1. Introduction
The interactions between environment and building heritage, i.e., monuments, archaeological sites, and historical and modern buildings have always been, and will continue to be, crucial for conservation issues. Moreover, the deterioration and damage of materials caused by weathering processes is still not completely understood since it is a highly complex phenomenon resulting from the interaction of both chemical and mechanical processes.
As natural stones are exposed to the modification of the environment and surrounding landscapes, concrete is also subjected to attack by multiple damaging factors, such as weathering, chemical aggression and abrasion, that may cause its deterioration in terms of a modification of the original form, quality and serviceability. Such weathering processes are always associated with water flow within the material, determined by wetting or infiltration, caused by meteoric precipitation or groundwater capillary rise, respectively.
An increased rate of extreme conditions due to climate change also constitutes a further threat, increasing the decaying rates and contributing to the occurrence of new degradation mechanisms. This happens because climatic changes can not only influence the frequency and intensity of hazardous events but can also worsen the physical, chemical and biological mechanisms causing the degradation of the structure and its materials. In recent years, research communities have started to discuss the impact of climate change on cultural heritage (CH). Since 2003, the number of papers on this subject increased significantly; see [
1,
2]. The changes in the climate system have been studied by climate scientists; see, for instance, [
3,
4,
5]. In particular, the studies presented by NOAA Climate.gov show that environmental changes in recent decades are leading to a progressive increase in the
concentration in the atmosphere, along with human emissions. The Intergovernmental Panel on Climate Change (
https://www.ipcc.ch/, accessed on 8 November 2022) takes into account possible scenarios of
emissions, showing that, in the near future (up until 2050), the
concentration will grow with different rates of increase and, in only a few cases, might be similar to that in 2015.
The environmental changes may affect CH buildings and artifacts due to the synergistic action of atmospheric agents and pollutants; see the recent studies in [
6,
7]. Such phenomena may determine an irreversible weakening of the mechanical strength and an increased vulnerability to the chemical aggression of porous materials through several mechanisms, such as freeze–thaw cycles, a change in precipitation, corrosion, salt crystallization cycles and an increased frequency of extreme events, just to mention a few; see [
8,
9,
10,
11,
12,
13]. Recently, predictive maintenance, consisting of anticipating future deterioration through appropriate diagnostic techniques, has emerged as a new tool for monitoring and protecting CH sites [
14]. Such a diagnosis is carried out by collecting and analyzing, with statistical tools, data on the constitutive materials of the work of art, as well as the action of environmental factors at the CH site. Our approach is within the model-driven framework on predictive maintenance, based on mathematical models for CH (equations describing deterioration processes and the effects of conservation practices)—see, for instance, [
15,
16,
17,
18]—coupled with data derived from laboratory experiments and/or gathered by sensors suitably placed at the CH site.
Our study is focused on concrete since it is the most widely used construction material worldwide [
19], as it is employed in the construction of buildings, stadiums, stairs, sidewalks and foundations; see
Figure 1.
Such a material has a porous structure and its durability is mainly due to its resistance to chemical aggression. In particular, one of the most important degradation phenomena for concrete is carbonation, caused by carbon dioxide in the atmosphere. As shown in
Figure 2, environmental changes in recent decades have determined a constant increase in the carbon dioxide concentration in the atmosphere. This fact may accelerate the damage process of building structures.
Concrete carbonation is a complex process whose study requires the cooperation of chemists, engineers and mathematicians in order to capture its features. It is caused by a sequence of chemical reactions that consume calcium hydroxide () and form calcium carbonate (). The above reactions are induced by carbon dioxide (), which is transported by water through the porous medium.
Although concrete is a long-lasting material, its degradation and the consequent weakening may be due to the sulfation of the cementitious matrix, freeze–thaw cycles and the corrosion of steel armor caused by carbonation. Indeed, a basic environment inside the non-carbonated concrete forms a thin film of oxide that protects the steel bars reinforcing the structures, and such a layer is maintained as long as the environment has a pH value of at least 13. In carbonated areas, the presence of carbon dioxide instead neutralizes the alkalinity of concrete with the solution within the pores, assuming a pH value inferior to 9. This causes the destruction of the protective layer and starts corroding the steel bars [
20].
