# American Institute of Mathematical Sciences

August  2022, 15(8): 2033-2052. doi: 10.3934/dcdss.2022092

## A moving boundary problem for reaction and diffusion processes in concrete: Carbonation advancement and carbonation shrinkage

 Istituto per le Applicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, via dei Taurini 19, 00185, Roma, Italy

* Corresponding author: Roberto Natalini

Received  February 2022 Published  August 2022 Early access  April 2022

The present work is devoted to modeling and simulation of the carbonation process in concrete. To this aim we introduce some free boundary problems which describe the evolution of calcium carbonate stones under the attack of ${CO}_2$ dispersed in the atmosphere, taking into account both the shrinkage of concrete and the influence of humidity on the carbonation process. Indeed, two different regimes are described according to the relative humidity in the environment. Finally, some numerical simulations here presented are in substantial accordance with experimental results taken from literature.

Citation: Gabriella Bretti, Maurizio Ceseri, Roberto Natalini. A moving boundary problem for reaction and diffusion processes in concrete: Carbonation advancement and carbonation shrinkage. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2033-2052. doi: 10.3934/dcdss.2022092
##### References:
 [1] W. Ashraf, Carbonation of cement-based materials: Challenges and opportunities, Construction and Building Materials, 120 (2016), 558-570. [2] G. Bretti, M. Ceseri, M. C. Ciacchella, R. Natalini, M. L. Santarelli and G. Tiracorrendo, A forecasting model for the porosity variation during the carbonation process, International Journal on Geomathematics, 2022. Submitted [3] G. Bretti, L. Gosse and N. Vauchelet, Diffusive limits of 2D well-balanced schemes for kinetic models of neutron transport, ESAIM Math. Model. Numer. Anal., 55 (2021), 2949-2980.  doi: 10.1051/m2an/2021077. [4] M. Ceseri and J. M. Stockie, A three-phase free boundary problem with melting ice and dissolving gas, European J. Appl. Math., 25 (2014), 449-480.  doi: 10.1017/S0956792513000430. [5] F. Clarelli, A. Fasano and R. Natalini, Mathematics and monument conservation: Free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168.  doi: 10.1137/070695125. [6] F. Freddi and L. Mingazzi, Phase-field simulations of cover cracking in corroded RC beams, Procedia Structural Integrity, 33 (2021), 371-384. [7] R. M. Furzeland, A comparative study of numerical methods for moving boundary problems, J. Inst. Math. Appl., 26 (1980), 411-429.  doi: 10.1093/imamat/26.4.411. [8] Lindsey L Climate Change: Atmospheric Carbon Dioxide, 2020. https://www.climate.gov/news-features/understanding-climate/climate-change-atmospheric-carbon-dioxide (Accessed: 18 March 2022). [9] S. Kashef-Haghighi, Y. Shao and S. Ghoshal, Mathematical modeling of CO2 uptake by concrete during accelerated carbonation curing, Cement and Concrete Research, 67, 20151–0. [10] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1. [11] K. Kovler and S. Zhutovsky, Overview and future trends of shrinkage research, Materials and Structures, 39 (2006), 827-847. [12] A. Leemann and F. Moro, Carbonation of concrete: The role of CO2 concentration, relative humidity and CO2 buffer capacity, Materials and Structures, 50 (2017), 1-4. [13] F. Matsushita, Y. Aono and S. Shibata, Calcium silicate structure and carbonation shrinkage of a tobermorite-based material, Cement and Concrete Research, 34 (2004), 1251-1257. [14] I. Monteiro, F. A. Branco, J. de Brito and R. Neves, Statistical analysis of the carbonation coefficient in open air concrete structures, Construction and Building Materials, 29 (2012), 263-269. [15] G. Pan and Q. Shen abd J. Li, Microstructure of cement paste at different carbon dioxide concentrations, Magazine of Concrete Research, 70 (2018), 154-162. [16] V. G. Papadakis, C. G. Vayenas and M. N. Fardis, Experimental investigation and mathematical modeling of the concrete carbonation problem, Chemical Engineering Science, 46 (1991), 1333-1338. [17] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4. [18] M. A. Peter, A. Muntean, S. A. Meier and M. Böhm, Competition of several carbonation reactions in concrete: A parametric study, Cement and Concrete Research, 38 (2008), 1385-1393. [19] B. Šavija and M. Luković, Carbonation of cement paste: Understanding, challenges, and opportunities, Construction and Building Materials, 117 (2016), 285-301. [20] A. Steffens, D. Dinkler and H. Ahrens, Modeling carbonation for corrosion risk prediction of concrete structures, Cement and Concrete Research, 32 (2002), 935-941. [21] Y. Sumra, S. Payam and I. Zainah, The pH of cement-based materials: A review, Journal of Wuhan University of Technology-Mater. Sci. Ed., 35 (2020), 908-924. [22] F. P. Torgal, S. Miraldo, J. A. Labrincha and J. De Brito, An overview on concrete carbonation in the context of eco-efficient construction: Evaluation, use of SCMs and/or RAC, Construction and Building Materials, 36 (2012), 141-150.

