## 2 Materials and Methods
### Modelization
It should be noted that the mean color intensity \(\overline{I}\) of the selected surface depends directly on the number of pixels with the dirt color, \(n_{dc}\):  
\[\overline{I}=\frac{\mathrm{n}_{\mathrm{dc}}I_{1}+\left(n_{p}-n_{dc}\right)I_{ 2}}{n_{p}}=n_{dc}\frac{\left(I_{1}-I_{2}\right)}{n_{p}}+I_{2} \tag{4}\]  
where \(I_{1}\) corresponds to the characteristic color of the dirt surface and \(I_{2}\) the characteristic color of the clean surface. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Order** & \(\overline{I}\) & \(sd^{2}\) & \(\omega_{1/2}\) & \(\omega_{1/4}^{r}\) & \(\omega_{1/4}^{I}\) & \(\frac{\omega_{1/4}^{r}}{\omega_{1/4}^{r}}\) \\ \hline
0 & \(I_{1}+\frac{I_{2}-I_{1}}{n_{p}}k_{1}t\) & \(\frac{k(I_{2}-I_{1})^{2}(n_{p}-kt)t}{n_{p}^{2}}\) & \(\frac{n_{p}}{k\sqrt{2}}\) & \(\frac{n_{p}}{k\sqrt{8}}\) & \(\frac{n_{p}}{k\sqrt{8}}\) & 1 \\ \hline
1 & \(I_{2}+\frac{\left(I_{1}-I_{2}\right)}{e^{2t}}\) & \(\frac{\left(e^{it}-1\right)\left(I_{2}-I_{1}\right)^{2}}{e^{2t}}\) & \(\frac{176}{k}\) & \(\frac{123}{k}\) & \(\frac{0.53}{k}\) & 2.3 \\ \hline
2 & \(\frac{kt(I_{2}-I_{1})^{2}n_{p}}{\left(ktn_{p}+1\right)^{2}}\) & \(\frac{kt(I_{2}-I_{1})^{2}n_{p}}{\left(ktn_{p}+1\right)^{2}}\) & \(\frac{5.66}{n_{p}}\) & \(\frac{4.83}{k\mu_{p}}\) & \(\frac{0.83}{k\mu_{p}}\) & 5.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak widths (\(\omega_{1/2}\)) and half peak widths (\(\omega_{1/4}\)) of variance versus time for different kinetic orders (\(\alpha\) in Equation (3)). \(n_{p}\) is the number of pixels in the image, \(I_{1}\) is the mean color intensity characteristic of the dirt layer, \(I_{2}\) is characteristic of the clean metal surface and \(k\) is the kinetic constant of the irreversible process of cleaning [44]. Figure 3: Simulation of the evolution of mean color intensity and variance against the elapsed time, in arbitrary measurement units, during a surface chemical reaction for a zero-order reaction (\(\alpha=0\) in Equation (3)) (**A**) and second order reaction (\(\alpha=2\) in Equation (3)) (**B**) [44]. In accordance with Table 2, these peaks will show a symmetrical shape only for kinetics of order 0 (\(\alpha=0\)), while for kinetics of higher orders, the right side of the peak appears wider than the left side [41,44]. In the second proposed mechanism, the cleaning product chemically attacks the layer of dirt, changing both the mechanical properties and the adhesion to the metal surface. The layer of dirt will eventually become easier to peel off from the metal surface and easier to break off. Therefore, this proposed mechanism necessarily adds to the chemical cleaning some mechanical actions when dirt is removed in portions. This second proposed mechanism also starts with a chemical attack on the surface of the deposited dirt layer that does not necessarily involve the dissolution of the dirt layer. In some cases, the chemical attack will involve a change in the physical-chemical properties of the layer of dirt which can be observed as a color change of the layer of dirt. Therefore, we would observe changes in color intensity from the original color to the color of the coating after the chemical attack and a peak in the evolution over time of the variance [44] similar to that simulated in Figure 3.  
Once the chemical attack has occurred, the mechanical action removes this layer of dirt from the surface of the metal, and both the mean color intensity and the variance will show irregular evolutions until the clean metal surface appears. These irregularities can be masked if the surface area studied is large enough and the pieces of dirt leaving the surface are not very large in size. Note that this process and its speed will be totally dependent on the type of mechanical action and its extent. As the layer of dirt is not dissolved, portions of dirt pass to the solution as a suspension or precipitate, depending on the relative density of the dirt.