Carbonation of cement is a process that involves multiple chemical reactions between the main components of set cement and carbon dioxide such that the carbon dioxide is permanently sequestered. 0.25 gigatons of carbon are estimated to be sequestered in cement-based material annually; however, this estimate is based on a simplified model of the carbonation process. By comparing this simplified model with existing data, we show that the model works best under highly controlled conditions in which the rate of carbonation is limited by diffusion of carbon dioxide into the concrete. To improve the modeling of carbonation, we developed a finite difference analysis that included the unsteady diffusion of carbon dioxide and the rate of reaction between carbon dioxide and the various cement components. This improved model allowed us to examine a range of conditions and provided a more complete estimate of the reactions, which will ultimately enable more accurate global carbon estimates.

Author: Miles Jones
California Institute of Technology
Mentors: Professor Melany Hunt
California Institute of Technology
Editor: Wesley Huang

Introduction

Concrete has been a vital part of modern society, as it is a versatile, cheap, and sturdy building material. However, the production of cement, the main component and “glue” found in concrete, results in the release of large amounts of carbon dioxide gas. Due to the immense amount of cement produced, it is estimated that 6 − 10% of all anthropogenic CO₂ released is from cement [1].Once cement is created, however, it absorbs CO₂ as the gas diffuses and reacts with the calcium hydroxide and a calcium silicate hydrate (CSH) gel. The reaction with calcium hydroxide is as follows:
Ca(OH)₂(s) + CO₂(g) → CaCO₃(s) + H₂O (1) 
The reactions between CSH and CO₂ are: 
(3CaO·2SiO₂·3H₂O)+3CO₂ → (3CaCO₃·2SiO₂·3H₂O) (2)
(3CaO · SiO₂) + 3CO₂ + H₂O → SiO₂ · H₂O + 3CaCO₃ (3)
(2CaO · SiO₂) + 2CO₂ + H₂O → SiO₂ · H₂O + 2CaCO₃ (4)
[2]. These processes can be modeled with a simple equation as done by Xi et al.:
d = k × √ t (5) 
Here, t refers to time, k is the carbonation rate coefficient, and d is how far into the cement the reaction has proceeded [3]. 
The coefficient can be expanded, such that: 
d = (2[CO₂]₀D_e,CO2t/[Ca(OH)₂(s)]⁰+3[CSH]⁰)^0.5 (6) 
[Ca(OH)₂(s)]⁰ and [CSH]⁰ are the initial concentrations of calcium hydroxide and CSH gel respectively, [CO₂]⁰ is the concentration of carbon dioxide at the edge of the material, and D_e,CO2 is a diffusion coefficient based on porosity and water content [2]. 

This allows for the estimation of not only how fast the reaction proceeds, but also the total amount of CO₂ absorbed by cement. Xi et al. estimate that around 0.24 gigatons of carbon are absorbed by this process each year by using equation (5) to find the depth of carbonation and thus the amount of carbon sequestered. They also estimate that the total amount of carbon sequestered by cement from 1930 to 2013 is 4.5 gigatons [3].

The method by which the aforementioned estimates were found are based on large assumptions, especially for equation (5). The reaction is assumed to take place instantly and the rate limiting factor is simply the diffusion of CO₂. Additionally, it assumes that the cement is fully hydrated, which is not necessarily true. These assumptions, as well as other issues that arise from using this equation, coupled with the accuracy of some of the coefficients used led us to question if there was a method of modeling this process that would result in higher accuracy of these estimates.

More complex models not only give better estimates to carbonation depth and amount of carbon stored, but also a more complete picture of the process. The models we developed in MATLAB, based on mass transfer and finite-difference principles give the concentration of CO₂, Ca(OH)₂, and CSH gel at each position and time. Upon comparisons with experimental data from Chang and Chen [4] we find that, given the same initial conditions, our models hold up well.This process has a large impact on global carbon models, as the CO₂ fossil fuel emission estimates decrease from 9.9 gigatons of carbon per year to 9.7 gigatons if included. While this difference (0.2 GTC) may seem small, it is of the same magnitude of the net carbon budget imbalance (0.3 GTC) [5]. Thus, we want the estimate of carbon that is sequestered through these reactions to be as accurate as possible.

