An Improved Fick Model for Predicting Carbonation Depth of Concrete

Abstract

Concrete carbonation can weaken its strength, cause the corrosion of steel reinforcement, and shorten its service life. Predicting the concrete carbonation depth is a critical aspect of assessing concrete durability. Currently, mathematical models for the concrete carbonation depth, exemplified by the Fick model, suffer from a low fitting accuracy and limited applicability due to the complexity and variability of concrete materials and service environments. In light of this, this work proposes an improved Fick model that incorporates a correction term to effectively enhance the curve fitting accuracy. The correction term in the improved model provides a reasonable adjustment for deviations in the development pattern of the concrete carbonation depth from the Fick model under different conditions, thereby broadening the applicability of the new model compared to the Fick model. Several sets of experimental data on the concrete carbonation depth are used to validate the universality and superiority of the new model. The results of the case studies indicate that the average prediction error and standard deviation of the new model are significantly smaller than those of the Fick model. For the first two examples, in most situations, the average prediction error and standard deviation of the new model are less than 50% of those of the Fick model, with the lowest average prediction error being only 4% and the lowest standard deviation being only 2% of the Fick model’s respective values. For the third example, the new model demonstrates superior predictive capability for the later-stage concrete carbonation depth compared to the Fick model and the ANN model. Specifically, for the carbonation depth of the concrete on the 56th day, the relative error between the predicted value of the new model and the measured value is only 2%, which is much smaller than the 27% of the Fick model and the 12% of the ANN model. These results demonstrate the unique advantage of the proposed model in predicting the carbonation depth, especially when only a limited amount of experimental data are available.

