2.2 Diffusionreaction synergism model
The transport of gases (such as carbon dioxide) and ions (such as calcium ions) in concrete in cementitious systems can be assumed through the conservation of mass and Fick’s second law [15, 18], as shown in Eq. (1).
\(\frac{\partial {c}_{i}}{\partial t}=div\left({D}_{i}\nabla {c}_{i}\right){R}_{i}\)

(6)

\({R}_{i}=\frac{\partial {c}_{b,i}}{\partial t}\)

(7)

\({c}_{t,i}={c}_{b,i}+{c}_{i}\)

(8)

where \({c}_{i}\) is the concentration of ion \(i\) in concrete; \({D}_{i}\) is the diffusion coefficient of ion \(i\) in concrete; \(t\) is the time; \({R}_{i}\) is the source term; \({c}_{b,i}\) is the deposition concentration of ion \(i\); \({c}_{t,i}\) is the total concentration of ion \(i\).
According to the law of conservation of mass and Fick’s second law, the transfer of carbon dioxide in concrete can be described by Eqs. (9)(12):
\(\frac{\partial {c}_{{CO}_{2}}}{\partial t}=div\left({D}_{{CO}_{2}}\nabla {c}_{{CO}_{2}}\right){R}_{{CO}_{2}}\)

(9)

\({R}_{{CO}_{2}}={r}_{CH}+3\bullet {r}_{CSH}\)

(10)

\({r}_{CH}={k}_{CH}^{0}{\text{S}}^{3.7}{c}_{CH}H{C}_{g}\)

(11)

\({r}_{CSH}={k}_{CSH}^{0}{\text{S}}^{3.7}{c}_{CSH}H{C}_{g}\)

(12)

where \({c}_{{CO}_{2}}\) is the molar concentration of carbon dioxide in concrete; \({D}_{{CO}_{2}}\) is the diffusion coefficient of carbon dioxide in concrete; \(t\) is the carbonation time of the concrete; \({r}_{CH}\) is the carbonization rate of calcium hydroxide; \({r}_{CSH}\) is the carbonization rate of hydrated calcium silicate; \(\text{S}\) is water saturation; \({c}_{CH}\) is the molar concentration of calcium hydroxide in a carbonated environment; \({c}_{CSH}\) is the molar concentration of hydrated calcium silicate in a carbonated environment; H (= 0.8317) is the ratio of the concentration of gaseous carbon dioxide in the pore to the concentration of liquid carbon dioxide [19]; \({C}_{g}\) is the concentration of gaseous carbon dioxide.
According to the law of conservation of mass and Fick’s second law, the transfer of calcium ions in concrete can be described by Eqs. (13)(15):
\(\frac{\partial {c}_{{Ca}^{2+}}}{\partial t}=div\left({D}_{{Ca}^{2+}}\nabla {c}_{{Ca}^{2+}}\right){R}_{\left\{{CO}_{2}, {Ca}^{2+}\right\}}\)

(13)

\({R}_{\left\{{CO}_{2}, {Ca}^{2+}\right\}}={R}_{{CO}_{2}}+{R}_{{Ca}^{2+}}\)

(14)

\({R}_{{Ca}^{2+}}=\frac{\partial {c}_{ca}^{s}}{\partial t}\)

(15)

