Long-term state-driven atmospheric corrosion prediction of carbon steel in different corrosivity categories considering environmental effects

1 Introduction

The long-term corrosion data of metal materials or structures under typical natural environments, such as atmospheric environments are essential to evaluate and predict the corrosion loss of metal materials or structures (Melchers 2013). The corrosion test under a natural operational environment is a reliable and effective approach to acquiring actual corrosion information.

For example, the ISO CORRAG Program (Knotkova et al. 2012) was initiated in 1986 and developed to expose both flat panels and wire helix specimens to the atmosphere throughout the world to obtain a data set on atmospheric corrosion that was carried out uniformly and with well-characterized samples. In addition, this program also accumulated data on the important atmospheric variables of temperature, relative humidity, sulfur dioxide, and sodium chloride deposition rates. In Europe, “the International Co-operative Programme on Effects on Materials including Historic and Cultural Monuments” (ICP Materials) has performed exposures investigating the influence of pollution and climate on corrosion since 1987 (Tidblad et al. 2012). In a network of more than 20 sites with different environmental characteristics, this program conducted repeated outdoor exposure tests of about 20 different materials and issued more than 60 official reports. Meanwhile, a dose-response function has been developed for ICP materials to quantitatively evaluate the effects of various pollutants and climate parameters on the atmospheric corrosion of important materials. The Ibero-American Map of Atmospheric Corrosiveness (MICAT) project was set up in 1988 and sponsored by the International Ibero-American Programme “Science and Technology for Development” (Morcillo 1995). Fourteen countries are currently involved in the project. Research is conducted both at laboratories and in a network of 72 atmospheric exposure test sites throughout the Ibero-American region, thus considering a broad spectrum of climatological and pollution conditions. This project basically follows the outline of the ISO CORRAG and ICP/UNECE projects, with the aim of a desirable link between the three projects.

However, the corrosion test under a natural operational environment is usually restricted by its expensive time and cost consumption (Cai et al. 2020), so some studies produce long-term corrosion data through accelerated tests in the laboratory (Kim et al. 2005). In recent decades, a main focus of researchers is how to accurately predict the long-term corrosion properties, including corrosion loss and remaining corrosion lifetime through limited corrosion data (either under natural or laboratory environment), without the need to conduct long-term corrosion tests. As such, considerable sample, material and cost resources can be saved.

The main approach for corrosion prediction is to construct physical or statistical corrosion models, particularly when the corrosion sample number is limited. A representative instance is the power function model, which is widely employed to describe corrosion loss in terms of exposure time (De la Fuente et al. 2007, 2011; Hou and Liang 2004), as introduced below

where C₁ are the undetermined coefficients, and n is a mass loss time exponent that characterizes the protective performance of corrosion products. C is the average corrosion mass loss or the average corrosion depth, and t is the corrosion time. Typically, C₁ represents the first year of corrosion in this model. Apparently, the power function model is intuitive and easy to apply. However, due to the variability of protective effects among different corrosion products, such a model is suitable to describe some long-term corrosion behaviors. To this end, some works proposed the bi-logarithmic law model, which is capable of long-term corrosion prediction up to 20–30 years (De la Fuente et al. 2011; Ma et al. 2010; Morcillo et al. 1993):

where C1′, n1, n2 are coefficients to be determined, t1 is the point in time when the quality of the atmospheric environment improves or deteriorates significantly, or when the physicochemical characteristics of the embroidery layer change very significantly [], C and C′ represent the average corrosion mass loss or the average corrosion depth before and after the moment, t1 respectively, and t is the corrosion time. Based on the real test data, the power function model is modified. This model verifies that there are an initial corrosion stage and a stable corrosion stage in the long-term corrosion process, which defined two successive corrosion phases, each of which contains several parameters that characterize critical corrosion features. However, the above-mentioned models have not considered the impact of the operational environment, which may lead to unsatisfied prediction outcomes. For instance, the corrosion rate of metal materials exposed to the dry, scorching atmosphere significantly differs from that under humid, windy marine environments.

In order to quantify the influence of environmental factors on corrosion, some studies constructed environment-related corrosion models considering multiple environmental factors based on power function, that is,

where C1″, n′ are the undetermined coefficients, f(TOW,SO2,RH,⋯) is the function related to the environmental factors, C is the average corrosion mass loss or average corrosion depth, and t is the corrosion time. Klinesmith et al. (2007) proposed a widely accepted corrosion model considering four atmospheric environmental factors (time of wetness, sulfur dioxide concentration, chloride deposition rate, air temperature) and applied it to the prediction of test results in ISO CORRAG. On this basis, a corrosion prediction model considering more complex environmental effects was proposed by Ji et al. (2021), which was applied to the corrosion prediction of weathering steel (Corten-A and Corten-B) in China. This type of model considers the influence of environmental factors on the corrosion process based on the power function model, but for long-term corrosion, there is a lack of consideration of the different corrosion stages. Other researchers employed machine learning approaches (Cai et al. 1999; Halama et al. 2011; Kenny et al. 2009; Li et al. 2022; Pei et al. 2020; Yan et al. 2020; Zhi et al. 2019), particularly artificial neural networks for corrosion prediction when condition monitoring is available or the training samples are abundant. This method is suitable for the analysis of complex systems with uncertainty, but it lacks the consideration of the physical meaning of the corrosion model and requires a large amount of data.

