## 2 Methods and Models
### Pore Fractal Model
#### 2.3.3 Density Function Model Based on the Carbon Dioxide Adsorption Method  
Assuming that the aperture is a monotonically increasing function expressed as follows:  
\[\mathrm{r}=\mathrm{r}(\mathrm{x}) \tag{15}\]  
The aperture distribution density function [27; 28] can be expressed as follows:  
\[\mathrm{J}(\mathrm{r})=\mathrm{F}(\mathrm{x})\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\mathrm{r}} \tag{16}\]  
\[\mathrm{x}=1/\mathrm{E} \tag{17}\]  
In the formula, \(\mathrm{E}\) is the adsorption characteristic energy, \(\mathrm{kJ}/\mathrm{mol}\) and \(\mathrm{F}(\mathrm{x})\) is the normalized distribution function of \(\mathrm{x}\). [PERSON] [29] represented the heterogeneity distribution function of microporous structures based on the gamma distribution function as follows [14; 30]:  
\[\mathrm{F}(\mathrm{x})=\frac{3\rho^{\upsilon}}{\Gamma\left(\frac{\upsilon}{ \upsilon}\right)}\mathrm{x}^{\upsilon-1}\mathrm{e}^{-\left(\rho\mathrm{x} \right)^{3}} \tag{18}\]where \(\rho\) is the scale parameter of the distribution function, kJ/mol; \(\upsilon\) is the shape parameter of the distribution function, dimensionless; and \(\Gamma(x)\) is the gamma function, and the equation is as follows:  
\[\Gamma(x)=\int_{0}^{+\infty}t^{x-1}e^{-t}dt \tag{19}\]  
[PERSON] et al. [31] established a functional relationship between the pore size and adsorption characteristic energy of micropores, with a pore width of 0.44 nm or more. The empirical formula is as follows:  
\[r=2852.5E^{-3}+15E^{-1}+0.014E-0.75 \tag{20}\]  
Or  
\[r=2852.5x^{3}+15x+0.014x^{-1}-0.75 \tag{21}\]  
By combining Equations (16)-(19) and (21), the expression for the density function of pore size distribution in heterogeneous solid materials can be obtained as follows [28,29]:  
\[J(r)=\frac{3\rho^{\upsilon}x^{\upsilon-1}e^{-(\rho x)^{3}}}{\Gamma\left(\frac{ \upsilon}{3}\right)(8557.5x^{2}-0.014x^{-2}+15)} \tag{22}\]  
The relevant parameters \(\rho\) and \(\upsilon\) in Equation (22) can be obtained from the D-R equation. [PERSON] et al. [32] believed that the adsorption process of micropores is the filling of their internal volume, which is different from the single-layer adsorption on the pore wall surface. The pore-filling degree can be used to characterize the size of the adsorption capacity, as follows:  
\[\frac{V}{V_{0}}=\exp\left[-\left(\frac{A}{\beta E}\right)^{2}\right] \tag{23}\]  
In the formula, \(V\) is the adsorption capacity of the adsorbate at relative pressure \(\frac{p}{P_{0}}\); cm\({}^{3}\)/g; \(V_{0}\) is the maximum adsorption capacity of the adsorbate, cm\({}^{3}\)/g; \(\beta\) is the affinity coefficient, with carbon dioxide taken as 0.38; and A is the adsorption potential, kJ/mol, expressed as follows:  
\[A=RTIn\frac{P_{0}}{P} \tag{24}\]  
In the formula, \(R\) is the gas constant, taken as 8.314 J/(mol\(\cdot\)K); T is the thermodynamic temperature, K; P is the adsorption equilibrium pressure, MPa; and \(P_{0}\) is the saturated vapor pressure, MPa. Substituting Equation (24) into Equation (23) yields the DR equation, as follows:  
\[\mathrm{J}(\mathrm{r})=\mathrm{F}(\mathrm{x})\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\mathrm{r}} \tag{16}\]0  
[PERSON] et al. [14,29,33] proposed, according to the DR equation, that the adsorption characteristics of micropores in heterogeneous solids satisfy the following integral form:  
\[\mathrm{J}(\mathrm{r})=\mathrm{F}(\mathrm{x})\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\mathrm{r}} \tag{16}\]1  
Combining Equation (12) with Equation (26) yields the following [28]:  
\[\mathrm{J}(\mathrm{r})=\mathrm{F}(\mathrm{x})\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\mathrm{r}} \tag{16}\]2  
Parameters \(\rho\) and \(\upsilon\) can be obtained according to Equation (27). [PERSON] et al. [28] found that there is a good linear relationship between the pore size distribution function and pore size of nanoscale micropores, as follows:  
\[\mathrm{J}(\mathrm{r})=\mathrm{F}(\mathrm{x})\frac{\mathrm{d}\mathrm{x}}{ \mathrm{d}\mathrm{r}} \tag{16}\]3  
In the formula, D represents the fractal dimension and C is a constant.