1. Introduction

With the increase in the service life of concrete structures, durability problems such as cracks, concrete carbonation, and chloride ion corrosion will appear in the structures [1,2,3]. In addition to buildings, bridges and other critical infrastructure are equally vulnerable to concrete cover, spalling, damage, and carbonation. References [4,5] highlight the importance of addressing existing concrete performance in bridges and other critical infrastructure. Carbonation and chloride ion corrosion are two unfavorable chemical reactions that frequently occur in concrete structures and constitute the two most important aspects of durability assessment. Generally, chloride ion corrosion is more prevalent in coastal and saline–alkali soil regions, whereas carbonation affects concrete structures in nearly all regions. This work focuses on addressing the prediction problem of the concrete carbonation depth. The mechanism of concrete carbonation is that CO₂ gas in the air permeates through the pores of hardened concrete and reacts with the alkaline substance (Ca(OH)₂) to produce carbonate (CaCO₃) and water. The hazard of concrete carbonation lies in its reduction in the concrete alkalinity, which results in the loss of protection for steel reinforcement, leading to rust expansion and subsequently causing concrete cracking and spalling. This significantly impacts the structural mechanical properties and durability. By accurately assessing the depth of carbonation, potential carbonation issues can be promptly identified and addressed with corresponding repair and protective measures to enhance the concrete’s resistance to carbonation. Furthermore, based on the time-varying pattern of the concrete carbonation depth, the remaining lifespan of concrete structures can be predicted. Therefore, accurately assessing the depth of concrete carbonation is of great significance in ensuring the safety of engineering structures and extending their service life. Feng [6] roughly divided the factors that affect concrete carbonation into the following three categories: construction, environment, and materials, which complicate the development law of the concrete carbonation depth. The time-varying law of the concrete carbonation depth is of great significance for the safety assessment of structures and the adoption of targeted preventive measures. In view of this, in recent years, scholars at home and abroad have conducted many studies on the issue of concrete carbonation and have proposed many models for predicting the depth of concrete carbonation, which can be roughly divided into two categories. The first type is an empirical model based on experimental data or actual structural measurements of the carbonation depth, fitted using methods such as neural networks or mathematical statistics. Felix et al. [7] studied the prediction of the carbonation depth of fly ash concrete using an artificial neural network (ANN) with a backpropagation algorithm. The results indicate that, after extensive training, the model can estimate the carbonation depth at any time during the service life of concrete and predict the onset time of the complete carbonation of the concrete protective layer. Paul et al. [8] developed an empirical model specifically designed to investigate the effects of mix proportions, environmental weathering conditions, and exposure time on the carbonation depth of concrete using innovative Automatic Neural Network Search (ANS) technology. The model was subsequently validated in a series of carefully designed experiments. Wei et al. [9] conducted a study on predicting the carbonation depth of concrete containing mineral admixtures based on machine learning. They considered the influence of 17 parameters and established a relevant experimental database. They compared backpropagation neural networks with support vector machines, and the results showed that the former had a higher prediction accuracy than the latter, and both had a higher prediction accuracy than existing mathematical curve prediction models. Niu et al. [10] conducted on-site testing and collected experimental data on reinforced concrete exposed to natural environments, and used numerical methods to fit and obtain a prediction model for the length of concrete carbonation zones. Silva et al. [11] used multiple linear regression and combined a large amount of experimental data to consider factors such as the concrete strength and aggregate water absorption, and proposed a universal model for predicting the carbonation coefficient. Qu et al. [12] studied the carbonation depth of recycled concrete under a chloride salt erosion environment, analyzed the influence of internal chloride ions, improved the existing prediction model, and enhanced its accuracy. Based on 433 sets of data, Qin et al. [13] first established a deterministic model for the carbonation depth, and then introduced uncertainty to build a probabilistic model. Bayesian and MCMC models were used to update the parameters and study the influence of the prior distribution on the posterior to improve the prediction accuracy. Based on a large amount of data, Liu et al. [14] established a carbonation coefficient model to analyze the carbonation of reinforced concrete, considering various factors such as the environment, materials, and curing. They used multiple nonlinear regression methods to fit the environmental exposure situation, verified the good applicability of the formula, and quantitatively analyzed the impact of each factor on the carbonation. Most of the first type of models are based on artificial neural networks for the analysis and prediction. Their advantage is that they can consider the influence of multiple factors on the carbonation depth of concrete by configuring an appropriate number of neurons in the input layer. However, the fitting capability of the ANN model primarily depends on the number of hidden layers and the number of neurons in each layer. Currently, the determination of these parameters is based mainly on experience, and there is a lack of formulae that have undergone a rigorous mathematical proof in this regard. Furthermore, the prediction capability of the ANN model is primarily influenced by whether there are sufficient training samples. Generally, for prediction problems within the vicinity of the training sample regions, the ANN model can provide highly accurate prediction results. However, for predictions far away from the training sample regions, the accuracy of the ANN model is often less than ideal, indicating that the generalization ability of the ANN model is still lacking. For instance, in the case of predicting the later-stage carbonation depth of concrete (which deviates significantly from the testing time points), the reliability of the prediction results obtained by the ANN model needs further verification.

