# Research on Defect Detection of Bare Film in Landfills Based on a Temperature Spectrum Model
## 2 Position Detection Principle  
The detection principle of bare film defects under a natural light source based on a heat conduction model is as follows: under conditions where natural heat sources are sufficient (as shown in Figure 1, where the sun serves as the natural heat source) and the minimum ambient temperature is 37 \({}^{\circ}\)C, the HDPE film after 30 min of exposure is laid over the soil layer and the HDPE film is cooled by sprinkling water (the water temperature was generally maintained at 20-25 \({}^{\circ}\)C). Due to the thermal diffusion coefficient of the wet soil layer, below the defect area is different from the unruptured area of the HDPE film. Defective and complete areas will have synchronized temperature difference characteristics, and the location of the defective areas can be identified from the captured infrared images. In the detection process, given the extensive coverage of HDPE films in landfill installations, the HDPE film was divided into equally sized areas (area size is 50 cm \(\times\) 50 cm) and the surface of a specific area cooled through water flow. An infrared thermal imaging camera was used to capture infrared images of this area until no temperature difference was detected. The above operation was carried out on the entire surface of the HDPE film by means of preset horizontal and longitudinal acquisition routes. The detection principle is shown in Figure 1.  
Infrared heat waves follow the heat conduction equation as they propagate through the HDPE film. To detect defects, only heat conduction in the depth direction is considered. When the thermal conductivity is determined, the heat conduction equation is:  
\[\
abla\cdot[k\
abla T(x,t)]-\rho c\frac{\partial T(x,t)}{\partial t}=-f(x,t) \tag{1}\]  
For the plane heat source available:  
\[f(x,t)=q\delta(x)\delta(t) \tag{2}\]  
Bringing Equation (2) into Equation (1) yields:  
\[\left\{\begin{array}{c}\frac{\partial^{2}T(x,t)}{\partial x^{2}}-\frac{1}{ a}\frac{\partial T(x,t)}{\partial t}=-\frac{q}{\lambda}\delta(x)\delta(t)\\ \alpha=\lambda/\rho c\end{array}\right. \tag{3}\]  
In Formula (3), \(x\) is the position in the depth direction, \(t\) is the time, \(T\) is the temperature, \(\alpha\) is the thermal diffusion coefficient of the medium, and \(\lambda\) is the thermal conductivity, where \(\lambda_{1}\) is the thermal conductivity of moist soil in the broken area; \(\lambda_{2}\) is the equivalent thermal conductivity of the area under the joint influence of the HDPE membrane and moist soil; and \(\lambda_{3}\) is the thermal conductivity of the unbroken area of the HDPE membrane. \(\rho\) is the density, \(c\) is the specific heat capacity, and \(q\) is heat flux density. When the value of \(\lambda\) is larger, the object reacts faster to the change in ambient temperature. The moist soil in the  
Figure 1: Infrared imaging detection of HDPE film defects.  
defect area acts as a heat-conducting medium, which has the largest thermal conductivity \(\lambda_{1}\) and the fastest response to environmental temperature changes. The humid area under the film is affected by the HDPE and the humid soil, and its equivalent \(\lambda_{2}\) value is less than \(\lambda_{1}\) and greater than \(\lambda_{3}\). Therefore, within a period of time after the sprinkling water cools down, the temperature rise trend is \(T_{1}<T_{2}<T_{3}\), as shown in Figure 2.  
The relative thermal value is approximated as the difference between the surface temperature of the complete area of the HDPE film (\(T_{3}\)) and the defective area (\(T_{1}\)), expressed by \(\Delta T\) as:  
\[\Delta T=T_{3}-T_{1} \tag{4}\]  
By analyzing the above formulas, we know that, due to the effect of water flow, when the HDPE film has defects, it will cause fluctuations in the thermal field on the surface of the film, which ultimately affects the infrared radiation detection process.