A notable feature of metal corrosion under an atmospheric environment is that the corrosion evolution process possesses typical two-stage behaviors (ISO 9224 2012; Panchenko and Marshakov 2016). Such behaviors are well captured by the power-linear function models according to corrosion test outcomes (Cai et al. 2018a,b; Panchenko et al. 2014), that is,

where C1‴, n″, k are the undetermined coefficients, t1 is the corrosion steady-state start time, C and C′ represent the average corrosion mass loss or average corrosion depth before and after the moment t1, respectively, and t is the corrosion time. To be specific, the power function is employed to describe the exponential corrosion behaviors at the first stage, while the linear function is applied to describe the steady corrosion behaviors at the second stage. Once the corrosion steady-state start time is determined, then such a corrosion kinetics model can accurately capture the two-stage corrosion process and has a good prediction effect on long-term corrosion, but it lacks consideration of environmental dynamic effects. According to the corrosivity category of ISO 9223 (2012), Díaz et al. (2012) found that the steady-state start time of metals is 6–8 year under less corrosive environments and 4–6 year under more corrosive environments. The heterogeneity among steady-state start time indicates that the time to reach a steady corrosion state varies according to a metal material type and operational environment.

It is noteworthy that the applications of the aforementioned atmospheric corrosion prediction approaches are challenged by three limitations. First, there lacks comprehensive power-linear atmospheric prediction models that can integrate multiple environmental factors, which restricts its applicability under different atmospheric environments. Data-driven approaches such as deep learning are possible solutions, but are not directly correlated to actual physical corrosion characteristics. Second, the environmental coefficients employed in most corrosion models are constant, which are unable to capture the dynamics of the environment during the long-term corrosion process. This may lead to an inaccurate prediction outcomes of long-term corrosion prediction. Third, the steady-state start time (linear corrosion pattern), particularly under a dynamic natural environment is seldomly addressed.

To fill these gaps, in this study, environmental factors are taken into account in the long-term atmospheric corrosion prediction model based on power linear function, and a novel method to describe the steady-state start time of corrosion is proposed based on the robustness of corrosion kinetics. Finally, the effectiveness of the proposed framework is verified by the prediction results of long-term atmospheric corrosion test data and comparison with the previous models.

To summarize, the method described in this paper can be used as a framework for long-term corrosion prediction modeling, including the calculation of steady-state start time, the modeling of environmental factors, the modeling of the two-stage corrosion process, the contributions of this work to long-term atmospheric corrosion prediction under atmospheric environment are outlined below:

1. A comprehensive power-linear corrosion model considering environmental effects is proposed.

2. Based on the robustness of corrosion kinetics, a novel method of determining corrosion steady-state start time is proposed.

3. Compared with previous models, the results of 40 years of corrosion loss prediction show that the proposed model is in good agreement with the experimental data and corrosion state.

4. Based on the corrosivity categories, the corrosion loss prediction of the proposed framework obtains sufficiently accurate output information with less input information.

The rest of section is structured as follows. Section 2 proposes a comprehensive power-linear function model considering dynamic environmental effects. Section 3 investigates the approach to determining the steady-state start time of the corrosion process. In Section 4, the long-term prediction of corrosion loss under different natural environments is conducted based on ISO CORRAGE program and field exposure test data. Section 5 concludes the paper.

2 State-driven corrosion kinetics model

As mentioned in Section 1, the kinetics of long-term metal corrosion loss of metal materials under an atmospheric environment follows a power-linear function based on atmospheric corrosion test outcomes (Melchers 2013). As specified in ISO 9224 (2012), the metal corrosion process possesses a power function pattern at the initial stage, and then a steady linear function when exposed for more than 20 years. Recent studies further summarized a three-phase division of the corrosion process, including the incubation phase, the transition phase, and the steady state phase. As corrosion emerges from the metal surface, the corrosion rate continues to decrease in the initial phase. The steady state is reached when the metal is completely covered by a layer of corrosion products of a certain thickness, and thereby the corrosion rate remains constant. Therefore, the corrosion loss kinetics is described by a power-linear function (Cai et al. 2018a,b; ISO 9224 2012; Panchenko and Marshakov 2016). Especially, the power function indicated in Eq. (1) is employed to capture the corrosion behavior at the former two phases, and a linear function to characterize the steady state as

where α represents the annual corrosion rate at the steady state; C0 is the corrosion loss obtained by extrapolation of the linear plot to t=0; α is a constant at the steady state.

Remark: To conclude, it is commonly accepted that the atmospheric corrosion process is the union of two corrosion phases, whose evolution can be effectively captured by the power-linear model. However, there is no explicit criterion to determine the steady-state start time, as it hinges heavily on the properties of relevant metals, atmospheric type, and corrosiveness. In particular, the impact of the operational environment is rarely reported in the power-linear model.

Motivated by the above-mentioned challenges, we provide a comprehensive power-linear corrosion prediction model that can integrate multiple atmospheric environmental factors, which defines the corrosion loss as follows

where a0 is an empirical coefficient, M represents the corrosion mass loss at the initial stage of corrosion, ts represents the steady-state start time, M′ is the corrosion mass loss at corrosion steady state, and κ is the corrosion rate in the steady state, and f(ϑ) is an extended function between the environmental factors and corrosion process.

Under the current model framework, the steady-state start time can be evaluated from the view of corrosion kinetics. Generally, considering the robustness of the corrosion model (Ji et al. 2021), the corrosion equivalent factor is a constant determined by the values of the stress levels and independent of other parameters when the degradation mechanism remains unaltered. The following is a brief introduction to this theory.

