coating of building with infrared thermography method (JG/T277-2012)” [32,33], an experiment wall with the size of 500 × 500 × 70 mm and two experimental walls with the size of 480 × 350 × 70 mm were man-made. The wall structure layers are composed of the prefabricated substrate (50 mm), leveling layer (10 mm), bonding layer (hollowing layer 5 mm) and finishing layer (5 mm).

Cement from the Conch brand M32.5 was used for making concrete and masonry. According to the “Standard for technical requirements and test method of sand and crushed stone (or gravel) for ordinary concrete (JGJ52-2006) [34]” and” Standard for construction quality acceptance of building decoration(GB 50210-2018) [35]”, the ratio of cement mortar for the leveling layer is 1:2, for the masonry layer is 1:2.7 and for bonding layer is 1:3, respectively. The surface schematic diagram of each structural layer on building facade is shown in Figure 4.

The shape and property defect fabrication process is shown in Figure 5. The whole manufacturing process of the facade models was carried out in ventilated and no-rain environment. After the basic layer was built, it was naturally air-dried for 5 days. Each layer was separately poured to dry, and the mold was removed after 3 days. After being maintained under natural conditions for 5 days, the experimental wall was transferred to the experimental site.

As shown in Figure 6, a Fluke Ti200 uncooled infrared thermal imager was used in the experiment. The resolution of the instrument is 640 × 480 and the working temperature range is −20~+650 °C, and the accuracy of temperature measurements can reach ±2 °C.

3.3. Qualitative Analysis of Passive Detection Infrared Thermography

• Model 1: Hollowing shape

Figure 7a is an infrared thermal image taken at 9:00 a.m. on 11 March 2024 with clear skies and an ambient temperature of 16 °C and without wind. The average temperature of the defect-free wall was 20.91 °C. It can be seen from Figure 7b that there are four obvious anomalies in the temperature field. The shapes of four obvious anomalies in temperature correspond to the shapes of the prefabricated defects, and the surface temperature of the defect area is obviously higher than the normal surface. The measured surface temperature of each defect area in Model 1 is shown in

Table 3. Surface temperature of each defect area in Model 1.

	Circular	Rectangle	Triangle	Square
$\mathrm{Maximum} \mathrm{temperature}/°\mathrm{C}$	32.80	32.94	29.18	30.17
$\mathrm{Average} \mathrm{temperature}/°\mathrm{C}$	29.10	29.11	26.89	26.87
$\mathrm{Minimum} \mathrm{temperature}/°\mathrm{C}$	21.83	20.90	21.82	20.90

. The maximum temperature difference between the defect and defect-free area was 12.03 °C, and the location is in the rectangular hollowing defect area.

Figure 7d shows the temperature curves with the different extraction paths. The surface temperature of the defect area is obviously higher than that of the normal area, and the highest point of surface temperature of the center of the defects is approximately symmetric. According to the lowest point of the temperature curve, the hollowing obviously affects the temperature distribution on the exterior surface, which has little effect on the edge area of the hollowing. The results show that the surface temperature field distribution is not affected by the shape of defects with similar areas under the excitation of external solar radiation. The larger the surface area of the hollowing defects with the same height, the larger the surface temperature field is with the more effect on surface temperature. The temperature field in four defect areas show low temperatures at the edge and high temperatures in the middle area. The reason for this is that the sun was on the left-front of the wall during the shooting process, so the radiation received by the wall was not uniform. Therefore, the peak temperature appears at the left of the center of the defect as the heat is accumulating.

• Model 2: Hollowing dimensions

Figure 8a is an infrared thermal image taken at 9:00 a.m. on 12 January 2024 with clear skies and an ambient temperature of 14 °C and without wind. The average temperature of the defect-free wall was 19.48 °C. It can be seen from the Figure 8b that there are four obvious anomalies in temperature field. The shapes of four obvious anomalies in temperature correspond to the shapes of the prefabricated defects, and the surface temperature of the abnormal areas is obviously higher than the normal surface. The measured surface temperature of each defect area in Model 2 is shown in

Table 4. Surface temperature of each defect area in Model 2.

Size/mm	50 × 50	75 × 75	100 × 100	125 × 125
$\mathrm{Maximum} \mathrm{temperature}/°\mathrm{C}$	25.60	27.69	27.72	28.01
$\mathrm{Average} \mathrm{temperature}/°\mathrm{C}$	24.06	24.86	25.36	25.51
$\mathrm{Minimum} \mathrm{temperature}/°\mathrm{C}$	21.92	21.89	20.67	21.32

. The maximum temperature difference between the defect and defect-free area was 8.53 °C, and the location is within the size of the 125 mm × 125 mm hollowing defect area.

