## 2 Materials and Methods
### Modelization  
First, let us consider two simplified and ideal models by which the cleaning process can take place and the expected digital image results for each process. The purpose of this section is to gain a better understanding of the experimental results from real samples. In the first model, the cleaning took place by the chemical attack of the detergent product with the consequent complete dissolution of the deposited soiling, layer by layer. As the dissolution of the outermost layers of dirt exposed a new layer of dirt with the same surface color and homogeneity, but with a decreasing thickness, the cleaning process showed an apparent latency period without any color change, despite the dissolution process taking place. At a certain moment, the last layer of dirt began to disappear, exposing the bare metal surface. We can define this time as \((t_{0,s})\), the first time at which the metal surface appears. This process results in changes of color homogeneity and intensity of the image due to the emergence of new colored pixels. Depending on the color of the dirt layer and the bright metal surface, the change in mean color intensity can be either positive or negative. As the metal surface becomes more exposed and at least two different colored surfaces appear at the same time, the variance always increases initially. Variance reaches a maximum value, which is interpreted as the point where the color dispersion is maximum. In other words, this is the time at which half of the metal surface is covered by the layer of dirt and the other half is clean. We call this time the half-life time of surface cleaning process \((t_{1/2})\). It is important to note that this time is not the half-life time of the complete cleaning process since it depends on the thickness of the dirt layer. From this point on, the dirt layer will continue to dissolve, and the color of the metal surface will increasingly prevail, making the surface more homogeneous and thus decreasing the variance. In an ideal situation, at the end, the entire surface is clean and this will be identified by the fact that the average color intensity no longer changes, and the variance has returned to a minimum and constant value. This is the final time of the cleaning process \((t_{f})\). Figure 3 shows two simulations of how these parameters (mean color intensity and color intensity variance) evolve over time for a cleaning process considering this dissolution mechanism and assuming that the analyzed region of the image of \(n_{p}\) pixels changes its color between a color intensity \(I_{1}\) characteristic of the dirt layer to a final color intensity \(I_{2}\), characteristic of the clean metal surface. We also suppose an irreversible process governed by a kinetic constant \(k\). Mechanical action, such as stirring the dissolution, can speed up the process by removing the layers of solvent that are more concentrated in the dirt from the metal surface. The shape of these curves is characterized by the equations collected in  
Figure 2: Workflow to obtain mean color intensities and variance from the color image acquired. Table 2[44;46]. The mean color intensity (\(\overline{I}\)), the variance \(sd^{2}\), the peak widths (\(\omega_{1/2}\)), the peak half-widths (\(\omega_{1/4}\) left and right side) and the ratio between the peak half-widths of variance peaks depends on the kinetic orders (0,1,2)[41;44]. It should be noted that this last ratio can be interpreted as a measure of the symmetry of variance peaks. Considering that the cleaning rate is directly related to the color change rate of the metal surface, we can propose a kinetic rate law for the cleaning process as follows:  
\[\textit{cleaning rate}=-\frac{\mathrm{d}\mathrm{n}_{\mathrm{dc}}}{dt}=\mathrm{ k}\cdot\mathrm{n}_{\mathrm{dc}}^{\mathrm{a}} \tag{3}\]  
In Equation (3) \(n_{dc}\) refers to the number of pixels with color of the dirt surface and \(\alpha\) the kinetic order for the cleaning process.