The second type is a theoretical model established based on the quantitative analysis of the carbonation reaction process. Among the several models that represent the time-varying relationship between the concrete carbonation depth, Fick’s first law is the most commonly used model. Fick’s first law [15,16] is one of the basic laws describing diffusion phenomena. The fundamental principle of this law states that the flux of diffusing material passing through a unit cross-sectional area perpendicular to the direction of diffusion per unit time is directly proportional to the concentration gradient at that cross-section. In simpler terms, the larger the concentration gradient, the greater the diffusion flux. Since the rate of concrete carbonation is primarily influenced by the concentration of CO₂ in the air and the diffusion rate of CO₂ within the concrete, the time-varying model of the carbonation depth based on Fick’s first law is considered to have a physical basis. This makes the Fick model the most commonly used in predicting the carbonation depth. Ta et al. [17] proposed a new meta-model for calculating the carbonation depth based on Fick’s first diffusion law analysis. This model only needs to input known data such as material variables and process parameters to predict the carbonation depth of concrete structures. Zhang et al. [18] improved the existing theoretical model for predicting the carbonation depth of recycled concrete by incorporating the water absorption of aggregates into the consideration factors, and verified it through carbonation experiments. The results showed that the improved model had a higher prediction accuracy. He et al. [19] established a calculation model for the oxygen diffusion coefficient in the carbonation zone based on Fick’s law. Jiang et al. [20] proposed an effective carbon dioxide diffusion coefficient for fatigue-damaged concrete based on residual strain, and established a theoretical model for the carbonation of fatigue-damaged concrete on this basis, which was verified through experiments. Li et al. [21] studied the carbonation effect of superabsorbent polymer (SAP) concrete, analyzed the key factors, and established an NS/SAP concrete carbonation coefficient model. This model reflects the influence of parameters such as the SAP content and the water-–cement ratio on the carbonation through regression analysis, and predicts the carbonation depth more accurately than existing models. He et al. [22] quantitatively analyzed the coupling effect between chloride erosion and carbonation. Based on the DLA principle, an integrated model was established considering multiple factors, and a DLA-based prediction method for concrete and steel reinforcement performance considering fatigue effects was proposed for the bridge durability analysis. Based on Fick’s law, Zhang et al. [23] introduced the carbonation rate coefficient determined by the permeability and the influence coefficient of actual environmental conditions, further optimized it, and proposed a prediction model for the carbonation depth of slag high-performance concrete. Chen et al. [24] studied the effects of the F-T cycle and continuous load on concrete carbonation, established and improved a carbonation theoretical model considering these factors, and improved the prediction accuracy. Possan et al. [25] proposed a mathematical model for estimating concrete carbonation depth and predicting the structures’ service life. The model consists of the following three sets of input variables: the concrete performance, exposure, and environmental conditions, and is constructed based on the conductivity equation (such as the first Fick’s law). Peng et al. [26] established a two-dimensional model of concrete carbonation based on CO₂ mass conservation, calculated the CO₂ concentration using the finite element method, considering the effects of time and temperature, and implemented it through Matlab programming. Wang et al. [27] gave a detailed overview of the prediction model of the carbonation depth for concrete structures.

Although many achievements have been made in the prediction of the concrete carbonation depth, it is still necessary to study more accurate mathematical models of the variation in the concrete carbonation depth over time. In view of this, the innovation of this work mainly lies in proposing an improved Fick model to more accurately predict the depth of concrete carbonation. Using experimental data from the existing literature, it was found that the new carbonation depth-age mathematical model has a better prediction accuracy than the existing Fick model. The framework of this study is organized as follows: Section 2 first briefly reviews the existing Fick model, and based on this, proposes an improved Fick model for the carbonation depth evaluation. Section 3 validates the proposed new model by using more than ten sets of data obtained from concrete carbonation tests in the existing literature. Finally, Section 4 summarizes the conclusions of this work.

2. Fick Model and the Improved Model

Currently, various mathematical models for the concrete carbonation depth have been proposed, among which the Fick model is the most representative one. This model assumes that the carbonation depth of concrete is proportional to the square root of time [28], namely:

$$h=x_0\sqrt{t}$$

(1)

where $h$ represents the carbonation depth of concrete, $t$ represents time, and $x_0$ can be termed as the carbonation coefficient, which is a comprehensive parameter reflecting the rate of concrete carbonation. There are two main methods to determine the value of “$x_0$”: the theoretical formula method based on gas diffusion theory [17,18,19,20,26,29], and the data regression analysis (also known as curve fitting) method based on the results of carbonation testing experiments [21,22,23,24,25,30]. The method used to determine the value of the model parameter in this work is the regression analysis, with the specific process outlined in Equations (3)–(7). The advantages of the Fick model are that its physical meaning is very clear, and only more than two sets of experimental data are required to perform curve fitting to obtain a specific model equation. The disadvantages of the Fick model are that it only contains one unknown fitting parameter, and the equation form is too simple to adapt to the development law of the concrete carbonation depth under different conditions, resulting in a low fitting accuracy in many cases.