Figure 8d shows the results of the temperature values along the different extraction paths. The surface temperature of the area with internal defects is obviously higher than that of the area without defects. The highest point of the surface temperature of the center of defects is approximately symmetric. The degree of influence decreases with the increase of distance, so it presents a continuous warming change from edge to center. The results show that under the external solar radiation excitation, the larger the defect surface area with the same shape and the same cavity thickness is, the greater the influence on the surface temperature. However, in the same group of experiments, the change rate of temperature differences in the different sizes of hollowing areas is almost the same, which shows that the change rate of temperature differences in hollowing area is its own inherent characteristic.

• Model 3: The cavity thickness of the hollowing

Figure 9a is an infrared thermal image taken at 9:00 a.m. on 13 January 2024 with clear skies and an ambient temperature of 14 °C and without wind. The average temperature of the defect-free wall was 18.96 °C. It can be seen from the Figure 9b that there are six obvious anomalies in the temperature field. The shapes of six obvious anomalies in temperature correspond to the shapes of the prefabricated defects, and the surface temperature of the abnormal areas is obviously higher than the normal surface. The measured surface temperature of each defect area is shown in

Table 5. Surface temperature of each defect area in Model 3.

Thickness/mm	3	5	7	9	11	13
$\mathrm{Maximum} \mathrm{temperature}/°\mathrm{C}$	24.10	26.53	28.87	29.26	31.83	29.64
$\mathrm{Average} \mathrm{temperature}/°\mathrm{C}$	22.10	23.49	25.52	25.92	27.53	26.31
$\mathrm{Minimum} \mathrm{temperature}/°\mathrm{C}$	18.68	18.37	18.84	19.04	19.87	19.48

. The maximum temperature difference between the defect and defect-free area was 12.87 °C, and the location was the hollowing cavity thickness of the 13 mm hollowing defect areas.

Figure 9d shows the results of the temperature values along the different extraction paths. The surface temperature of the defect area is obviously higher than the no-defect area, and the highest point of the surface temperature of the defect center is approximately symmetric. The results show that the thickness of the defect has a great influence on the distribution of the temperature field on the wall surface under external solar radiation excitation. The larger the thickness, the higher the surface temperature when the cavity thickness of the defect is 3 mm, 5 mm, 7 mm, 9 mm and 11 mm; this rule is followed, but when the cavity thickness is 13 mm, a temperature anomaly occurs: the maximum temperature, average temperature and minimum temperature of the building facade are lower than the defects of the cavity thickness of 11 mm. Considering that the hollowing defect with a cavity thickness of 13 mm has a large heat capacity and a small thermal conductivity, it plays the role of insulation at the edge of the medium layer. After being stimulated by solar radiation for a short period of time, the heat received does not reach its heat capacity. The cavity thickness of the 11 mm hollowing defect heat capacity is small, and when the heat conduction has been more than the defect can hold, the heat will lead to a small amount of excess heat conduction to the interior. Most of the heat transferred to the building facade is eventually transmitted to the air as heat radiation and where it is finally received by the infrared thermal imager.

4. Analysis of Influencing Factors on Infrared Characteristics of Building Facade

In this experiment, the temperature at the intact position of the wall surface was measured by a heat flow meter with a sampling frequency of 5 min per time, and the infrared temperature field of the wall surface was measured by an infrared thermal imager with a sampling time of 8:00–10:00, and the sampling frequency was 5 min per time.

4.1. Model 1: Hollowing Shape

As shown in Figure 10, for hollowing defects with similar size, the shape differences have little effect on the distribution of surface temperature field. Under the thermal excitation of solar radiation, the variation trend of the surface temperature with and without hollowing defects was consistent. The maximum and average surface temperature in the defect area was obviously higher than that in the normal area, and the temperature difference was more than 5 °C. There was no significant difference between the minimum temperature of the defect and the normal area.

The main reason for the above phenomenon is that, along the direction of the defect width, there was a continuous temperature change from the hollowing edge to the center, and there was no significant difference between the surface temperature of the defect edge area and the normal defect-free area. This is also one of the main reasons that the edge of the infrared thermal image was blurred and the contour could not be accurately extracted.

