# Quaternion Processing Techniques for Color Synthesized NDT Thermography
## 4 Results  
### Data Used in the Evaluation  
Computationally generated thermal data has been used in the first analysis stage of this study, in order to avoid the influence of imperfections inherent to real situations on the behavior of the analyzed methods. The absence of effects related to the noise produced by the IR sensor, the heterogeneities of the stimulation on the surface of the inspected objects and the uncertainties about the characteristics of the base material and the defects, enables the analysis of the fundamental characteristics of the developed methods and to determine their characterization capabilities. The generation of synthetic data has been carried out by simulating an ideal thermographic NDT test. The equation that governs the heat transfer process conducted in NDT thermographic inspections has been defined by means of a heterogeneous 3D isotropic thermal diffusion model and the consideration of a series of simplifying hypotheses, such as opaque, heterogeneous and isotropic material, with thermophysical properties independent of temperature, no internal heat sources, and incident energy being diffuse and homogeneously distributed along the external surface. This model is mathematically represented by Equation (24). \[\frac{\partial T(x,t)}{\partial t}-\
abla\cdot(\alpha(x)\
abla T(x,t))=F(x,t) \tag{24}\]  
\[x\in\mathbb{R}^{3}\,,\;t\geq 0\,,\;\alpha(x)\in\mathbb{R}_{+}\]  
where \(T\) is the temperature at each material point, \(\alpha\) is the thermal diffusivity of the material \(\alpha=k/(\rho c_{p})\), \(k\) is the thermal conductivity, \(\rho\) is the density of the material, \(c_{p}\) is the specific heat at constant pressure and \(F\) represents the heat sources. The Equation (24) states the balance among the heat generated inside the material \((F(x,t))\), the net heat flux in the material \((\rho(x)c_{p}(x)\partial T(x,t)/\partial t)\) and the heat stored inside the material \((\
abla\cdot(k(x)\
abla T(x,t)))\). The type of material modeled consisted of a Carbon Fiber Reinforced Polymer (CFRP). This is a composite material with excellent mechanical properties and relationship between resistance and weight, widely used in the aeronautical industry. The sample consisted of several layers with the fibers oriented in a specific direction, stacked and bonded by an epoxy resin. The thermal characteristics of this material were modeled by three different values of thermal diffusivity, one for each coordinate axis. These diffusivity values have been estimated as the mean value of the data obtained from various bibliographic sources [39; 40], so that the values finally used have been \(\alpha_{x}=7.0\pm 2\cdot 10^{-7}\) m\({}^{2}\)/s, \(\alpha_{y}=3.5\pm 2\cdot 10^{-7}\) m\({}^{2}\)/s and \(\alpha_{z}=1.0\pm 0.3\cdot 10^{-6}\) m\({}^{2}\)/s. This sample was designed with 18 internal defects of sizes \(10\times 10\) mm\({}^{2}\) and \(5\times 5\) mm\({}^{2}\) located at different depths from 1 mm to 9 mm (Figure 4a). These defects were modeled with thermal properties simulating delaminations with \(\alpha=1.25\pm 2\cdot 10^{7}\) m\({}^{2}\)/s. This diffusivity value corresponds to Polytetrafluoroethylene (PTFE), whose use to simulate delaminations is widespread. Optical Step Heating Thermography (OSHT) has been modeled for experimental capacity reasons. The stimulation conditions simulated a constant heating of the inspected  
Figure 4: Modeled NDT thermographic inspections: (**a**) position of defects with depth indicated numerically [mm], (**b**) thermal stimulation considered in the model, (**c**) scaled temperature values obtained with the theoretical model in absence of noise after 5 s in the heating process, (**d**) scaled temperature values obtained with the theoretical model in the presence of noise after 5 s in the heating process. sample, applied to its external surface by means of halogen lamps, generating a heat flow into the material (Figure 4b).