Take the model with t≤ts in Eq. (6) as an example, that is:

In fact, the corrosion loss may not only be limited to the influence of environmental factors but will also be affected by random factors, such as measurement errors, surface parameters of materials, individual differences of materials, and so on. The scatter can be modeled by distributions. Thus, considering the influence of data scatter, adding an error term ε, which follows a normal distribution with a mean of 0 and a standard deviation of σ, then Eq. (7) can be expressed as:

Therefore, given the failure threshold of corrosion loss shown as γ, a cumulative distribution function for the corrosion life of the material can be shown as:

where Φ(⋅) is the cumulative distribution function (CDF) of the standard normal distribution.

Considering the degradation process equivalence, this corrosion model can be seen as a corrosion equivalent model. When the environmental stress levels are δp and δq, respectively, for a given reliability R, that is, the probability that the corrosion loss of the material does not exceed the threshold γ is R, one has:

and the reliable life of materials under δp and δq are tR,p and tR,q, respectively. The ratio of reliable life is:

where Kp,q is the corrosion equivalent factor between the pth environmental stress level and the qth environmental stress level. Generally, Kp,q is only required to be related to the environmental stress levels while it is independent of the product working time and reliability, so from Eqs. (10) and (11), it is easy to have:

under the condition of degradation process equivalence, the corrosion models of the form Eq. (8) should have a robust parameter b by Eq. (12). In other words, based on the robustness of corrosion kinetics, the power-linear corrosion models constructed in Eq. (6) should have a robust parameter b in the corrosion steady state phase. Thus, the steady-state start time can be considered by analyzing the variation of parameter b.

The method described in this paper can be used as a framework for long-term corrosion prediction modeling, including the calculation of steady-state start time, the modeling of environmental factors, and the modeling of the two-stage corrosion process. The following two cases in Section 3 take carbon steel as an example to illustrate the process of corrosion prediction modeling. In fact, it is not limited to carbon steel. If there are corrosion test data and environmental data of other materials, this method may be applied to the long-term corrosion prediction of corresponding materials.

3 Determination of corrosion steady-state start time

The steady-state start time has been widely discussed. Notably, the steady-state start time varies significantly due to: (a) the properties of metals; (b) the type of atmosphere; and (c) the corrosiveness. Therefore, there exhibits no explicit standard for the stabilization time of long-term corrosion. In other words, the demarcation point between the corrosion initial stage and the corrosion steady state remains quite uncertain. For instance, Díaz et al. (2012) found that the stabilization time of low alloy steel (weathering steel) is 6–8 years in a less corrosive environment (categories C2–C3 according to ISO 9223 (2012)) and 4–6 years in a more corrosive environment (categories C3–C5). On the other hand, the steady-state start time is 10 years according to ISO 9224 (2012). De la Fuente et al. (2011) observed that the corrosion rate of low alloy steel tends to be stable after exposure to all types of atmosphere for 4 to 6 years.

In the study of the method for determining the steady-state start time, Morcillo et al. (2013) proposed a general principle for the determination of corrosion steady-state start time, according to which the steady state is attained when both the annual decrease and annual variation of corrosion rate are less than 10%.

On this basis, Panchenko et al. (2014) introduced an empirical approach to determine the steady-state start time, where the study of corrosion kinetics parameters indicates a 6 years distance to reach steady. According to their model, the corrosion loss increases linearly after the steady-state start time, which can be indicated as:

where Kt is the mass loss in time t and K0 represents the mass loss obtained by extrapolation of linear relationship Eq. (13) to t=0. Accordingly, the corrosion rate δ during the linear phase is formulated as

it can be seen that when t→∞,δ→α. δ is not only the annual increment of mass loss at the steady state, but also the limit corrosion rate of metals. Hereby, the stationary corrosion rate α is the limit corrosion rate in long-term corrosion.

Assuming that the steady state begins at the ith year of the test. Then the change of corrosion rate for i∼i+1 and i+1∼i+2 year periods can be expressed from Eq. (14):

and the yearly variation in corrosion rate after the start of the stationary stage is

Assuming that the stabilization time of the layer of corrosion products is 4 years, 5 years, 6 years, 7 years and 8 years, respectively, then the results of Eq. (16) should be 1.50, 1.40, 1.33, 1.29, and 1.25, respectively, which can be used to confirm the steady-state start time.

Therefore, a novel method of determining corrosion steady-state start time needs to be proposed to verify the effect of the comprehensive power-linear model addressed in Section 2, this section analyzes the results of 8 years corrosion tests of the carbon steel within both the China program (Case 1) (Hou and Liang 2004; National Materials Corrosion and Protection Data) and the ISO CORRAG international program (Case 2) (Knotkova et al. 2012).

3.1 Case 1: long-term corrosion program of field exposure in China

There are significant differences in the corrosion process of materials in different climatic regions, which affects the corrosion steady-state start time of the materials. Since the 1950s, China has established test sites in several typical cities, and a network of experimental sites covering various climatic regions across the country. Beginning in 1984, a long-term program of exposing various materials to the atmosphere was implemented at typical sites of China over 16 years. Test sites covered typical atmosphere types, including temperate, subtropical, industrial, marine, and rural atmospheres, as well as humid and dry environments (Hou and Liang 2004). A long-term corrosion program with field exposure of carbon steel in China will provide data to support this study.

In order to obtain the specific steady-state start time, it is not enough to know the corrosion mass loss of 1 years, 2 years, 4 years and 8 years. For example, the calculation shows that the material has reached the corrosion steady-state in the range of 4–8 years, but the specific steady-state start time remains unknown. Therefore, the corrosion mass loss of 3 years, 5 years, 6 years and 7 years are supplemented by cubic spline interpolation in this paper, as shown in Table 1.

Table 1:

Testing sites and corrosion loss (μm) of carbon steel in China.