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##### References:
 [1] W. Ashraf, Carbonation of cement-based materials: Challenges and opportunities, Construction and Building Materials, 120 (2016), 558-570. [2] G. Bretti, M. Ceseri, M. C. Ciacchella, R. Natalini, M. L. Santarelli and G. Tiracorrendo, A forecasting model for the porosity variation during the carbonation process, International Journal on Geomathematics, 2022. Submitted [3] G. Bretti, L. Gosse and N. Vauchelet, Diffusive limits of 2D well-balanced schemes for kinetic models of neutron transport, ESAIM Math. Model. Numer. Anal., 55 (2021), 2949-2980.  doi: 10.1051/m2an/2021077. [4] M. Ceseri and J. M. Stockie, A three-phase free boundary problem with melting ice and dissolving gas, European J. Appl. Math., 25 (2014), 449-480.  doi: 10.1017/S0956792513000430. [5] F. Clarelli, A. Fasano and R. Natalini, Mathematics and monument conservation: Free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168.  doi: 10.1137/070695125. [6] F. Freddi and L. Mingazzi, Phase-field simulations of cover cracking in corroded RC beams, Procedia Structural Integrity, 33 (2021), 371-384. [7] R. M. Furzeland, A comparative study of numerical methods for moving boundary problems, J. Inst. Math. Appl., 26 (1980), 411-429.  doi: 10.1093/imamat/26.4.411. [8] Lindsey L Climate Change: Atmospheric Carbon Dioxide, 2020. https://www.climate.gov/news-features/understanding-climate/climate-change-atmospheric-carbon-dioxide (Accessed: 18 March 2022). [9] S. Kashef-Haghighi, Y. Shao and S. Ghoshal, Mathematical modeling of CO2 uptake by concrete during accelerated carbonation curing, Cement and Concrete Research, 67, 20151–0. [10] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1. [11] K. Kovler and S. Zhutovsky, Overview and future trends of shrinkage research, Materials and Structures, 39 (2006), 827-847. [12] A. Leemann and F. Moro, Carbonation of concrete: The role of CO2 concentration, relative humidity and CO2 buffer capacity, Materials and Structures, 50 (2017), 1-4. [13] F. Matsushita, Y. Aono and S. Shibata, Calcium silicate structure and carbonation shrinkage of a tobermorite-based material, Cement and Concrete Research, 34 (2004), 1251-1257. [14] I. Monteiro, F. A. Branco, J. de Brito and R. Neves, Statistical analysis of the carbonation coefficient in open air concrete structures, Construction and Building Materials, 29 (2012), 263-269. [15] G. Pan and Q. Shen abd J. Li, Microstructure of cement paste at different carbon dioxide concentrations, Magazine of Concrete Research, 70 (2018), 154-162. [16] V. G. Papadakis, C. G. Vayenas and M. N. Fardis, Experimental investigation and mathematical modeling of the concrete carbonation problem, Chemical Engineering Science, 46 (1991), 1333-1338. [17] L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4. [18] M. A. Peter, A. Muntean, S. A. Meier and M. Böhm, Competition of several carbonation reactions in concrete: A parametric study, Cement and Concrete Research, 38 (2008), 1385-1393. [19] B. Šavija and M. Luković, Carbonation of cement paste: Understanding, challenges, and opportunities, Construction and Building Materials, 117 (2016), 285-301. [20] A. Steffens, D. Dinkler and H. Ahrens, Modeling carbonation for corrosion risk prediction of concrete structures, Cement and Concrete Research, 32 (2002), 935-941. [21] Y. Sumra, S. Payam and I. Zainah, The pH of cement-based materials: A review, Journal of Wuhan University of Technology-Mater. Sci. Ed., 35 (2020), 908-924. [22] F. P. Torgal, S. Miraldo, J. A. Labrincha and J. De Brito, An overview on concrete carbonation in the context of eco-efficient construction: Evaluation, use of SCMs and/or RAC, Construction and Building Materials, 36 (2012), 141-150.
The geometrical depiction of the setting we considered for the molar mass conservation, eq. (2). The grey area indicates the non-carbonated region while the light blue area indicates the carbonated one
