## 3 Model Building  
### Mathematical Model of Ion Leaching and Diffusion in Fly Ash Micro-Nano Pores  
As the medium for CO2 mineralization reaction, fly ash primarily consists of tiny particles emitted after coal combustion. These fly ash particles are mostly spherical in shape, with internal pores and cracks. Figure 2 showcases the microstructure of fly ash, captured using a scanning electron microscope (SEM). The black part in the image represents the pores, while the white part represents the fly ash particles. Fly ash, being a porous medium material, contains matrix, pore, and crack structures of varying sizes, exhibiting a multi-scale characteristic in space. It is assumed that the fly ash particles are isotropic spheres. When a cross-section is taken through the center of the sphere, the spherical fly ash reveals a multi-scale structure, with larger circles encompassed by smaller circles, as illustrated in Figure 3.  
Figure 2: SEM of fly ash particle structure. Alkali metal oxides, such as Ca and Mg, present in fly ash, dissolve in water, resulting in the generation of alkaline metal ions that accumulate on the surface of fly ash particles. When CO\({}_{2}\) dissolves in water, it ionizes into CO\({}_{3}\)\({}^{2-}\). These two substances react to form carbonates, with all dissolved CO\({}_{2}\) participating in the reaction during the process. CO\({}_{2}\) actively participates in the reaction, while the alkaline ions diffuse from the interior to the exterior. Moreover, CO\({}_{2}\) reacts with alkaline ions on the surface of the fly ash particles, leading to the formation of carbonate crystals. Based on the aforementioned analysis, a mathematical model was developed to describe the dissolution and diffusion process of fly ash particles. The model incorporates the following assumptions: (1) Fly ash is considered a continuous medium. (2) Fly ash particles are assumed to be spherical and isotropic. (3) The leaching and diffusion of alkaline ions adhere to [PERSON]'s second law of diffusion and the _law_ of conservation of mass. (4) During the leaching process, alkaline ions dissolve and diffuse from the interior to the exterior of the pore surface. (5) All leached alkaline ions participate in the reaction. The continuity mass conservation equation for particle ion dissolution and diffusion is presented in Equation (5). \[\left\{ \begin{array}{l}\frac{\partial\mathbb{C}}{\partial t}=D\Big{(}\frac{ \partial^{2}\mathbb{C}}{\partial t^{2}}+\frac{2}{r}\frac{\partial\mathbb{C}}{ \partial r}\Big{)}\\ \frac{\partial\mathbb{C}}{\partial r}|_{r=0}=0,(r=0,t\geq 0)\\ C|_{r=r_{0}}=C_{a}=0,(r=r_{0},t\geq 0)\\ C|_{t=0}=C_{0},(t=0,0\leq r\leq r_{0})\end{array} \right. \tag{5}\]  
In the formula, \(C\) represents the dissolution and diffusion mass concentration of alkaline ions, which is a function of time (\(t\)) and distance (\(r\)), denoted as \(C=C(r,t)\). The unit of \(C\) is g/cm\({}^{3}\). The variable \(t\) represents time and is measured in seconds. The variable \(r\) represents the diffusion path and is measured in centimeters. \(D\) represents the diffusion coefficient, with a unit of cm\({}^{2}\)/s. The variable \(r\) represents the particle radius, measured in centimeters. \(C_{a}\) denotes the ion mass concentration at the particle surface during the diffusion process, with a unit of g/cm\({}^{3}\), which serves as the boundary condition. \(C_{0}\) represents the initial mass concentration of alkaline ions, with a unit of g/cm\({}^{3}\), and it represents the initial condition. Substituting the variables in Equation (5), supposing \(u=(C-C_{a})r\). Then, Equation (5) can be rewritten as follows:  
\[\left\{ \begin{array}{l}\frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{ \partial r^{2}}\\ u|_{r=0}=0,(r=0,t\geq 0)\\ u|_{r=r_{0}}=0,(x=L,t\geq 0)\\ u|_{t=0}=(C-C_{a})r,(t=0,0\leq r\leq r_{0})\end{array} \right. \tag{6}\]  
Figure 3: Schematic diagram of fly ash particle structure.