Corrosivity category	Testing sites	Years
		1	2	3	4	5	6	7	8
C3	Qionghai	31.0	58.0	89.5	126.0	168.3	217.0	272.6	336.0
C3	Beijing	34.0	55.0	74.0	91.0	106.3	120.0	132.1	143.0
C3	Wuhan	51.0	82.0	112.3	141.0	167.1	189.8	208.1	221.0
C4	Guangzhou	61.0	104.0	143.3	179.0	211.1	239.8	265.1	287.0
C4	Qingdao	96.0	161.0	220.8	276.0	327.3	375.3	420.6	464.0
C4	Jiangjin	94.0	176.0	246.8	308.0	361.0	407.3	448.5	486.0

This case shows the corrosivity category of 6 sites, which are classified according to the standard of ISO 9223 based on the first year corrosion exposure data of standard carbon steel (A3) at each site (Tian et al. 1995), as shown in Figure 1. The red dots represent C3 sites, that is Qionghai (QH), Beijing (BJ) and Wuhan (WH), and the green dots represent C4 sites, that is Guangzhou (GZ), Qingdao (QD) and Jiangjin (JJ). It can be seen that most C4 sites are located in the coastal area or along the river, the climate environment is rainy, the relative humidity is large, and it is affected by atmospheric pollutants (Cl⁻), while the C3 sites are mostly inland, and the Qionghai has been classified into C3 grade because of the low corrosion loss in the first year, but from the point of view of its long-term corrosion loss, it is in fact close to the long-term corrosion loss in C4 sites (Ding et al. 2020).

Figure 1:

Corrosivity category distribution of 6 sites on China map.

The steady-state start time for Case 1 is calculated in this section by two approaches proposed in reference Morcillo et al. (2013) and Panchenko et al. (2014), as shown in Figure 2 and Table 2. It can be seen that Morcillo’s criterion determined the steady-state start time of 2–3 years for C3 sites and 5–6 years for C4 sites, according to the principle whether the corrosion rate variation of the carbon steel reduces to 10%, as shown in Figure 2. The carbon steel exposed to C3 sites reached the corrosion steady state within 2–3 years, because the corrosion rates in 3 years, 5 years, 6 years and 7 years are obtained by interpolations, which are not real experimental data and will affect the calculation to a certain extent, but a more stable criterion is needed. As for the approach introduced by Panchenko et al. (2014), the steady-state start time for Case 1 is not available as shown in Table 2, assuming that the stabilization time of the layer of corrosion products is 4 years, 5 years, 6 years and 7 years, Eq. (16) should be equal to 1.50, 1.40, 1.33 and 1.29. Therefore, a stable corrosion steady state determination criterion based on corrosion kinetics is proposed in this section.

Figure 2:

Variation of corrosion rate of carbon steel in C3 and C4 sites.

Table 2:

Calculated by δi−δi+1δi+1−δi+2 in C3 and C4 sites.

i	1	2	3	4	5	6	7
C3	21.5833	1.0909	1.1000	1.1111	1.1250	1.1429	0.0115
C4	3.0391	1.1240	1.1416	1.1649	1.1975	1.2460	0.0876

From Eq. (6), the initial stage corrosion model can be established as the form of M=a0tbf(ϑ), while the extended function f(ϑ) can be determined by various environmental factors. In Case 1, the key environmental factors include relative humidity (RH), temperature (Tem), sulfur dioxide (SO₂), chloride deposition (Cl⁻), and precipitation (Prec), so f(ϑ) can be indicated as fRH,Tem,SO2,Cl,Prec.

3.1.1 The extended function of RH and Tem

It is evident that RH has a considerable effect on the corrosion process. Due to the existence of critical relative humidity (Lapuerta et al. 2008; Sherwood et al. 2016), corrosion would even not occur under a low humidity environment. At the same time, the effect of Tem on the corrosion process also needs consideration, because Tem and RH jointly determine the time of wetness (TOW) under the atmospheric environment, while TOW is a crucial triggering factor of the atmospheric corrosion process (Cai et al. 2018a,b; Klinesmith et al. 2007). Because of the complex relationship between these factors, some studies and standards use Tem and RH as parameters to replace the effect of TOW in the corrosion model (ISO 9223 2012; Mikhailov et al. 2004). In Case 2, the model with TOW as a parameter will be discussed.

Under an atmospheric environment, the effect of RH and Tem is commonly expressed as a power function in the dose-response functions (DRFs) (Guttman and Sereda 1968; ISO 9223 2012; Knotkova et al. 2012), which is a generally accepted form that can better reflect the effects of the environment. In the immersed marine environment, Soares et al. (2005) found that the effect of environmental factors on corrosion can be expressed as:

(17)f(η)=ρ⋅ηr+κ,

where ρ and κ are two coefficients. f(η) is a function of environmental factor η, ρ can be viewed as an adjustment factor, and ηr is the ratio between the actual value of immersed marine environmental factor η and the nominal value. For example, the oxygen content Ox is often the main influential factor in the immersed marine environment and Oxr can be indicated as the ratio of actual oxygen content and nominal oxygen content. However, when the actual value for immersed marine environmental factor η is equal to the nominal value, that is, ηr=1, the actual corrosion loss should also be equal to the nominal value. In general, the nominal value can be simply regarded as the environmental mean of multiple environmental sites.