Spatial evolution of $c$ for the case of concentration at $3\%$ at different times. The depicted results are obtained for grid spacing $\Delta \zeta = 0.1, \Delta t = 10^{-7}$
Relationship between carbonation depth and $\sqrt{t}$ for different concentration of pollutant. Plot of experimental data (red circles), theoretical slope (black dotted line) $K \sqrt{t}$ taken from [15] against numerical results obtained with the proposed model (magenta solid line) at time $t = 28$ days for $3\%$ (top figure) and $20\%$ (bottom figure) concentration of carbon dioxide
Carbonation front $\sigma$ and $\sigma_0 = \omega \sigma$
The evolution of carbonation shrinkage strain according to simulation results
The physical and chemical constants used in our mathematical model. Most of these represent typical values of characteristic properties of the materials under consideration. The value of $n_c$ has been found in [15,Table 7], that of $c^*$ in [8] and, that of $W^*$ in [5,Table 2]
 Parameter Meaning Value Dimension $\mu_c$ Molar Density of $CaCO_3$ $0.027086$ $mol\cdot cm^{-3}$ $\mu_h$ Molar Density of $Ca(OH)_2$ $0.02984$ $mol\cdot cm^{-3}$ $n_h$ Portland Cement porosity $0.2$ - $n_c$ Calcium Carbonate porosity $0.05$ - $t^*$ reference time (1 year) $3.1536 \cdot 10^{7}$ $s$ $c^*$ reference density ($CO_2$) $4.098 \cdot 10^{-4}$ $g \cdot cm^{-3}$ $W^*$ reference density ($H_2 O$) $17.3\cdot 10^{-6}$ $g \cdot cm^{-3}$ $M_w$ Molar weight (water) $18.01528$ $g\cdot mol^{-1}$ $M_c$ Molar weight ($CaCO_3$) $100.0869$ $g\cdot mol^{-1}$ $M_{cd}$ Molar weight ($CO_2$) $44.01$ $g\cdot mol^{-1}$
 Parameter Meaning Value Dimension $\mu_c$ Molar Density of $CaCO_3$ $0.027086$ $mol\cdot cm^{-3}$ $\mu_h$ Molar Density of $Ca(OH)_2$ $0.02984$ $mol\cdot cm^{-3}$ $n_h$ Portland Cement porosity $0.2$ - $n_c$ Calcium Carbonate porosity $0.05$ - $t^*$ reference time (1 year) $3.1536 \cdot 10^{7}$ $s$ $c^*$ reference density ($CO_2$) $4.098 \cdot 10^{-4}$ $g \cdot cm^{-3}$ $W^*$ reference density ($H_2 O$) $17.3\cdot 10^{-6}$ $g \cdot cm^{-3}$ $M_w$ Molar weight (water) $18.01528$ $g\cdot mol^{-1}$ $M_c$ Molar weight ($CaCO_3$) $100.0869$ $g\cdot mol^{-1}$ $M_{cd}$ Molar weight ($CO_2$) $44.01$ $g\cdot mol^{-1}$
Values for the parameter $\sigma^*$, $D_c$ and, $D_w$, determined by the procedures detailed. The values listed for $D_w$ are the lower bounds determined by equation (67). Note that the environmental $CO_2$ is converted from percentage concentration to $g/cm^3$ by applying formula (65)
 Parameter Value Dimension Environmental $CO_2$ $\sigma^*$ $2.87$ $cm$ $3\%$ $D_c$ $5.8\cdot 10^{-2}$ $cm^2\cdot sec^{-1}$ $3\%$ $D_w$ $1.1\cdot 10^{-1}$ $cm^2\cdot sec^{-1}$ $3\%$ $\sigma^*$ $5.69$ $cm$ $20\%$ $D_c$ $3.7\cdot 10^{-2}$ $cm^2\cdot sec^{-1}$ $20\%$ $D_w$ $4.5\cdot 10^{-1}$ $cm^2\cdot sec^{-1}$ $20\%$
 Parameter Value Dimension Environmental $CO_2$ $\sigma^*$ $2.87$ $cm$ $3\%$ $D_c$ $5.8\cdot 10^{-2}$ $cm^2\cdot sec^{-1}$ $3\%$ $D_w$ $1.1\cdot 10^{-1}$ $cm^2\cdot sec^{-1}$ $3\%$ $\sigma^*$ $5.69$ $cm$ $20\%$ $D_c$ $3.7\cdot 10^{-2}$ $cm^2\cdot sec^{-1}$ $20\%$ $D_w$ $4.5\cdot 10^{-1}$ $cm^2\cdot sec^{-1}$ $20\%$
Slope of the curve (4) for the two concentrations of ${CO}_2$ in the accelerated carbonation process. Data taken from [15]
 $CO_2$ concentration $K$ dimensions $3\%$ $1.5$ $mm\;d^{-1/2}$ $20\%$ $2.98$ $mm\;d^{-1/2}$
 $CO_2$ concentration $K$ dimensions $3\%$ $1.5$ $mm\;d^{-1/2}$ $20\%$ $2.98$ $mm\;d^{-1/2}$
Average percentage error (68) between simulated and experimental carbonation depth
 Carbon dioxide concentration Average $\%$ error $3\%$ $0.1\%$ $20\%$ $0.4\%$
 Carbon dioxide concentration Average $\%$ error $3\%$ $0.1\%$ $20\%$ $0.4\%$
Numerical accuracy of the approximation scheme in the computation of $w$
 Norm $\gamma_w$ $L^1$ $1.0$ $L^\infty$ $1.0$
 Norm $\gamma_w$ $L^1$ $1.0$ $L^\infty$ $1.0$
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