To overcome the shortcomings of the Fick model, this paper proposes a new carbonation depth curve model, as follows:

$$h=x_0\sqrt{t}+x_1{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(2)

where $x_1$ represents the newly introduced unknown fitting parameter, which, along with $x_0$, needs to be obtained through curve fitting using experimental data from carbonation depth tests. This new curve model is equivalent to adding a correction term to the Fick model. This correction term has the following mathematical characteristics: when $t=0$, the correction term $x_1{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$ also equals 0; when $t\to \infty$, the correction term will equal $x_1$, that is, $x_1{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}\in [0,x_1]$. Therefore, this correction term can be regarded as the deviation from the Fick model’s law in the development pattern of the concrete carbonation depth due to variations in different conditions, such as the concrete mix ratio, admixtures, strength, temperature, humidity, etc. The physical significance of this correction term is a reasonable adjustment to the deviation from the Fick model’s law in the development pattern of the concrete carbonation depth, reflecting the influence of different conditions on the development pattern. The new model with the added correction term will have a higher fitting accuracy and a wider range of application compared to the Fick model, and can be applied to predict the development and variation in the concrete carbonation depth under various conditions.

Next, the calculation method for the fitting parameters of the new model will be elaborated. Assuming that we have measured the experimental data of the concrete carbonation depth at $m$ different time points (where $m\ge 2$), denoted as $h_1,\cdots ,h_m$, by substituting these experimental data into Equation (2) and solving them simultaneously, we can obtain the following system of linear equations:

$$y=\mathsf{\Pi }\cdot x$$

(3)

where $y$ represents the column vector composed of the carbonation depths at times $t_1,\cdots ,t_m$, $\mathsf{\Pi }$ denotes the coefficient matrix of the system of linear equations, and $x$ represents the column vector of fitting parameters consisting of $x_0$ and $x_1$, specifically

$$y={(h_1,h_2,\cdots ,h_m)}^T$$

(4)

$$x={(x_0,x_1)}^T$$

(5)

$$\mathsf{\Pi }=\left[\begin{array}{cc}\sqrt{t_1}& {\displaystyle \frac{e^{t_1}-e^{-t_1}}{e^{t_1}+e^{-t_1}}}\\ \sqrt{t_2}& {\displaystyle \frac{e^{t_2}-e^{-t_2}}{e^{t_2}+e^{-t_2}}}\\ ⋮& ⋮\\ \sqrt{t_m}& {\displaystyle \frac{e^{t_m}-e^{-t_m}}{e^{t_m}+e^{-t_m}}}\end{array}\right]$$

(6)

From Equation (3), we can obtain the estimated values of each fitting parameter as follows:

$$\widehat{x}={({\mathsf{\Pi }}^T\mathsf{\Pi })}^{-1}{\mathsf{\Pi }}^T\cdot y$$

(7)

where the superscript “$T$” denotes the transpose of a matrix. Based on the specific numerical values of the fitting parameters obtained from Equation (7), substituting them into Equation (2) will yield a specific curve model that can be used to predict the concrete carbonation depth at any given time. To evaluate the fitting accuracy of the model, the mean and standard deviation of the predicted relative errors are used. To this end, the relative error of the model prediction is defined as follows:

$$e_{t_m}=\left|\widehat{h_m}-h_m\right|/h_m$$

(8)

where $e_{t_m}$ represents the relative error between the model’s predicted value and the experimental test value at time $t_m$, and $\widehat{h_m}$ represents the model’s predicted value of the carbonation depth at time $t_m$. Furthermore, the mean $\overline{e}$ and standard deviation $\sigma$ of the relative errors are calculated as follows:

$$\overline{e}={\displaystyle \frac{e_{t_1}+\cdots +e_{t_m}}{m}}$$

(9)

$$\sigma =\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^m{(e_{t_i}-\overline{e})}^2}}{m-1}}}$$

(10)

Obviously, the smaller the mean value $\overline{e}$ and standard deviation $\sigma$ of the relative errors, the higher the fitting accuracy of the curve model, and the more accurate the predicted concrete carbonation depth will be.

3. Case Study

Taking the test data of the concrete carbonation depth in references [31,32] as examples, the proposed new prediction model for the carbonation depth was verified, and the calculation results were compared with those of the Fick model to illustrate the advantages of the model in this paper.