In fact, the model with dose function can be applied to the case of less environmental data, but since the nominal value of the environment is closer to the real total, it will also perform better in the calculation of corrosion prediction and can effectively attenuate the effect of order-of-magnitude differences between different environmental factors, so it is a better choice to use the nominal value of the environment as the input of environmental variables in the case of multiple environmental site data. Therefore, after referring to the immersion corrosion model in the marine environment, a linear function with the nominal value is proposed to express it (Ji et al. 2021). Therefore, in Case 1, the extended function of RH and Tem yields the following two forms:

(18)f(Tem)=exp(c1⋅Tem),

(19)f′Tem=1−c2⋅Temr+c2,

(20)f(RH)=(d1⋅RH)e1,

(21)f′(RH)=(1−d2)⋅RHr+d2,

where c1, c2, d1, e1 and d2 are the coefficients; Temr and RHr can be viewed as the ratio between the actual time of wetness and nominal time of wetness.

3.1.2 The extended function of SO₂

The pollutants in the atmosphere are mainly caused by the combustion of industrial coal, diesel fuel, natural gas, and other sulfur-containing fuels. Among them, SO₂ is considered to be the most serious gaseous pollutant to the atmospheric corrosion process. Many studies have verified that SO₂ has a serious effect on the atmospheric corrosion process.

Different from TOW (RH and Tem), SO₂ is not an essential factor in the corrosion process, in other words, even if the effect of SO₂ is equal to zero, the corrosion mass loss is not equal to zero. However, it is undeniable that the presence of SO₂ has a certain acceleration on the corrosion process. Therefore, the extended function of SO₂ can be expressed in the following two forms:

(22)f(SO2)=(1+f1⋅S)g1,

(23)f′(SO2)=(1−e2)⋅Sr+e2,

where f1, g1 and e2 are the coefficients, Sr represents the ratio between actual sulfur dioxide deposition and nominal sulfur dioxide deposition.

3.1.3 The extended function of Cl⁻

Under the marine environment, salinity is considered to be an important factor affecting the corrosion process (Wang et al. 2021; Yang et al. 2022). In contrast, under the atmospheric environment, chloride deposition is deemed to come from the marine atmosphere, and researchers concluded a direct relationship between chloride deposition and the coastline (Feliu et al. 1999; Yang et al. 2020). Similarly, chloride is not an immediate influencing factor in the corrosion process, but its presence also accelerates the corrosion process. Thus, the extended function of Cl⁻ is expressed in the following two forms:

(24)f(Cl)=(1+h1⋅Cl)j1,

(25)f′(Cl)=((1−f2)⋅Clr+f2),

where h1, j1 and f2 are the coefficients, Clr is the ratio between actual chloride deposition and nominal chloride deposition.

3.1.4 The extended function of Prec

Excessive Prec leads to a decrease in temperature and an increase in relative humidity. The effect of Prec in the corrosion process is receiving increasing attention and becoming an important environmental factor in predictions of atmospheric corrosion (Ji et al. 2021; Panchenko et al. 2019; Pei et al. 2020; Zhang and Yang 2020). The extended function of Prec is structured in the following two forms:

(26)f(Prec)=(1+k1⋅Prec)l1,

(27)f′(Prec)=(1−g2)⋅Precr+g2,

where k1, l1 and g2 are the coefficients, Precr can be seen as the ratio between actual precipitation and nominal precipitation.

Summing up the foregoing influencing factors, the corrosion model at the initialization stage under Case 1 is formulated via the following two forms.

Model I:

(28)M1=a1⋅tb1⋅exp(c1⋅Tem)⋅(d1⋅RH)e1⋅(1+f1⋅S)g1⋅(1+h1⋅Cl)j1⋅(1+k1⋅Prec)l1,

Model II:

(29)M2=a2⋅tb2⋅((1−c2)⋅Temr+c2)⋅((1−d2)⋅RHr+d2)⋅((1−e2)⋅Sr+e2)⋅((1−f2)⋅Clr+f2)⋅((1−g2)⋅Precr+g2).

Notably, considering the error term of the experimental data, the robustness of the parameter b is reflected by its variance. Thus, Model I and II integrating the variance can be rewritten as:

(30)ξi=lnMi+εi,i=1,2,

where the error term εi∼N(0,σi2), and the log-likelihood function for ξi is given by:

(31)Li=∑υ=1n1/2πσi×exp[(−1/2σi2)×(ξiυ−lnMiυ)2],i=1,2,

where υ is the environmental stress level, n is the number of samples. By maximizing the log-likelihood function Li, the maximum likelihood estimates of the model parameters can be obtained, there are θˆ1=(aˆ1,bˆ1,cˆ1,dˆ1,eˆ1,fˆ1,gˆ1,hˆ1,jˆ1,kˆ1,lˆ1,σˆ12), θˆ2=(aˆ2,bˆ2,cˆ2,dˆ2,eˆ2,fˆ2,gˆ2,σˆ22), respectively. Combining the corrosion data of the carbon steel and environmental data collected from the 6 sites, the model parameters can be estimated through maximum likelihood estimation (MLE) (Ji et al. 2021). For example, the Fisher information matrix for Model I can be formulated as:

(32)U1(θ1)=−(E(∂2lnL1∂a12) ⋯E(∂2lnL1∂a∂(σ12))⋮⋱⋮E(∂2lnL1∂(σ12)∂a1)⋯E(∂2lnL1∂(σ12)2))12×12,

and the variance-covariance matrix for Model I V1(θ1) can be derived by:

(33)V1(θ1)=U1−1(θ1),

and the variance of the parameter b can be calculated by Eq. (33). Since test samples are field exposed for just 1 year, the corrosion steady state is unlikely to be achieved. Therefore, the variance of parameter b for two models within 2–8 years can be calculated.

