## 2 The 3D Moisture Simulation Program for Cement Concrete Pavement Panels  
### Principles and Equations  
The variation of moisture in a certain area inside concrete can be expressed as humidity diffusion and cement hydration. Based on the fact that the phases of moisture inside the concrete are in thermodynamic equilibrium, the variation of the relative humidity inside the concrete and the variation of the moisture content are approximately in a linear relationship, as explained by the principle of adsorption and desorption, expressed as  
\[\frac{\partial h}{\partial t}=\frac{\partial h_{d}}{\partial t}+\frac{ \partial h_{s}}{\partial t}+\kappa\frac{\partial T}{\partial t} \tag{1}\]  
In the Equation (1), \(\partial h/\partial t\) is the total relative humidity change in the panel, \(\partial h_{d}/\partial t\) is the humidity change due to diffusion, \(\partial h_{s}/\partial t\) is the relative humidity change due to self-drying, and \(\kappa\partial T/\partial t\) is the change in relative humidity under the influence of temperature. The following are the expressions and differential equations for the three components of Equation (1).  
(1) Humidity diffusion model  
Scholars have conducted numerical studies of humidity diffusion based on Fick's law and the law of mass conservation [25]. As shown in Figure 1, [PERSON]'s law states that the rate of transport of diffusing substances per unit area through a region is proportional to the concentration gradient in that region, such that \(h_{d}=h-h_{s}-\kappa\cdot\partial T\) transforming Equation (1) into  
\[\frac{\partial h_{d}}{\partial t}=div(D\cdot grad\cdot h_{d}) \tag{2}\]  
In Equation (2), the humidity diffusion coefficient \(D\) is a function of \(h\), expressed as  
\[D(h)=D_{0}\left\{\alpha_{0}+\sqrt[1-\alpha_{0}]{1+[(1-h)/(1-h_{c0})]^{n}}\right\} \tag{3}\]  
In Equation (3), \(D_{0}\) is the moisture diffusion coefficient of saturated concrete; \(n\) is the regression coefficient of the moisture nonlinear diffusion equation, indicating the rate of decline of \(D(h)\), usually taken as 6-16; \(\alpha_{0}=D_{\min}/D_{0}\), \(D_{\min}\) is the diffusion coefficient at \(h=0\); \(h_{c0}\) is the relative humidity value of \(D\) at \((D_{0}-D_{\min})/2\), usually taken as 0.75-0.8.  
(2) Temperature correction model  
When the water content inside the pores remains constant, the saturated vapor pressure will change with temperature, indirectly affecting the relative humidity \(h\) and the humidity diffusion coefficient \(D_{0}\) of saturated concrete. The influence of temperature on \(D_{0}\) and \(h\) can be expressed as [21]  
\[\kappa=0.0135h(1-h)/(1.25-h) \tag{4}\]  
\[\frac{D_{T}}{D_{0}}=\frac{T}{T_{0}}\exp\left(\frac{Q}{RT}-\frac{Q}{RT_{0}}\right) \tag{5}\]  
In Equation (4), \(\kappa\) is the moisture diffusion coefficient, \(T\) is the absolute temperature (\({}^{\circ}K\)); \(T_{0}\) is the reference temperature, usually chosen as room temperature, \(D_{T}\) and \(D_{0}\) are the saturation diffusion coefficients of concrete at \(T\) and \(T_{0}\), respectively, \(Q\) is the activation energy of hydration.  
(3) Self-drying model  
The relationship between the change in relative humidity due to self-drying \(\Delta h_{as}\) and the degree of hydration \(\alpha\) can be expressed as  
\[\Delta h_{as}=\frac{a+\alpha}{b}\frac{1}{2} \tag{6}\]  
In Equation (6), \(\Delta h_{as}\) is the change in relative humidity due to self-drying; \(\alpha\) is the degree of hydration; \(a\) and \(b\) are the shape parameters.  
Figure 1: Humidity dispersion model.