4.2. Model 2: Hollowing Dimensions

As shown in Figure 11, the trend of surface temperature changes with and without a defect is consistent under the thermal excitation of solar radiation. The maximum and average temperatures of the abnormal area surfaces are always higher than the normal defect-free ones, and there is no significant difference between the minimum temperature of the abnormal and normal defect-free area surfaces.

The larger the defect size of the hollow with the same shape and the same cavity thickness, the greater its impact on the surface temperature, which manifests as a greater temperature difference from the defect-free area. The surface temperature of defects with the side lengths of 7.5 cm, 10 cm and 12.5 cm had the same change trend, and the temperature difference was not large. However, it was obviously higher than that of the defects with a side length of 5 cm.

4.3. Model 3: Hollowing Cavity Thickness

Figure 12 shows that under the thermal excitation of solar radiation, the trend of surface temperatures is the same with and without defect, both of which are increasing. The maximum and average surface temperatures of the defect areas were always higher than those of normal defect-free ones. The difference between the maximum temperature and the average temperature of the defect surface with a thickness of 5 mm and the normal defect-free area was less than 5 °C. The difference between the maximum temperature and the average temperature of the defect surface with the thickness of 13 mm cavity and that of the normal defect-free area was more than 7 °C.

When the thickness of the hollowing defect is within a certain range, the larger the thickness, the higher the surface temperature. When the thickness reaches a certain value, the surface temperature of the hollowing defect with the same shape and size has little difference with the change of the cavity thickness. As shown in Figure 12, the maximum temperature of the hollowing defect area with a cavity thickness of 13 mm is always less than that of the cavity defect area with a cavity thickness of 11 mm, and so is the average temperature. It is verified that the ascending speed of surface temperature is not affected by the wall size or the thickness of hollowing defect under the action of solar radiation.

5. Experimental Verification of Defect Depth Formula

For concrete building facade materials, the weather conditions were selected as sunny days, and the experimental wall was placed in a relatively closed (can ignore the impact of wind speed) environment. The experiment consisted of the following steps:

(1)

Turn on the infrared thermal imager in advance to make it stable and cover the experimental wall model with tarpaulin to ensure that the test object is in a relatively stable state;

(2)

Observe the surface temperature change in the defect areas and remove the tarpaulin after it is stable. The initial time is the heating time of solar radiation;

(3)

When the heating time is 0 s, 30 s, 60 s and 90 s, take a infrared thermal image of the experimental wall (when the tarpaulin is removed, the infrared thermal imager will enter the auto focus function, which takes about 5 s);

(4)

The temperature data of the defect and non-defect areas are extracted by the software Fluke Connect, and the formula is verified by subsequent experiments.

The actual depth of the hollowing defect of the experimental wall was 5 mm. The defect depth theoretical calculated values are shown in

Table 6. Theoretical depth in Model 1.

Defect Shape	Time (s)	Defect Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{m}}$ (°C)	Defect-Free Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{s}}$ (°C)	Temperature $\mathbf{Difference} ∆\mathbf{T}$ (°C)	Defect Depth (mm)
Circular	5	27.57	19.82	7.75	2.167
	30	29.87	20.98	8.89	5.076
	60	28.32	21.73	6.59	8.620
	90	32.86	23.08	9.78	8.796
Rectangle	5	27.58	19.82	7.76	2.165
	30	29.73	20.98	8.75	4.724
	60	30.95	21.73	9.22	7.172
	90	32.56	23.08	9.48	8.949
Triangular	5	26.19	19.82	6.37	2.382
	30	27.05	20.98	6.07	6.100
	60	30.84	21.73	9.11	7.222
	90	29.84	23.08	6.76	10.526
Square	5	25.05	19.82	5.23	2.581
	30	26.84	20.98	5.86	6.186
	60	28.42	21.73	6.69	8.407
	90	30.52	23.08	7.44	10.094

,

Table 7. Theoretical depth in Model 2.

Defect Shape	Time (s)	Defect Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{m}}$ (°C)	Defect-Free Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{s}}$ (°C)	Temperature $\mathbf{Difference} ∆\mathbf{T}$ (°C)	Defect Depth (mm)
5 × 5 × 5	5	17.05	14.89	2.16	3.107
	30	18.05	15.01	3.04	6.921
	60	18.91	15.56	3.35	9.600
	90	19.33	17.21	2.12	13.728
7.5 × 7.5 × 5	5	17.49	14.89	2.60	2.954
	30	18.81	15.01	3.80	6.413
	60	19.71	15.56	4.15	8.905
	90	20.85	17.21	3.64	11.814
10 × 10 × 5	5	17.39	14.89	2.50	2.987
	30	18.76	15.01	3.75	6.45
	60	20.06	15.56	4.50	8.628
	90	20.83	17.21	3.62	11.845
12.5 × 12.5 × 5	5	16.42	14.89	1.53	3.373
	30	17.97	15.01	2.96	7.023
	60	19.91	15.56	4.35	8.745
	90	21.00	17.21	3.79	11.67

and

Table 8. Theoretical depth in Model 3.