In reference [31], carbonation tests were conducted on ordinary concrete with a strength grade of C25 at temperatures of 20 °C and 30 °C. The mix ratio of the C25 concrete (cement/water/sand/river gravel) was 1:0.61:1.85:2.64. After 28 days of curing, the concrete specimens were placed in an artificial carbonation test chamber with a CO₂ concentration of (20 ± 3)% and a humidity of 70% for carbonation. The carbonation depth data of the concrete specimens were measured on the 3rd, 7th, 14th, 28th, 56th, and 84th days after the test, respectively. The fitting curves of the experimental data using the Fick model and the new model proposed in this paper are shown in Figure 1. In the figure, the symbol “o” represents the experimental test values, the symbol “◁” represents the predicted values of the Fick model, and the symbol “*” represents the predicted values of the new model. The fitted equations are presented in

Table 1. Fitted curve equations for carbonation tests at 20 °C and 30 °C.

Carbonation Conditions	Fick Model	New Model
20 °C Carbonation	$h=3.0868\sqrt{t}$	$h=3.0270\sqrt{t}+0.3817{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
30 °C Carbonation	$h=4.6126\sqrt{t}$	$h=4.2462\sqrt{t}+2.3403{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$

, and the average values and standard deviations of the relative errors of the fitting results are shown in

Table 2. Average relative error and standard deviation of the fitted carbonation test results at 20 °C and 30 °C.

Carbonation Conditions	Statistical Indicators	Fick Model	New Model
20 °C Carbonation	Mean error value $\overline{e}$	0.0180	0.0078
	Standard deviation $\sigma$	0.0191	0.0035
30 °C Carbonation	Mean error value $\overline{e}$	0.0667	0.0267
	Standard deviation $\sigma$	0.0770	0.0339

.

As can be seen from Table 2, under the carbonation condition of 20 °C, the average error of the new model is approximately 43% of that of the Fick model, and the standard deviation is approximately 20% of the Fick model. Under the carbonation condition of 30 °C, the average error of the new model is approximately 43% of that of the Fick model, and the standard deviation is approximately 45% of the Fick model. Clearly, for the C25 concrete in this example, the fitting accuracy of the new model is significantly better than that of the Fick model under both the temperature carbonation conditions, indicating that the new model has better versatility and accuracy.

In the literature [32], standard carbonation tests, freeze–thaw carbonation tests, dry–wet carbonation tests, and combined freeze–thaw and dry–wet carbonation tests were conducted on fly ash concrete with fly ash contents of 0%, 10%, and 20%. The mix proportions (fly ash/cement/water/river sand/crushed stone) for the concrete with fly ash contents of 0%, 10%, and 20% were 0:345:163.2:703:1145, 35:310:163.2:696:1135, and 70:275:163.2:693:1130, respectively. During the tests, one dry–wet cycle involved soaking the specimens for 16 h, followed by drying for 6 h, and finally allowing them to stand for 2 h. For the specimens undergoing dry–wet cycling, the entire test period included 20 of such dry–wet cycles. For the specimens undergoing freeze–thaw cycling, the test period was set to 50 freeze–thaw cycles. As for the more complex combined freeze–thaw and dry–wet cycling tests, one cycle consisted of eight freeze–thaw cycles immediately followed by seven dry–wet cycles, leading to specimens with combined freeze–thaw and dry–wet damage. The specimens were placed in a carbonation chamber for carbonation, and the carbonation depth of the concrete specimens was measured on the 7th, 14th, 21st, and 28th days, respectively. For the concrete with a 0% fly ash content, under the four carbonation conditions, the fitting curves of the experimental data using the Fick model and the new model proposed in this paper are shown in Figure 2. The fitting equations are presented in

Table 3. Curve equations for fitting the carbonation depth of concrete with a 0% fly ash content.

Carbonation Conditions	Fick Model	New Model
Standard carbonation	$h=1.6455\sqrt{t}$	$h=2.3339\sqrt{t}-2.9632{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Freeze–thaw cycling carbonation	$h=2.1653\sqrt{t}$	$h=3.4302\sqrt{t}-5.4449{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Dry–wet cycling carbonation	$h=1.8127\sqrt{t}$	$h=2.6244\sqrt{t}-3.4943{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Coupled carbonation	$h=2.1941\sqrt{t}$	$h=3.4271\sqrt{t}-5.3076{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$

, and the average values and standard deviations of the relative errors of the fitting results are shown in

Table 4. Average relative error and standard deviation of fitting the carbonation depth of concrete with a 0% fly ash content.