In the literature, many experiments for quantitatively evaluating the effects of the carbonation process on cement materials can be found. Since, in natural conditions, the carbonation process is very slow, most of the experiments are carried out in an accelerated regime. For fixed conditions of the
concentration, temperature, humidity and time, the samples are subjected to carbonation in a sealed chamber [
21]. At the end of the test, the specimens are split, cleaned and sprayed with a phenolphthalein pH indicator, which is an organic compound that is colorless in an acid environment (carbonated part) and turns pink in a basic environment (non-carbonated part); see, for instance, [
22,
23]. SEM microscope observations and X-rays allow for the quantification of the effect of carbonation by identifying the presence of
formed and residual
not yet reacted with
. From the comparison of the sample before and after carbonation, the material, initially rich in calcium hydroxide, transforms into being mainly composed of
, clearly indicating that the reaction with
has taken place [
24,
25]. Porosity represents a fundamental parameter for process control that allows us to evaluate the effect of carbonation on the structure of cement. To this aim, gammadensimetry and mercury intrusion analysis can be carried out.
Carbonation has attracted research interests also within the mathematical community and there is a huge amount of literature addressing the mechanism of carbonation with different mathematical models; see, for instance, [
26,
27,
28,
29,
30,
31,
32,
33,
34].
In [
35], we introduced a mathematical model describing porosity variation as the result of several intermediate chemical reactions triggered by the penetration of carbon dioxide that diffuses and is transported into the pores by water that is present.
The present paper describes the application of a mathematical model toward the simulation of CH degradation under climate change scenarios. Such scenarios are introduced in
Section 2. The mathematical-based simulation algorithm is presented in
Section 3.
Section 4 is devoted to describe the fitting procedure of the mathematical-based simulation algorithm against laboratory data obtained in natural conditions of the carbon dioxide concentration. Then, in
Section 5, we present different pollution scenarios produced by the simulation algorithm with the forecast of porosity variation occurring during the carbonation process. Moreover, a theoretical study on the effects of climate changes in terms of pollution levels on the carbonation constant K was carried out for the analyzed scenarios.
3. The Mathematical-Based Simulation Algorithm
Here, we refer to the mathematical model recently introduced in [
35]. Such a model describes the movement of water (indicated by the letter
w) within the porous stones; carbon dioxide (
a) dissolved in water; the carbonate ion (
b) produced by the dissolution of carbon dioxide; the evolution of calcium hydroxide (
i), which reacts with carbonate to produce calcium carbonate; the evolution of calcium carbonate (
c), produced by the above reaction and later dissolving; finally, the evolution of calcium ion (
e) produced by the dissolution of calcium carbonate. The model will describe the subsequent change in porosity, i.e., the fraction of the volume of voids over the total volume of the porous sample, (
). The complete model reads as follows:
The terms
represent the chemical reactions, defined according to the following equations:
where the terms
indicate the molar masses of the respective substances, and the
ωs describe the single chemical reactions and are defined as follows:
where
,
and
are the reaction rates that represent the key parameters of our model, since their values take into account the different time scales of the chemical reactions involved in the carbonation process. In particular,
represents the reaction coefficient between carbon dioxide and water,
is the reaction coefficient between calcium hydroxide and carbonate ions and
is the dissolution rate of calcium carbonate.
Remark 2. Since porosity varies as a consequence of carbonation, a couple of words are in order to explain how we model it. Porosity changes following two processes:
Following the same arguments as in [35], we can represent porosity with the following expression:where: is the porosity of the non-carbonated concrete;
is the porosity of (totally) carbonated concrete;
represents how porosity varies when calcium carbonate dissolves.
3.1. Initial and Boundary Conditions
The mathematical model is endowed with initial and boundary conditions for the unknown variables. Here, we consider a concrete specimen in the domain
, with
. The left boundary
identifies the surface in contact with the environment, with which the exchange of humidity, carbon dioxide and carbonate occurs. For the unknowns, we impose initial conditions of the form
Notice that no boundary conditions are needed for the variables
i,
c and
e since these are solutions of ODEs and their evolution in space only depends on the evolution of the other variables. For the unknowns
and
b, we impose a zero flux condition at the right boundary
, i.e.,
At
, we assign a Dirichlet condition for water, i.e.,
where the constant
is the value of the (time-dependent) environmental moisture content computed as
where
SVD is the saturated vapor density in [g/cm
3] computed as in [
27]:
where temperature
T and relative humidity
can be time-dependent if we consider real environmental settings. If instead, we refer to the experimental settings such as those reported in [
22], we assume a relative humidity of
% and temperature
°C.
Then, for
a, we assume that the flux at
depends on the difference between the internal and external carbon dioxide concentration:
where
is an unknown constant describing the penetration rate of carbon dioxide estimated from the calibration against data and
is the value of the external carbon dioxide concentration.
