## 2 Corrosion Depth Prediction of Carbon Steel  
### Construction of Predictive Model  
In this paper, carbon steel corrosion data from different atmospheric exposure tests worldwide were used as samples. The data sets were obtained from the ISO-CORRAG program [26] and China Gateway to Corrosion and Protection ([[http://www.ecorr.org/](http://www.ecorr.org/)]([http://www.ecorr.org/](http://www.ecorr.org/)), accessed on 1 July 2021), including the latitude and longitude of the exposure site, the corrosion depth of the carbon steel and the corresponding natural exposure time. According to the different geographical locations of the exposure sites (as shown in Figure 3), the atmospheric environments were divided into a marine atmosphere (M), industrial atmosphere (I), urban atmosphere (U), rural atmosphere (R), urban-marine atmosphere (U-M) and industrial-marine atmosphere (I-M). Exposure tests to the carbon steel atmosphere at some of the sites are shown in Figure 4. The exposure test protocols were designed in strict compliance with \"ISO 9226:2012, Corrosion of Metals and Alloys\" [26]. The corrosion depth \(D\) (\(\upmu\)m) was used to represent the corrosion loss of steel. Scholars have proposed various time-varying models for corrosion, mainly including linear, power,  
Figure 4: Atmospheric exposure test. (**a**) Industrial atmosphere. (**b**) Marine atmosphere. Figure 3: Geographic distribution of atmospheric exposure sites in the ISO-CORRAG program. logarithmic, exponential, nonlinear-linear and bimodal models [3]. However, according to the results of the long-term exposure tests, it was found that the corrosion loss of metals under different atmospheric environments basically follows the power-linear function law [27]: In the first stage, the metal corrosion loss shows a power function growth law; in the second stage, the corrosion loss grows linearly, i.e., it shows a stable development trend. The expressions are as follows:  
\[\left\{ \begin{array}{l}D_{1}=D_{0}\times{t_{1}}^{n}\\ D_{2}=D_{1}+\alpha{t_{2}}\end{array} \right. \tag{1}\]  
where \(D_{1}\) indicates the first stage corrosion depth (\(\upmu\)m), \(D_{2}\) indicates the second-stage corrosion depth (\(\upmu\)m), \(D_{0}\) indicates the first-year corrosion depth (\(\upmu\)m) and \(t_{1}\) and \(t_{2}\) indicate the first-stage and second-stage corrosion ages (year), respectively. According to the existing research [8] conclusions, carbon steel enters into the corrosion stabilization stage (second stage) when its corrosion age is 6 years, \(\alpha\) indicates the second-stage corrosion rate (\(\upmu\)m/year), \(\alpha=d(D_{2}-D_{1})/dt_{2}\) and \(n\) is a coefficient that characterizes the protective properties of corrosion products. According to the Tammann theory [27], the metal corrosion rate is controlled by the rate of oxygen across the corrosion products (rust layer) to the metal surface, so the thicker the rust layer, the more difficult it is for oxygen to reach the metal surface, and the smaller the corrosion rate. Considering the correlation between \(n\) values on the corrosiveness of the atmospheric environment, double logarithmic fits were performed according to the test data of each exposure site to obtain the \(n\) values and the corresponding \(D_{0}\) values. Figure 5 shows a double logarithmic plot of the corrosion depth vs. exposure time for carbon steel at some typical sites--specifically, marine (M) atmosphere for Tokyo, rural (R) atmosphere for Ahtari, urban (U) and industrial (I) atmospheres for Otaniemi and urban-marine (U-M) and industrial-marine (I-M) atmospheres for Ponteau Martigues. It can be seen that the corrosion rate of carbon steel is the fastest, and the first-year corrosion depth \(D_{0}\) is also the largest for the exposure site in the urban-marine and industrial marine atmospheres. The first-year corrosion depth of carbon steel at the exposure site in the rural atmosphere is the smallest.