Carbonation Conditions	Statistical Indicators	Fick Model	New Model
Standard carbonation	Mean error value $\overline{e}$	0.1253	0.0407
	Standard deviation $\sigma$	0.1104	0.0117
Freeze–thaw cycling carbonation	Mean error value $\overline{e}$	0.1955	0.0492
	Standard deviation $\sigma$	0.2109	0.0324
Dry–wet cycling carbonation	Mean error value $\overline{e}$	0.1543	0.0411
	Standard deviation $\sigma$	0.1247	0.0258
Coupled carbonation	Mean error value $\overline{e}$	0.1931	0.0095
	Standard deviation $\sigma$	0.2249	0.0056

.

As shown in Table 4, for concrete with a 0% fly ash content, under standard carbonation conditions, the average error of the new model is approximately 28% of the Fick model’s, and the standard deviation is about 12% of the Fick model’s. Under freeze–thaw carbonation conditions, the average error of the new model is approximately 24% of the Fick model’s, and the standard deviation is about 14% of the Fick model’s. Under dry–wet carbonation conditions, the average error of the new model is approximately 25% of the Fick model’s, and the standard deviation is about 17% of the Fick model’s. Under the coupled freeze–thaw/dry–wet carbonation conditions, the average error of the new model is approximately 4% of the Fick model’s, and the standard deviation is about 2% of the Fick model’s. Clearly, for concrete with a 0% fly ash content, under these four carbonation conditions, the fitting accuracy of the new model is significantly better than that of the Fick model, demonstrating the better versatility and accuracy of the new model.

For concrete with a 10% fly ash content, under the four carbonation conditions, the fitting curves of the experimental data using the Fick model and the new model proposed in this paper are shown in Figure 3. The fitting equations are presented in

Table 5. Curve equations for fitting the carbonation depth of concrete with a 10% fly ash content.

Carbonation Conditions	Fick Model	New Model
Standard carbonation	$h=1.8289\sqrt{t}$	$h=2.2892\sqrt{t}-1.9814{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Freeze–thaw cycling carbonation	$h=2.4471\sqrt{t}$	$h=3.5897\sqrt{t}-4.9184{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Dry–wet cycling carbonation	$h=1.9734\sqrt{t}$	$h=2.5912\sqrt{t}-2.6590{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Coupled carbonation	$h=2.6372\sqrt{t}$	$h=4.0841\sqrt{t}-6.2284{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$

, and the average values and standard deviations of the relative errors of the fitting results are shown in

Table 6. Average relative error and standard deviation of fitting the carbonation depth of concrete with a 10% fly ash content.

Carbonation Conditions	Statistical Indicators	Fick Model	New Model
Standard carbonation	Mean error value $\overline{e}$	0.0751	0.0082
	Standard deviation $\sigma$	0.0710	0.0044
Freeze–thaw cycling carbonation	Mean error value $\overline{e}$	0.1711	0.0559
	Standard deviation $\sigma$	0.2477	0.0230
Dry–wet cycling carbonation	Mean error value $\overline{e}$	0.1008	0.0161
	Standard deviation $\sigma$	0.0957	0.0122
Coupled carbonation	Mean error value $\overline{e}$	0.2092	0.0434
	Standard deviation $\sigma$	0.2759	0.0210

.

As can be seen from Table 6, for concrete with a 10% fly ash content, under the standard carbonation condition, the average error of the new model is approximately 8% of that of the Fick model, and the standard deviation is approximately 7% of that of the Fick model. Under the freeze–thaw carbonation condition, the average error of the new model is approximately 38% of that of the Fick model, and the standard deviation is approximately 13% of that of the Fick model. Under the dry–wet carbonation condition, the average error of the new model is approximately 17% of that of the Fick model, and the standard deviation is approximately 11% of that of the Fick model. Under the coupled freeze–thaw/dry–wet carbonation condition, the average error of the new model is approximately 25% of that of the Fick model, and the standard deviation is approximately 10% of that of the Fick model. Clearly, for concrete with a 10% fly ash content under these four carbonation conditions, the fitting accuracy of the new model is significantly better than that of the Fick model, further demonstrating the versatility and accuracy of the new model.

