# Analysis of Factors for Compacted Clay Liner Performance Considering Isothermal Adsorption
## 3 Scheme for Analyses and Discussion of Results
### Effect of Different Types of Sorption Isotherms with the Same Maximum Adsorption Capacity  
Three types of isotherms: upper convex, linear, and lower concave, with the same maximum adsorption capacity, are compared below, represented by the Langmuir model,  
Figure 16: Dimensionless outflow curves of the Langmuir model with different values of \(Q_{0}\): (**a**) \(T\) as abscissa; (**b**) \(T_{\rm R}\) as abscissa. Figure 17: Dimensionless outflow curves for Langmuir models with different C\({}_{0}\): (**a**) \(T\) as abscissa; (**b**) \(T_{\rm R}\) as abscissa. Figure 15: Dimensionless outflow curves for the Langmuir model with different values of \(Q_{0}\): (**a**) \(T\) as abscissa; (**b**) \(T_{\rm R}\) as abscissa. the linear model, and the Freundlich model, respectively. The specific adsorption parameters are shown in Figure 18. The other parameters used in the calculation are given in Table 1.  
Figure 19 shows the pore water concentration profiles. As shown in Figure 19, there is a common intersection point in the three concentration profiles: above the intersection point, the concentration front of the upper convex (Langmuir model) is the deepest, and the concentration front of the lower concave (Freundlich model) is the shallowest; below the intersection point, the concentration front of the upper convex (Langmuir model) is the shallowest, and the concentration front of the lower concave (Freundlich model) is the deepest. With increasing time, the intersection-point position on the profile gradually moves downward. As a whole, the concentration front of the upper convex (Langmuir model) is the narrowest, followed by the linear model, and the lower concave (Freundlich model) is the widest. The slopes (partition coefficients) for each isotherm are as follows: in the low pore water concentration segment, the upper convex type (Langmuir model) > linear model > lower concave type (Freundlich model), that is, the upper convex adsorption rate is largest and the lower concave adsorption rate is smallest; in the high pore water concentration segment, the upper convex type (Langmuir model) < linear model < lower concave type (Freundlich model), that is, the upper convex adsorption rate is smallest and the lower concave adsorption rate is largest. Therefore, there is an intersection point in the pore water concentration front of different models. Figure 20a,b show the outflow concentration curves with \(T\) as the abscissa and the outflow curves with \(T_{\text{R}}\) as the \(x\)-axis, respectively. As shown in Figure 20a, when \(T\) is taken as the abscissa, there is a common intersection point in the three curves, and  
Figure 19: Pore water profiles in soil for the three isotherms. Figure 18: Three types of sorption isotherms. before the intersection point, the order of the outflow concentration is lower concave (Freundlich model) > linear > upper convex (Langmuir model), with the opposite after the intersection point. The lower concave (Freundlich model) outflow concentration appeared first, followed by the linear model, with the upper convex (Langmuir model) appearing latest. The upper convex (Langmuir model) reached equilibrium first, the linear model in the middle, and the lower concave (Freundlich model) latest. This is consistent with the distribution of pore water concentrations shown in Figure 19. As shown in Figure 20, when \(T_{\text{R}}\) is taken as the abscissa, the outflow curves of different isotherms do not coincide and there are intersections. Comparing Figure 20 with Figure 3a, it can be seen that compared with the linear model, the Langmuir model corresponds to a narrower concentration front, which is similar to the effect of a smaller \(D_{\text{h}}\) (i.e., \(P_{\text{L}}\)), and the Freundlich model corresponds to a wider concentration front, which is similar to the effect of a larger \(D_{\text{h}}\) (i.e., \(P_{\text{L}}\)), that is, the nonlinear adsorption model has a similar effect on the shape of the outflow curve as changing \(D_{\text{h}}\).