As can be seen from Figure 3, for the carbon steel in Case 1, the variance of parameter b shows a gradually stable trend with time, which corresponds to the corrosion process from the initial corrosion stage to the steady-state corrosion. Given the variation threshold of variance for parameter b, it corresponds to the corrosion process reaching steady-state corrosion. Therefore, it is possible to give a threshold of 0.005 for the variation of variance for parameter b when the corrosion of the carbon steel reaches the steady-state corrosion, which is consistent with the results calculated by Morcillo’s criterion in C4 sites, as shown in Figures 3 and 4, and the steady-state start time is about 6–7 years for C3 sites and 5–6 years for C4 sites.

Figure 3:

Comparison of variance of parameter b of carbon steel in C3 and C4 sites by model I.

Figure 4:

Comparison of variance of parameter b of carbon steel in C3 and C4 sites by model II.

3.2 Case 2: ISO CORRAG

The ISO CORRAG program is a field exposure corrosion test carried out on 53 sites located in 14 countries, including the short-term tests (1 year) and the long-term tests (8 years). Since this paper focuses on long-term corrosion, the long-term test results of 21 sites for the carbon steel are selected.

Similar to Case 1, to obtain the specific steady-state start time, the corrosion mass loss of 3 years, 5 years, 6 years and 7 years is supplemented by cubic spline interpolation in this paper, as shown in Table 3. This case shows the corrosivity categories of the 21 sites according to the standard of ISO 9223 (2012). However, since the amount of data in C5 is limited, which only has one site (Auby), this case only analyzes C2–C4 sites. The corrosivity categories distribution of 20 sites on the world map are shown in Figure 5, where the blue dot represents C2, the red dot represents C3, and the green dot represents C4. In Case 2, the environmental factors include TOW, SO₂, and Cl⁻, so f(ϑ) can be indicated as f(TOW,SO2,Cl−).

Table 3:

Testing sites and corrosion loss (μm) of steel with flat specimens in ISO CORRAG.

Corrosivity category	Code	Testing sites	Years
			1	2	3	4	5	6	7	8
2	CND 1	Boucherville	24.0	32.7	33.8	31.1	28.2	28.8	36.5	55.2
2	SF 3	Ahtari	12.8	24.8	31.6	34.8	36.1	37.1	39.4	44.8
2	N 3	Birkenes	19.7	37.8	48.1	52.8	54.2	54.4	55.8	60.6
2	N 6	Svanwik	20.2	33.8	42.2	46.8	49.0	50.1	51.5	54.6
3	CS 1	Kaiperske Hory	26.0	35.5	44.1	51.5	57.6	62.1	64.8	65.6
3	CS 2	Praha-Bechovice	47.4	76.2	87.7	88.9	86.8	88.5	101.0	131.2
3	SF 1	Helsinki	33.3	60.0	75.0	82.0	84.6	86.6	91.6	103.2
3	SF 2	Otaniemi	25.6	43.8	54.0	58.8	60.6	62.0	65.5	73.6
3	F1	Saint Denis	37.2	59.6	72.6	79.3	82.9	86.5	93.3	106.4
3	JAP 1	Choshi	43.3	68.2	96.0	125.0	153.3	179.1	200.6	216.0
3	JAP 2	Tokyo	39.5	59.8	76.9	91.2	103.0	112.5	120.2	126.4
3	N 1	Oslo	25.2	41.2	48.1	49.2	47.5	46.2	48.6	57.6
3	N 5	Bergen	27.9	39.8	49.3	56.4	61.3	63.9	64.5	63.0
3	E 3	Lagoas	26.9	48.6	61.5	68.4	72.3	76.2	82.8	95.2
3	E 4	Baracaldo	43.9	66.8	76.2	77.2	75.2	75.4	82.9	103.2
3	S2	Kattesand	35.2	54.0	68.7	80.3	89.8	98.1	106.2	115.2
4	CS 3	Kopisty	70.7	108.0	117.4	111.0	100.7	98.5	116.4	166.4
4	N 2	Borregaard	61.7	103.0	124.9	134.0	136.6	139.2	148.4	170.4
4	N 4	Tannanger	59.6	84.4	104.8	121.0	133.2	141.7	146.6	148.2
4	S3	Kvarnvik	61.6	79.4	95.6	110.0	122.6	133.2	141.7	148.0
5	F8	Auby	106.3	150.0	190.6	227.2	259.1	285.4	305.3	318.0

Figure 5:

The corrosivity category distribution of 20 sites on the world map.

As shown in Tables 4 and 5, both traditional approaches mentioned in Case 1 cannot provide an exact corrosion steady-state start time for Case 2. However, according to the approach proposed in this paper, that is, from the perspective of the variation of variance for parameter b, it is still possible to evaluate the steady-state start time of carbon steel for Case 2 at C2–C4 sites.

Table 4:

Variation of corrosion rate in C2–C4 sites.

	1–2 years (%)	2–3 years (%)	3–4 years (%)	4–5 years (%)	5–6 years (%)	6–7 years (%)	7–8 years (%)
C2	31.68	49.20	42.69	49.29	61.87	339.10	146.29
C3	41.15	35.32	37.34	20.30	17.14	31.01	61.99
C4	52.21	43.94	32.12	18.20	36.65	69.25	97.65

Table 5:

Calculated by δi−δi+1δi+1−δi+2 in C2–C4 sites.

i	1	2	3	4	5	6	7
C2	0.9425	2.2692	1.5108	1.5714	0.4784	0.5279	0.5940
C3	1.9798	1.4623	2.9367	1.4860	0.6667	0.3819	0.3827
C4	2.4864	2.4395	2.6009	0.6069	0.8356	0.4190	0.4941

3.2.1 The extended function of TOW

As mentioned above, TOW is an important triggering factor in the atmospheric corrosion process (Cai et al. 2018a,b; Knotkova et al. 2012). Some studies and standards use RH and Tem as parameters to replace the effect of TOW in the corrosion model (ISO 9223 2012; Mikhailov et al. 2004), and this approach is used in Case 1.