Methods

Comparison of Simple Models with Data
For the comparison of “simpler” models which use equations (5) and (6), we utilize data from numerous sources. Multiple papers that took measurements of the carbonation depth at differing times were used for data points. Then, the groups of data with similar strengths and storage (indoors vs. outdoors) conditions were grouped together and compared to the results from (5) and (6), using similar initial conditions. These comparisons were done in Microsoft Excel by inputting the data and calculating the predictions.

Development of Models
To develop the more advanced models, MATLAB was utilized. These models are based on unsteady one-dimensional diffusion in a semi-infinite medium. Both forwards and backwards finite differencing methods were used to create two separate models, each being derived from Fick’s Law of Diffusion and an analysis of the mass diffusing in and out of each position, as well as that used in the reaction at each point. The forwards method estimates the concentration of the species at a certain position based on concentrations from the previous section. The backwards method is more complex, and estimates concentrations from those in the following section. The initial equation is follows:
∂C/∂t = ∂/∂x(D C/∂x) − k_CHCC_CH − 3k_CSHCC_CSH (7) 
This is the unsteady mass conservation equation for CO₂, with partial derivatives of the concentration of CO₂, as well as the following mass conservation equations for first Ca(OH)₂ then CSH: 
∂C_CH/∂t = −k_CHCC_CH (8) 
∂C_CSH/∂t = −k_CSHCC_CSH (9) 

In equations (7)-(9) C, C_CH, and C_CSH are the molar concentrations of CO₂, Ca(OH)₂, and CSH respectively, and k_CH and k_CSH are the rate constants for Ca(OH)₂ and CSH. 

For both methods, these derivations lead to a set of equations that could be used to iterate through a matrix of initial concentrations of CO₂, Ca(OH)₂, and CSH and give their final concentrations at each position after a time determined by the timestep and number of iterations. The forwards finite differencing method is easier to derive and implement, but is limited in that if the timestep is too large, it becomes unsteady and is unusable. The backwards method has no such limitation, but requires more complex operations such as matrix inversions. 

These initial models were then improved with the addition of equations that calculate new diffusion coefficients at each position depending on the initial porosity as well as reaction progression. There were two methods of the backwards finite differencing model that include this: one with a central difference calculation and the other with an average of the diffusion coefficients on each side of each position.

Comparison of Complex Models with Data
Many comparisons were done between the complex models and the simpler models with experimental data from recent literature. All of these comparisons took place in MATLAB, and they can be seen below in the Figures section. The data was then plotted with the estimate(s) from the models.

Results

In Figure 1, we combined numerous sets of data and models to show that, in certain cases, the simple estimates can prove to be decent estimates. All of the experimental data and models here are from high strength concrete stored indoors. The light blue line is the estimated depth from (5) with a carbonation rate coefficient of k = 1.5 mm/yr^0.5, and the dark blue line is the estimated depth from (6). We can clearly see that the data is close to the estimates from the two models in this case.

In comparison to Figure 1, Figure 2 demonstrates how these simple equations do not always work. All of the features are for similar strength classes, but they are not from controlled environments as in the first figure. These estimates and data are all from outside locations and coefficients. As the conditions were different for each, one can see that the slopes vary greatly, and the data does not fit close to them. The figure has estimates from (5) with a carbonation rate coefficient of k = 10.8 mm/yr^0.5 as the yellow line and estimates from (6) as the dark blue line. The estimated value from Xi et al. [3]. is uses their k value for outside high strength mortar. Additionally, the grey circles are from Papadakis et al. [6] and the orange squares are from Nagataki et al. [8].

