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

Gabriella Bretti; Maurizio Ceseri; Roberto Natalini

doi:10.3934/dcdss.2022092

Article Contents

2022, Volume 15, Issue 8: 2033-2052. Doi: 10.3934/dcdss.2022092

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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
^* Corresponding author: Roberto Natalini

Received: January 31, 2022

Published: August 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 76S05, 35R37; Secondary: 65N06.

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Figure 1. 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

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Figure 2. 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} $

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Figure 3. 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

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Figure 4. Carbonation front $ \sigma $ and $ \sigma_0 = \omega \sigma $

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Figure 5. The evolution of carbonation shrinkage strain according to simulation results

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Table 1. 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} $

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Table 2. 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\% $

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Table 3. 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} $

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Table 4. Average percentage error (68) between simulated and experimental carbonation depth

Carbon dioxide concentration	Average $ \% $ error
$ 3\% $	$ 0.1\% $
$ 20\% $	$ 0.4\% $

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Table 5. Numerical accuracy of the approximation scheme in the computation of $ w $

Norm	$ \gamma_w $
$ L^1 $	$ 1.0 $
$ L^\infty $	$ 1.0 $

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