Remark 3. Most often, in experimental works, the concentration of carbon dioxide is given as non-dimensional units as a percentage of the substance within the mixture; however, in our settings, we need the value expressed in dimensional units. Given a concentration of carbon dioxide at , is computed in g/cm3 as:where is the temperature expressed in Kelvin and is the gas constant. Note that, if we consider laboratory conditions reported in [
22], we have the
and
constant.
We assume the null-flux condition at the left boundary for carbonate:
The model described above is able to describe the evolution of the carbonation process in the time interval , with .
3.2. Numerical Algorithm
For the simulations, we assumed the model parameters listed in
Table 2.
The interval is discretized with a step , with . We also set as the approximation of the function w at the height and at the time .
We can assume that:
with the boundary values set as follows:
As shown in [
35], if we define
the numerical algorithm is the following:
with a suitable discretization of the boundary conditions for
w,
a and
b described above. Note that the scheme above is convergent under the CFL condition
. More details are reported in [
35].
4. Model Validation and Calibration
In the present section, we show the numerical validation of the mathematical model described above. A fine tuning of key model parameters, i.e., the reaction coefficients , and , was carried out in a time window of one year in order to represent realistic situations. A common approach usually adopted in order to study the carbonation of concrete is to consider specimens exposed to carbon dioxide concentrations of up to 20%, the so-called accelerated conditions. On the contrary, our aim here was to validate and calibrate the mathematical algorithm over one year for a natural concentration of carbon dioxide in the air, i.e., 0.03%.
In order to validate and calibrate the mathematical model, we referred to data from a carbonation experiment described in [
22] performed on a type Portland cement specimen. As described in the work by Pan et al., concrete specimens characterized by a w/c ratio of 0.53 are first polymerized at a temperature of 20 °C and RH of 70% for 24 h; then, they are placed in a seasoning environment at 20 ± 3 °C and 90% RH for 28 days and, successively, samples are placed in a drying oven at 50 °C for 48 h. The carbonation test is executed at
T = 20 °C and 70% humidity for
concentrations 0.03%, 3% and 20%. In what follows, we considered only the data corresponding to the carbon dioxide concentration of 0.03%. For more details on the experimental setting, the reader may refer to the original paper [
22].
In the following numerical tests, we assumed the values of the parameters, taken from the literature or calibrated against data, reported in
Table 2.
Since the kinetic coefficients refer to an aqueous solution under ideal conditions, such coefficients were estimated by the simulation algorithm against data, together with the diffusivity coefficient . Indeed, in the presence of a significantly lower concentration of , we expect that the diffusivity within the material is higher.
In
Figure 5, we report the plots of the profiles of the quantities obtained by the mathematical-based algorithm described in
Section 3 assuming parameters reported in
Table 2. In particular, the left picture shows the system behavior at
, when the diffusion of gaseous
has already taken place in the cement matrix and the carbonic acid formation reaction has just begun, whereas, on the right, the situation at
365 days is depicted.
Looking at the curve profiles of the main quantities involved in the carbonation process and depicted in
Figure 5, we obtained a qualitative validation of the model, since it perfectly describes the real phenomenon according to the following aspects:
Carbon dioxide a (magenta line) enters within the pores and is rapidly consumed;
Water w (blue line) is consumed by the reaction, as expected;
The carbonate ion b described by the yellow curve is the sum of two reactions, i.e., the dissolution of carbon dioxide in water and the reaction with calcium hydroxide to form ;
Calcium hydroxide i shows an “S” shape (green line): near the face in contact with carbon dioxide, calcium hydroxide dissolves by the chemical reaction with carbonate ion; far from there, it is still close to the initial datum since the calcium ion has not yet penetrated sufficiently within the stone;
On the other hand, the calcium carbonate c (red line) reaches its maximum at the left end of the specimen (in contact with ) and decreases towards the other end, where the concentration of carbonate ions is very low; moreover, the dissolution of calcium carbonate has already started;
The calcium ion e (black line), due to the rapid dissolution of calcium hydroxide, only participates in the dissociation reaction of calcium carbonate and it is consequently consumed.
Finally, it is worth noting that, due to the carbonation process, the porosity profile
(cyan line) depicted in the bottom picture of
Figure 5 shows an increasing behavior.
Now, we present the numerical procedure for a fine tuning of model parameters against laboratory data available from the literature. In particular, we used data in ([
22], Table 5).