Defect Shape	Time (s)	Defect Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{m}}$ (°C)	Defect-Free Surface $\mathbf{Temperature} {\mathbf{T}}_{\mathbf{s}}$ (°C)	Temperature $\mathbf{Difference} ∆\mathbf{T}$ (°C)	Defect Depth (mm)
5 × 5 × 3	5	20.17	14.09	6.08	2.049
	30	22.71	15.48	7.23	5.976
	60	24.93	16.53	8.40	6.637
	90	24.85	18.62	6.23	9.927
5 × 5 × 5	5	20.7	14.09	6.61	1.964
	30	23.32	15.48	7.84	5.832
	60	24.57	16.53	8.04	6.576
	90	25.95	18.62	7.33	9.159
5 × 5 × 7	5	19.41	14.09	5.32	2.206
	30	21.68	15.48	6.2	5.239
	60	22.65	16.53	6.12	7.721
	90	23.55	18.62	4.93	10.936
5 × 5 × 9	5	20.09	14.09	6	2.066
	30	21.63	15.48	6.15	5.262
	60	23.5	16.53	6.97	7.198
	90	23.77	18.62	5.15	10.755
5 × 5 × 11	5	18.13	14.09	4.04	2.499
	30	20.64	15.48	5.16	5.257
	60	21.6	16.53	5.07	8.421
	90	22.5	18.62	3.88	11.88
5 × 5 × 13	5	16.55	14.09	2.46	2.954
	30	18.36	15.48	2.88	7.103
	60	19.71	16.53	3.18	9.944
	90	20.07	18.62	1.45	15.157

.

According to the data of the thermal image, the calculated depth values of 14 different types of hollowing defects presupposed in the experiment can be obtained; the defect depth ranges from 1.964 mm to 15.157 mm. It can be seen from the Table 6, Table 7 and Table 8 that the actual defect depth is 5 mm, which corresponds to the calculated values of the defect depth with a heating time between 5 s and 30 s. According to the data collected in this time period, the depth is calculated and the result is closest to the actual depth value.

The reasons for this can be summarized as follows:

(1)

In the heat conduction process of building facades, the wall model is considered to be one-dimensional heat conduction, while the actual heat transfer process is carried out simultaneously in three directions, and some heat loss will be ignored by simplification methods [36];

(2)

In the process of heat transfer deduction, it is assumed that the hollowing defect is completely insulated. The heat is transferred from the outside to the inside and accumulates on the outside surface in the defect area due to the thermal excitation of solar radiation. Although the thermal conductivity of the hollowing defect is far less than the thermal conductivity of cement mortar, it is not completely insulated, which will cause some errors;

(3)

The calculation formula is obtained by the numerical integration trapezoid formula, and the longer the time is, the greater the error caused by the formula. Therefore, under certain conditions, with the increase of time, the difference between the calculated defect depth and the actual situation is larger.

In summary, the theoretical depth calculation formula is modified by experiments as shown in the following equation:

$$d=\sqrt{\alpha t\left[\ln \left(T_{s\left(5\mathrm{s}~30\mathrm{s}\right)}\right)-\ln \left(∆T_{\left(5\mathrm{s}~30\mathrm{s}\right)}\right)\right]}$$

(22)

6. Conclusions

The hollowing defects of the building facade not only reduces building insulation performance leading to large amounts of energy loss but also threatens the safety of pedestrians and property around the building. This paper has derived the formula to calculate the depth of hollowing defects based on the basic principle of energy radiation and heat transfer theory, and experimental models of the hollowing defects were built, by which the thermal images of hollowing under different sunlight radiation conditions were obtained by a thermal imager. The relationship between hollowing shape, size, cavity thickness and temperature distribution was analyzed, and the correctness of the calculation formula for hollowing defect depths was validated. The method proposed in this paper has provide a solution to the difficulty of characterizing hollowing defects, relying on subjective experience at present in practical engineering applications, and can lay the foundation for achieving high-precision quantitative detection of the hollowing defect areas on building facade.