For the concrete with a fly ash content of 20%, under four different carbonation conditions, the fitting curves of the experimental data using the Fick model and the new model proposed in this paper are shown in Figure 4. The fitted equations are presented in

Table 7. Curve equation for fitting the carbonation depth of concrete with a 20% fly ash content.

Carbonation Conditions	Fick Model	New Model
Standard carbonation	$h=1.9974\sqrt{t}$	$h=2.4998\sqrt{t}-2.1626{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Freeze–thaw cycling carbonation	$h=2.4026\sqrt{t}$	$h=2.7883\sqrt{t}-1.6605{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Dry–wet cycling carbonation	$h=2.1448\sqrt{t}$	$h=2.8530\sqrt{t}-3.0486{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$
Coupled carbonation	$h=2.7340\sqrt{t}$	$h=3.9448\sqrt{t}-5.2121{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$

, and the average and standard deviation of relative errors of the fitting results are listed in

Table 8. Average relative error and standard deviation of fitting the carbonation depth of concrete with a 20% fly ash content.

Carbonation Conditions	Statistical Indicators	Fick Model	New Model
Standard carbonation	Mean error value $\overline{e}$	0.0739	0.0225
	Standard deviation $\sigma$	0.1038	0.0111
Freeze–thaw cycling carbonation	Mean error value $\overline{e}$	0.0674	0.0481
	Standard deviation $\sigma$	0.0897	0.0294
Dry–wet cycling carbonation	Mean error value $\overline{e}$	0.0926	0.0212
	Standard deviation $\sigma$	0.1338	0.0125
Coupled carbonation	Mean error value $\overline{e}$	0.1680	0.0617
	Standard deviation $\sigma$	0.2283	0.0270

.

As can be seen from Table 8, for concrete with a 20% fly ash content, under standard-carbonation conditions, the average error of the new model is approximately 36% of that of the Fick model, and the standard deviation is approximately 14% of the Fick model; under freeze–thaw carbonation conditions, the average error of the new model is approximately 76% of that of the Fick model, and the standard deviation is approximately 37% of the Fick model; under dry–wet carbonation conditions, the average error of the new model is approximately 26% of that of the Fick model, and the standard deviation is approximately 12% of the Fick model; under coupled freeze–thaw/dry–wet carbonation conditions, the average error of the new model is approximately 41% of that of the Fick model, and the standard deviation is approximately 14% of the Fick model. Clearly, under these four carbonation conditions for concrete with a 20% fly ash content, the fitting accuracy of the new model is significantly better than that of the Fick model, once again demonstrating the better versatility and accuracy of the new model.

Next, the test data on the carbonation depth of bridge tower concrete from the literature [33] is utilized to further validate the proposed new model. The purpose of this example is to compare the later-stage prediction capabilities of the proposed new model, the Fick model, and the backpropagation ANN model [7] for the carbonation depth. In the literature [33], standard carbonation tests were conducted on C50 concrete samples incorporating fly ash and mineral powder. The mix proportion (cement/fly ash/mineral powder/sand/stone/water) was 329:94:47:743:1114:146. The water-reducing agent content was 1.25%. Carbonation tests were conducted according to standard specifications on the 3rd, 7th, 14th, 28th, and 56th days, respectively. The specific values for these carbonation depths can be found in reference [33], or as presented in

Table 9. Carbonation depth of bridge tower concrete.

Time	3d	7d	14d	28d	56d
Carbonation depth	0.2 mm	1.0 mm	2.5 mm	4.0 mm	6.6 mm

. In Table 9, the data from the first four time points (3d, 7d, 14d, and 28d) are used for training the ANN model, and the curve fitting of the Fick model and the improved Fick model, while the data from the 56th day are used to compare the predictive capabilities of each model. The structure of the ANN model is shown in Figure 5. For this example, the number of neurons in the input layer, denoted as $n_i$, and the number of neurons in the output layer, denoted as $n_o$, are both equal to one. The number of neurons in the hidden layer, denoted as $n_h$, is determined empirically. For this example, due to the small number of samples, the value of $n_h$ can be set to three, four, or five, respectively.