Similar to Case 1, the extended function of TOW in Case 2 yields the following two forms:

(34)f(TOW)=(c3⋅TOW)d3,

(35)f′(TOW)=(1−c4)⋅TOWr+c4,

where c3, d3 and c4 are the coefficients, TOWr denotes the ratio between the actual time of wetness and nominal time of wetness.

3.2.2 The extended function of SO₂

Similar to Case 1, the extended function of SO₂ yields the following two forms

(36)f(SO2)=(1+e3⋅S)f3,

(37)f′(SO2)=(1−d4)⋅Sr+d4,

where e3, f3 and d4 are the coefficients, Sr represents the ratio between actual sulfur dioxide deposition and nominal sulfur dioxide deposition.

3.2.3 The extended function of Cl⁻

Similar to Case 1, the extended function of Cl⁻ is structured as

(38)f(Cl)=(1+g3⋅Cl)h3,

(39)f′(Cl)=((1−e4)⋅Clr+e4),

where g3, h3 and e4 are the coefficients, Clr is the ratio between actual chloride deposition and nominal chloride deposition.

To sum up, the initial stage corrosion under Case 2 is characterized by the following two models:

Model III:

(40)M3=a3⋅tb3⋅(c3⋅TOW)d3⋅(1+e3⋅SO2)f3⋅(1+g3⋅Cl)h3,

Model IV:

(41)M4=a4⋅tb4⋅[((1−c4)⋅TOWr+c4)⋅((1−d4)⋅Sr+d4)⋅((1−e4)⋅Clr+e4)].

Similar to Case 1, the variance for the parameter b in 2–8 years can be calculated. As shown in Figure 6, the steady-state start time is about 8 years for C2 sites, 5–6 years for C3 sites and 4–5 years for C4 sites, and in fact, this result is in agreement with the results of the tests (De la Fuente et al. 2011; Díaz et al. 2012). The model comparisons and key information about each model are shown in Table 6.

Figure 6:

Comparison of variance of parameter b of carbon steel in C2–C4 sites by model III & model IV.

Table 6:

Model comparison of different cases.

Model type	Expression	Modeling source	Environment factor function type	Environment factor parameter type	Application of the model
Model 1	Eq. (28)	Case 1	Power function & exponential function	A c t u a l   v a l u e	a
Model 2	Eq. (29)		Linear function	A c t u a l   v a l u e   N o m i n a l   v a l u e	b
Model 3	Eq. (38)	Case 2	Power function & exponential function	A c t u a l   v a l u e	a
Model 4	Eq. (39)		Linear function	A c t u a l   v a l u e   N o m i n a l   v a l u e	b

1. a, the environment parameter input of the model is the actual value, which can be used in the case of less environment factors; b, the environment parameter input of the model is the ratio of the real value to the nominal value, which can effectively attenuate the effect of order-of-magnitude differences between different environmental factors.

For the variation of the variance of parameter b, it is found that 0.005 is an appropriate value through a large number of calculations. The change of less than 0.005 has little reference significance, and it can be considered that this parameter has reached a relatively stable state, and the result of corrosion steady-state start time given in Case 1 by this threshold is consistent with that of Morcillo’s criterion. At the same time, in Case 2, when Morcillo’s criterion and Panchenko’s method are not applicable, the proposed method can give the exact steady-state start time of corrosion, and the results of corrosion steady-state start time given by this threshold are also consistent with the results of the tests (De la Fuente et al. 2011; Díaz et al. 2012). Thus, it can be proved that the method for determining the steady-state start time proposed in this paper has better universality and accuracy.

4 Long-term prediction based on the comprehensive corrosion model

4.2 Corrosion loss prediction based on corrosivity category

Following ISO 9224 (2012), the general power-linear model (ISO model) is used for long-term corrosion prediction, that is:

(54){D(t≤20)=rcorrtn1 D(t>20)=rcorr[20n1+b(20n1−1)(t−20)],

where D is the corrosion loss, rcorr is the corrosion rate at the first year, which can be seen as the corrosion loss in the first year, n1 is the exponent and equals to 0.523 for carbon steel, and the linear function stage is t>20 in this model. This steady-state start time ensures that the corrosion steady state has been achieved for metal materials under general conditions, but this is perhaps due to the inability to find an accurate corrosion steady-state start time. Therefore, the ISO model and the proposed model will be used for corrosion prediction of the carbon steel in each corrosivity category, and the power function model (P model) Eq. (1) will be introduced at the same time, where the calculation method of n is considered to adopt the calculation method by Cai et al. (2018a,b), that is:

(55)n=ln(M8/M1)ln8.

We compare corrosion predictions of Case I and Case II for up to 40 years between several corrosion models and real data of the carbon steel, as illustrated in Figures 7 and 8.

Figure 7:

Prediction of corrosion of carbon steel in C3 and C4 sites in China.

Figure 8:

Prediction of corrosion loss of carbon steel in C2, C3 and C4 sites in ISO CORRAG.

As can be seen from Figure 7, for C3 and C4 sites in China, the ISO model basically deviates from the real data of the carbon steel, while the proposed model and P model both obtain good prediction results. The deviation in the later years of corrosion prediction is due to the continuous decrease of the corrosion rate of the power function, while the proposed model maintains the corrosion steady state.