where \({c}_{{Ca}^{2+}}\) is the molar concentration of calcium ions in concrete; \({D}_{{Ca}^{2+}}\) is the effective diffusion coefficient of calcium ions in concrete; \({c}_{ca}^{s}\) is the molar concentration of solidphase calcium. The solid phase calcium content and the calcium ion concentration in the liquid phase can be calculated according to the solidliquid equilibrium curve. The relevant formula is as follows:
$${c}_{ca}^{s}\left(x,t\right)=\left\{\begin{array}{c}\left[\frac{2}{{{x}_{1}}^{3}}{{c}_{{Ca}^{2+}}}^{3}\left(x,t\right)+\frac{3}{{{x}_{1}}^{2}}{{c}_{{Ca}^{2+}}}^{2}\left(x,t\right)\right]\left\{{c}_{CSH,0}{\left[\frac{{c}_{{Ca}^{2+}}\left(x,t\right)}{{C}_{satu}}\right]}^{1/3}\right\}\\ 0\le {c}_{{Ca}^{2+}}\left(x,t\right)\le {x}_{1}\\ {c}_{CSH,0}{\left[\frac{{c}_{{Ca}^{2+}}\left(x,t\right)}{{C}_{satu}}\right]}^{1/3} \\ {x}_{1}<{c}_{{Ca}^{2+}}\left(x,t\right)\le {x}_{2}\\ {c}_{CSH,0}{\left[\frac{{c}_{{Ca}^{2+}}\left(x,t\right)}{{C}_{satu}}\right]}^{1/3}+\frac{{c}_{CH,0}}{{\left({C}_{satu}{x}_{2}\right)}^{3}}{\left[{c}_{{Ca}^{2+}}\left(x,t\right){x}_{2}\right]}^{3} \\ {x}_{2}<{c}_{{Ca}^{2+}}\left(x,t\right)\le {C}_{satu}\end{array}\right.$$
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where \({c}_{CSH,0}\) is the initial concentration of hydrated calcium silicate in concrete; \({c}_{CH,0}\) is the initial concentration of calcium hydroxide in concrete; x1 (= 2mol/m3) is the concentration of dissolved calcium when CSH gel in concrete is rapidly dissolved into silica gel (SiO2); x2 (=(\({C}_{satu}\)3) mol/m3) is the concentration of dissolved calcium that CSH gel begins to dissolve after calcium hydroxide is completely dissolved.
In this section, the initial and boundary conditions for the diffusionreaction models (1) and (4) are as follows:
$$\text{I}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l} \text{c}\text{o}\text{n}\text{d}\text{i}\text{t}\text{i}\text{o}\text{n}\text{s}\left(\text{t}=0\right): x\in \varOmega \left\{\begin{array}{c}{c}_{{CO}_{2}}=0 \\ {c}_{{Ca}^{2+}}={C}_{satu}\\ {c}_{CH}={c}_{CH,0} \\ {c}_{CSH}={c}_{CSH,0}\end{array}\right.$$
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$$\text{B}\text{o}\text{u}\text{n}\text{d}\text{a}\text{r}\text{y} \text{c}\text{o}\text{n}\text{d}\text{i}\text{t}\text{i}\text{o}\text{n}\text{s}: x\in \varGamma \left\{\begin{array}{c}{c}_{{CO}_{2}}={C}_{s} \\ {c}_{{Ca}^{2+}}=0\end{array}\right.$$
18
where \({C}_{s}\) is the carbon dioxide concentration on the exposed surface of concrete. This value is calculated using the method of Mi et al. [19].
2.3 Temperature field model
2.3.1 Temperature gradient
Once there is a temperature difference between the internal and external environment of concrete, a temperature gradient appears. At present, there are many methods to explain the temperature gradient in concrete. Among, Damrongwiriyanupap et al. [20] added a specific term to the heatflow equation represented by Fourier’s law to describe the effect of temperature. According to the method of Samson and Marchand [10], a simplified energy balance equation is used to model the temperature field:
$$\rho {C}_{p}\frac{\partial T}{\partial t}=div\left(\kappa \nabla T\right)$$
19
$$\kappa ={\kappa }_{ref}\left(0.0015\left(T{T}_{ref}\right)\right)$$
20
where, \(\kappa\) is the thermal conductivity of porous materials; \({\kappa }_{ref}\) is the measured value at the laboratory temperature \({T}_{ref}\) (296 K). Assuming the density \(\rho\) and the thermal capacity \({C}_{p}\) as constants.
2.3.2 Effect of temperature on the diffusion coefficient
The increase of the temperature leads to an increase in molecular activity, resulting in an increase in the diffusivity of gaseous CO2. It is commonly assumed that the temperature dependence of the diffusion coefficient follows the Arrhenius relationship [21, 22]:
$${H}_{{CO}_{2}}\left(T\right)=exp\left(\frac{U}{R}\left(\frac{1}{{T}_{ref}}\frac{1}{T}\right)\right)$$
21
where \(U\) is the diffusion activation energy. The activation energy for CO2 diffusion in concrete was determined through experiments [23] (= 39000 J/mol), \(R\) is the gas constant (8.314 J/mol/K), \({T}_{ref}\) is the reference temperature (298 K), and \(T\) is the realtime temperature (K).
The ion diffusivity is temperature controlled and evaluated by the Arrhenius equation. According to Yokozeki et al. [24], the effect of temperature on diffusion coefficient can be described as:
$${H}_{{Ca}^{2+}}\left(T\right)=exp\left(b\left(\frac{1}{T}\frac{1}{{T}_{ref}}\right)\right)$$
22
where \(b\) is 2090 for Ca2+.
2.3.3 Effect of temperature on the chemical reaction rate and calcium leaching
The formation rate of calcium carbonate from the reaction between carbon dioxide and calcium hydroxide is temperature dependent. At present, there has been extensive research on the formation rate of calcium carbonate [25–28]. Khunthongkeaw and Tangtermsirikul [29] proposed a secondorder relationship for the reaction rate constants:
$${H}_{k}=exp\left(\frac{{E}_{0}}{RT}\right)$$
23
where \({H}_{k}\) is the reaction rate constant for the reaction between carbon dioxide and solidphase calcium at the target temperature; \({E}_{0}\) is the activation energy (40000 J/mol) [29].
The dissolution process of solidphase calcium also depends on temperature. For calcium hydroxide and hydrated calcium silicate, the influence of temperature on the solidliquid equilibrium curve can be derived according to the inverse ratio of solubility to temperature [10, 30]. By interpolating the available experimental data provided by Yokozeki et al. [16], the apparent activation energy was determined to describe the entire calcium leaching process in the global Arrhenius equation, as shown below:
$$\frac{C}{{C}_{0}}=exp\left(\frac{{E}_{l}}{R}\left(\frac{1}{T}\frac{1}{{T}_{ref}}\right)\right)$$
24
where \({C}_{0}\) is the concentration of liquidphase calcium, and \({E}_{l}\) is the apparent activation energy (=3222 J/mol). A negative value indicates that the solubility of hydrates decreases with increasing temperature.