Figure 3 is a comparison of the two different modeling methods. The red line demonstrates the moving front of the reaction as predicted by (5) and the blue line is the estimate from the backwards differencing model with the more accurate transition from fully reacted to not yet reacted. The initial conditions for both are the same. The figure specifically focuses on the concentration of Ca(OH)₂, one of the main reactants in the reactions. One can see that the method provided by (5) models as an instantaneous change, whereas the more accurate model has a more gradual curve, which is more correct.

This next figure is the first comparison between different backwards finite differencing methods with diffusion coefficient calculations. The red is the central difference method and the blue is the averaging method. The central difference method estimates the first and second derivative at a certain point with the values before and after this point, and then uses them to estimate the diffusion coefficient at that point. The method of averaging is much simpler, and finds the average between each point and the one two ahead of it, and uses that value for the point in between. Additionally, the points are data from Chang and Chen [4]. The Y-axis is the concentration of Ca(OH)₂, one of the main reactants involved in the process. All of these last 4 graphs are just showing the differences in the accuracies of the two methods for certain cases, but note that both methods are very close to reality.

Here is a very similar figure to Figure 4, but instead of taking place after 8 weeks, the data and estimates are after 16 weeks. Note how the data is much closer to the average estimate instead of the central difference estimate, opposite to Figure 4. Once again, the red is the central differencing estimate and the blue is the estimate from the averaging method of calculating diffusion coefficients, with the points being experimental data from Chang and Chen [4].

Similar to Figures 5 and 6, Figure 7 compares the central difference and averaging method for diffusion coefficients, with data from Chang and Chen [4]. However, the Y-axis is the concentration of CaCO₃, the main product of the carbonation process. We have the red line as the estimate from central difference method, the blue line as the estimate from the average method, and the points are the experimental data.

As mentioned above, this figure is similar to Figure 6, but is after 16 weeks, not 8. The lines and points are from the same sources as above. Note again the change in which method fits the data best. Once again, the data is from Chang and Chen [4].

Discussion/Conclusions

The simple models, based on (5) and (6), are adequate at predicting the carbonation depth in controlled environments where the assumptions that those equations make are true. However, they fail in uncontrolled situations where the environment is not kept the same. The more advanced models we develop solve many of these shortcomings by giving accurate estimates for a wider range of environments. Moving forward, the models themselves can be improved to contain more initial conditions to include more of the calculations involved in finding total carbon sequestered.

Acknowledgements

I would first like to thank my mentor, Professor Melany Hunt, for the instrumental insight and assistance that I was provided. I would also like to thank graduate student Ricardo Hernandez for his assistance with the development of the code, specifically the functions to calculate the porosity and diffusion coefficients. Additionally, I am extremely grateful for Dr. and Mrs. Harris for providing funds so that I was able to partake in this research. Lastly, I would like to thank the SFP department for the opportunity to engage in this research.

References

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[3] F. Xi et al., “Substantial Global Carbon Uptake by Cement Carbonation,” Nature Geoscience, vol. 9, no. 12, pp. 880–883, 2016. 

[4] C.-F. Chang and J.-W. Chen, “The Experimental Investigation of Concrete Carbonation Depth,” Cement and Concrete Research, vol. 36, no. 9, pp. 1760–1767, 2006. 

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[6] V. G. Papadakis, C. G. Vayenas, and M. N. Fardis, “Fundamental Modeling and Experimental Investigation of Concrete Carbonation,” ACI Materials Journal, vol. 88, no. 4, 1991. 

[7] K. Tuutti, “Corrosion of Steel in Concrete,” Diss. – Stockholm – Tekniska hogskolan, 1982. 

[8] S. Nagataki, M. A. Mansur, and H. Ohga, “Carbonation of Mortar in Relation to Ferrocement Construction ,” ACI materials journal, vol. 85, no. 1, pp. 17–25, 1988.