From Figure 8, because the corrosion steady-state start time of the ISO model is 20 years, it can be seen from the prediction results of the C4 sites that the ISO model has been quite different from the real data since the 5 years, and because of the existence of the corrosion steady state, the corrosion rate of the P model is still at a state of continuous decrease in the long-term prediction of corrosion loss, which seems to be unreasonable, and it also lacks the quantification of environmental impacts. Based on quantifying the influence of environmental factors, the prediction results of the proposed model for each corrosivity category are relatively robust.

Based on the observations of field exposure experiments, most researchers have found that long-term corrosion will eventually reach a steady state, and this is exactly what we want to reflect in corrosion prediction. Therefore, the corrosion loss kinetics is described by a power-linear function in this paper. To compare with previous studies, this paper considers the corrosion prediction for up to 40 years, although it may not be comparable with the real data, the purpose is actually to reflect the rationality of the two-stage corrosion prediction model and the importance of determining the exact corrosion steady-state start time.

In fact, the model proposed in this paper has a good prediction effect at each site of C2, C3, and C4 on the prediction of the first 16 years of real data. Since the steady-state start time of carbon steel at C2, C3 and C4 sites are less than 8 years calculated in this manuscript, according to the corrosion loss kinetics, it can be considered that the corrosion mass loss after 8 years should increase linearly. Thus, it can be considered that the linear extrapolation result of the model is still valid, that is, the prediction of 40 years is still valid.

4.3 Corrosion loss prediction of different sites based on corrosion state

In order to prove the effectiveness of the proposed model, this section employs the models in Sections 4.1 and 4.2 to predict the corrosion loss of the carbon steel in different sites of Case 1 for 16 years according to different corrosivity categories.

Substituting the environmental factors, ts, Mt1, and Mts into Eqs. (44)–(47), respectively, the results of corrosion loss prediction of the carbon steel for 16 years in each site are illustrated in Figure 9, and the relative error is illustrated in Figure 10. Based on the calculation of Section 3, it can be determined that the steady-state start time of the C3 site is 6–7 years and the steady-state start time of the C4 site is 5–6 years in Case 1. In order to ensure that the corrosion steady state is reached, the ts of C3 site is 7 years and the ts of the C4 site is 6 years in the calculation. The original data is the corrosion mass loss of the carbon steel for 16 years in Case 1 which can be found in NMCPDC (National Materials Corrosion and Protection Data Center), and the experimental data points are 1 year, 2 years, 4 years, 8 years, 16 years, respectively. Therefore, the corrosion mass loss of 3 years, 5 years, 6 years, 7 years, 9 years, 10 years, 11 years, 12 years, 13 years, 14 years and 15 years was supplemented by cubic spline interpolation in this paper.

Figure 9:

Prediction of corrosion loss of carbon steel in 6 sites in China.

Figure 10:

Relative error of prediction of carbon steel in 6 sites.

Since the prediction results of the two models are almost identical, the prediction results of both models are combined and shown in Figure 9, it can be seen that the proposed model has achieved good results for each site. Perhaps after 10 years, the corrosion prediction of JJ sites is slightly offset compared with other sites, but the maximum error is still less than 30%. In fact, the test data of the JJ site after 10 years are closer to those of the QH site 10 years later, but the JJ site belongs to the C4 corrosivity grade, while QH belongs to the C3 corrosivity grade, and theoretically, the C4 site should have higher corrosion loss. Although the proposed model overestimates the corrosion loss of the JJ site after 10 years, it also shows that the model is reasonable for the corrosion prediction of the same corrosivity grade.

As can be seen from Figure 10, without considering the corrosion loss in the first year and the corrosion loss in the exactly steady-state start time used in the calculation of model parameters, the average relative error in the prediction results for the remaining years is only 10% for C3 sites (QH, BJ, WH) and 10.3% for C4 sites (GZ, QD, JJ). Therefore, it can be considered that the proposed model obtains sufficiently accurate output information with less input information.

5 Conclusions

In this paper, a comprehensive power-linear model considering the environmental effect is proposed. Based on the robustness of corrosion kinetics, a novel method of determining corrosion steady-state start time is proposed. Finally, corrosion loss prediction of the carbon steel based on corrosivity category and corrosion state is obtained with the confirmed model. The conclusions are as follows:

1. The extended function of the environmental effect in the proposed model is not limited to fixed environmental factors but can be taken by the experimental atmospheric environment.

2. Compared with the traditional model, the proposed model can take into account the effect of environmental factors in an integrated manner, which is more suitable for evaluating the performance of metals under different corrosive environmental conditions.

3. Two forms of environmental extended functions are proposed. The model with a dose function is suitable for cases with little environmental data, and the model considering nominal values of environmental variables as inputs is recommended when environmental data are sufficient.

4. Based on the robustness of corrosion kinetics, the robustness of parameter b in the proposed model can be used as a criterion for evaluating corrosion of metal materials to reach a steady state, and it is possible to give a threshold of 0.005 for the variation of variance for parameter b when the corrosion reaches the steady-state corrosion.

5. A novel method of determining corrosion steady-state start time is proposed, while case calculations reveal that the corrosion steady-state start time is different for metals materials in different corrosivity categories environments (C2, C3, C4), and this method can be more accurate in obtaining the steady-state start time.

6. The results show that the proposed model quantifies the influence of environmental factors, and its prediction results are more robust and reasonable compared with the traditional power model and ISO model in 40 years corrosion loss prediction.

7. By comparing with the test data of the carbon steel, the maximum relative error is controlled within 30%, and the average relative error is only 10% in 16 years corrosion loss prediction for each corrosivity category, so it can be considered that the proposed model obtains enough accurate output information with less input information.