Reporting the concentration of calcium hydroxide and calcium carbonate at the natural exposure of carbon dioxide in the air corresponding to 0.03% for one year. As can be noticed from experimental data, the content of calcium hydroxide always increases with depth, and calcium carbonate shows an opposite trend. Then, a qualitative validation of model outcomes was obtained looking at the relationship between the carbonation depth and the content of calcium hydroxide and calcium carbonate.
In
Figure 6, plots of the profiles derived from experimental data (line-circles) taken from [
22] for a carbon dioxide concentration of 0.03% and the related numerical results obtained by the model (line-points) using parameters in
Table 2 at time
days are shown. As can be observed from the left picture in
Figure 6, with a fine tuning of model parameters describing reaction rates, not only the qualitative behavior but also the quantitative values of the calcium hydroxide profile are very close to the experimental ones after one year, and the same occurs for the calcium carbonate depicted in the right picture of
Figure 6.
In conclusion, a qualitative and quantitative validation of the model is obtained. Indeed, the main feature of our model is its capability to reproduce the mechanism of the creation and consumption of on one hand and, on the other hand, the ability to predict the time evolution of the system, even including the long-time behavior.
The forecasting algorithm was implemented in Matlab ⓒ and the computational time taken for a simulation on the complete model with fixed parameters until time days was approximately 2000 s on an Intel(R) Core(TM) i7-3630 QM CPU 2.4 GHz.
6. Conclusions
The present paper focused on the forecasting capability of the mathematical model of carbonation - and the related simulation algorithm presented in
Section 3. The principal aim of our work was to obtain a reliable simulation algorithm to be used as a numerical forecasting tool for predicting the effects of climate changes (i.e.,
emission levels, global warming) on the conservation of Portland cement in terms of the porosity variation and penetration depth of the carbonation front.
In this framework, a preliminary qualitative validation of the model followed by a quantitative calibration of its key parameters, i.e., the reaction rates of the chemical reactions involved in the carbonation process, was successfully carried out in
Section 4. Indeed, the pictorial representation of the curve profiles of the quantities mainly involved in the carbonation process allows for a first qualitative validation of the model. Then, a quantitative calibration of the model in the case of exposure to a natural carbon dioxide concentration (0.03%) over a year was carried out by comparing laboratory data available from the literature and model outcomes, for both calcium hydroxide and calcium carbonate concentrations, thus resulting in a fine tuning of model parameters. Moreover, a comparison between real vs. controlled laboratory settings was proposed, suggesting no effects in terms of porosity variation. Once calibrated, the simulation algorithm was used as a forecasting tool for predicting damage scenarios in order to evaluate the effects of the possible future increase in the carbon dioxide concentration, as well as in the temperature, as predicted in climate change scenarios reported in
Section 2. Finally, since our model does not explicitly describe the advancement in the carbonation front, some theoretical estimates on the effects of climate changes on the penetration of carbon dioxide within cement is proposed in
Section 5. In more detail, a study on the effects of climate changes in terms of pollution levels and temperature rise on the carbonation constant
K was carried out for the analyzed scenarios, and a substantial agreement with both the experimental and numerical outcomes was obtained.
Then, we can summarize the procedure by the steps reported below:
A qualitative validation and fine tuning of model parameters against laboratory data;
A numerical simulation of different damage scenarios quantifying the modification of the front position;
A theoretical verification of experimental and numerical findings for the analyzed scenarios.
This represents, to our knowledge, the first study on the effects of climate changes on the porosity of Portland cement.
Our future work will follow two interconnected lines of research.
First, we will further enhance the forecasting tool by applying it to different kinds of concrete and with a combination of multiple damaging factors. To this aim, not only data available from literature but ad hoc laboratory experiments and sensor measurements are needed to acquire data (chemical/physical properties such as porosity analyses, gas penetration rates and the presence of other harmful substances). The second line of research will focus on the impact of climate change on CH sites. Indeed, climate change implies modifications in several environmental aspects, such as temperature, precipitation, moisture content, wind intensity, sea level rise and the occurrence of extreme events. All of these changes will affect the degradation mechanisms of monuments or other artifacts in several ways. Thus, we will use our forecasting tool to study how the variation in the above environmental processes impacts cultural assets. This will be interesting from a scientific point of view, as well as for heritage management practice. The final goal of our research in the future is indeed to build a digital-twin prototype of the monitored object in order to develop a mathematical tool for predictive maintenance. Using mathematical algorithms to predict damages from chemical actions and from other mechanisms for building heritage can bring economic savings and allow for the optimal planning of conservative intervention strategies.