## 4 Experimental Validation of a Volumetric Hot Wire Warmed by the Joule Effect  
To demonstrate the efficiency of the method, the proposed 3D inverse processing is applied on an experimental case based on volumetric heat source generated by the Joule effect. For that, a chromel wire is molded in a PVC block. The thermal source is created via the Joule effect to warm up the sample; electric current travels on the chromel (nickel chrome) wire over 7.5 s. The temperature was measured by the IR camera on the black-painted rear face of the PVC block, as depicted in Figure 8.  
Figure 8: Experimental setting. Figure 7: (**a**) 400 measured temperatures at the surface on all the pixels. (**b**) Probability superior to 99% considering \(T_{surf}\). (**c**) 400 temperatures on all the pixels at the surface after subtraction of the source \(p_{1}\). (**d**) Probability superior to 99% considering \(T_{surf_{2}}\). In this experiment, the diameter of the chromel wire was 200 \(\,\upmu\)m, and its geometric shape was similar to an \"M\". It was placed on the rear face of a PVC sample 1 \(\,\upmu\)m thick. To avoid thermal losses on the same side that the sample, an insulating foam was bonded to the surface. The chromel wire was on a small area of the PVC sample; therefore, it can be considered semi-infinite along the \(x\) and \(y\) directions. The chromel wire was bonded to the back of the PVC sample. To estimate the whole 3D effect, the method of the images was used; the sample was then considered to be 2 mm thick with the wire located not on its back but in the middle of the plane. The PVC was assumed to be homogeneous and orthotropic with well-known thermal properties given as follows: a density of \(\rho=1180\)\(\,\)kg\(\cdot\)m\({}^{-3}\), a heat capacity of \(C_{p}=1000\)\(\,\)J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), and a diffusivity of \(a=1.2\times 10^{-7}\)\(\,\)m\({}^{2}\cdot\)s\({}^{-1}\). The voltage of the generator was \(U=3.5\)\(\,\)V with an intensity of \(I=1.13\)\(\,\) A. Measurements were carried out by an IR camera with an acquisition frequency of 200 \(\,\)Hz, and the size of the pixel at the surface sample was 290 \(\,\upmu\)m \(\,\times\,\) 290 and \(\upmu\)m.  
Figure 9a shows the thermogram (the mean of the temperature field on all the pixels of the surface as a function of time) as well as the measured temperature at the surface at \(t=0\) s Figure 9b and at \(t=7.5\) s Figure 9c. At \(t=0\) s, i.e., at the very beginning of the experiment, only noise was measured by the IR camera, as shown in Figure 9b. Indeed, the diffusion of the temperature had not yet taken place. During the experiment, the temperature rose throughout the material; at the surface, the geometrical figure formed by the wire is clearly visible. In theory, all the source points formed by the wire are of the same intensity. In addition, since the wire is placed on a plane parallel to the surface, the temperature rise at the line of the wire should be identical in all respects. It can be noted in Figure 9c that this is the case, except in two places where the temperature rise is higher. At the end of the experiment, we can note in Figure 9c that the maximum variation of temperature is approximately 20 degrees. Figure 9: (**a**) Thermogram as a function of time (mean of the measured temperature at the surface). (**b**) Surface temperature at \(t=0\) s and at (**c**) \(t=7.5\) s.  
As explained in the previous section, all the pixels of the surface (containing a temperature as a function of time) are treated simultaneously and considered as independent 1D problems, where the goal is to reconstruct the source in depth using the temperature at the surface point.