Table 10. Curve equation obtained by fitting the carbonation depth data of the bridge tower concrete at days 3, 7, 14, and 28.

Bridge Tower Concrete Carbonation	Fick Model	New Model
Equation	$h=0.6445\sqrt{t}$	$h=1.0938\sqrt{t}-1.7421{\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$

provides the specific curve equations for the Fick model and the new model obtained after data fitting. Using the fitted curve equations shown in Table 10 and the trained ANN model, the predicted carbonation depth values of the concrete on the 56th day are presented in

Table 11. Comparison between the measured and predicted carbonation depth values of the bridge tower concrete on the 56th day.

Time: The 56th Day	Measured Value	Predicted Value
		Fick Model	New Model	ANN with ${\mathit{n}}_{\mathit{h}}$ = 3	ANN with ${\mathit{n}}_{\mathit{h}}$ = 4	ANN with ${\mathit{n}}_{\mathit{h}}$ = 5
Carbonation depth (mm)	6.6	4.8228	6.4428	4.5008	5.7763	4.8487
Relative error of prediction	/	27%	2%	32%	12%	27%

. As can be seen from Table 11, for the carbonation depth of concrete on the 56th day, the relative error between the predicted value of the new model and the measured value is only 2%, which is much smaller than the 27% of the Fick model and the 12% of the ANN model with $n_h$ = 4. This result indicates that the ANN model only achieves a high prediction accuracy when there are sufficient training samples. For problems with only a limited number of test data, like in this example, the prediction accuracy of the ANN model is inferior to that of the proposed new model. This once again demonstrates the unique advantage of the proposed model in predicting the carbonation depth when only a small amount of experimental data are available.

4. Conclusions

Due to the complexity of concrete materials and external environments, the traditional Fick model struggles to accurately predict the carbonation depth of concrete under different conditions, necessitating modifications to reflect the impact of various conditions on concrete carbonation behavior. This paper proposes an improved Fick model that introduces a correction term to overcome the limitations of the original Fick model, thereby enhancing the prediction accuracy of the carbonation depth. Based on theoretical analysis and case studies, the main conclusions are as follows. (1) The physical significance of the correction term in the improved Fick model is to make reasonable adjustments to deviations in the development law of the concrete carbonation depth from the Fick model’s predictions, reflecting the influence of different conditions on the development law of the concrete carbonation depth. This enables the improved Fick model to have a broader application scope than the original Fick model. (2) According to comparative studies, the average prediction error and standard deviation of the new model are significantly smaller than those of the Fick model. For the first two cases, in most situations, the average prediction error and standard deviation of the new model are less than 50% of those of the Fick model, with the lowest average prediction error being only 4% and the lowest standard deviation being only 2% of the Fick model’s respective values. (3) For the third case, the new model demonstrates superior predictive capability for the later-stage carbonation depth of concrete compared to the Fick model and the ANN model. For the carbonation depth of concrete on the 56th day, the relative error between the predicted value of the new model and the measured value is only 2%, which is much smaller than the 27% of the Fick model and the 12% of the ANN model. The results demonstrate the unique advantage of the proposed model in predicting the carbonation depth, especially when only a limited amount of experimental data are available. (4) Since the model presented in this paper is a modification of the Fick model, it may not necessarily be applicable to issues where the development pattern of the carbonation depth deviates significantly from the Fick model. Due to the complexity of concrete materials and environments, the application fields of this model still require further research for verification. (5) In practical engineering, the testing of the concrete carbonation depth exhibits a certain degree of randomness. However, once specific values are assigned to the fitting parameters in the model presented in this paper, it becomes a deterministic model that fails to capture the randomness in carbonation testing. Introducing interval random numbers into this model may help further enhance its predictive capabilities, which could serve as a direction for future research.