## 2 Materials and Methods
### Fundamental Equations of R Technique  
The fundamental equations of the R technique are listed as follows:  
1. Intra-Particle Diffusion \[\left(\frac{\partial q_{m}}{\partial t}\right)=\left(\frac{D_{S}}{r^{2}}\right) \frac{\partial}{\partial t}\left(r^{2}\frac{\partial q_{m}}{\partial r}\right) q_{m}=q_{0}\,\text{at}\,t=0\] (1)
2. Fluid-to-Solid Film Transfer \[\rho_{S}\left(\frac{\partial q_{t}}{\partial t}\right)=k_{F}a_{P}(c_{t}-c_{S})\] (2)
3. Interface Transfer \[\left(\frac{\partial q_{t}}{\partial t}\right)=-D_{S}a_{P}\left(\frac{ \partial q_{m}}{\partial r}\right)_{r=r_{P}}q_{m}=f(c_{s})\,\text{at}\,r=r_{p}\] (3)
4. Mass Balance within Vessel \[\left(\frac{\partial q_{t}}{\partial t}\right)=-\left(\frac{V}{m}\right) \left(\frac{\partial c_{t}}{\partial t}\right)\] (4)
5. Average Amount Adsorbed \[q_{t}=\frac{\int_{0}^{r_{P}}4\pi q_{m}r^{2}dr}{4/3\pi r_{P}^{3}}\] (5)
6. Equilibrium Relationship \[q_{S}=k(c_{S})^{1/n}\] (6)  
The above equations (Equations (1)-(6)) are converted into dimensionless ones (Equations (7)-(12)) by means of the dimensionless variables defined below. Dimensionless Variables:  
\[T=\left(\frac{D_{S}}{r_{P}^{2}}\right)t;\ R=\left(\frac{r}{r_{P}}\right);\ C_{T}=\left(\frac{c_{t}}{c_{0}}\right);\ C_{S}=\left(\frac{c_{S}}{c_{0}} \right);\ Q_{T}=\left(\frac{q_{t}}{q_{0}}\right);\ Q_{m}=\left(\frac{q_{m}}{q _{0}}\right)\]  
Dimensionless Equations:  
1. First Item Intra-Particle Diffusion \[\left(\frac{\partial Q_{m}}{\partial T}\right)=\left(\frac{1}{R^{2}}\right) \frac{\partial}{\partial R}\left(R^{2}\frac{\partial Q_{m}}{\partial R} \right);\ Q_{m}=f(C_{s})\,\text{at}\,R=1\] (7)
2. Fluid-to-Solid Film Transfer \[\left(\frac{\partial Q_{T}}{\partial T}\right)=3\,\text{Bi}\,(C_{T}-C_{S}); \ \text{Bi}\ =\left(\frac{k_{F}\,r_{P}}{D_{S}\,\beta\,\rho_{S}}\right);\ \beta=\left(\frac{q_{0}}{c_{0}}\right)\] (8)
3. Interface Transfer \[\left(\frac{\partial Q_{t}}{\partial T}\right)=-3\left(\frac{\partial Q_{m}} {\partial R}\right)_{R=1}Q_{m}=f(C_{s})\,\text{at}\,R=1\] (9)4. Mass Balance within Vessel \[\rho_{S}\left(\frac{\partial q_{t}}{\partial t}\right)=k_{F}a_{P}(c_{t}-c_{S})\]0 (10)
5. Average Amount Adsorbed \[\rho_{S}\left(\frac{\partial q_{t}}{\partial t}\right)=k_{F}a_{P}(c_{t}-c_{S})\]1 (11)
6. Equilibrium Relationship \[\rho_{S}\left(\frac{\partial q_{t}}{\partial t}\right)=k_{F}a_{P}(c_{t}-c_{S})\]2 (12)  
The calculation method for the R technique is as follows: The core of the calculation involved solving Equation (7). The Crank-Nicolson method was used as the solution method. In this case, boundary values were required. At the boundary point, \(R=0\), the relational expression \(\partial Q_{m}/\partial R=0\) was used based on the assumption of point symmetry. At \(R=1\), the value of \(Q_{m}\) was determined by solving Equations (8) and (9), and the equilibrium was determined using Equation (12). The calculation program was created using a C++ compiler. The typical simulation results of the R technique's dimensionless concentration decay curves (RCD) based on Equations (7)-(12) under various conditions are presented in Figure 1 (Freundlich exponent \(n=4\), 5, 6; Biot number \(=4\), 40; fluid/solid ratio (\(m\beta/V=m\ q_{0}/VC_{0}\)) \(=0.443\ -\ 1.196\)). The final RCD curve concentrations mainly depend on the values of the fluid/solid ratio (\(m\beta/V\)) and the Biot number (Figure 1a). For example, the final dimensionless concentration (\(T=1\)) values reach 0.6, 0.4, and 0.2 when the fluid/solid ratios are 0.443, 0.721, and 1.104, respectively. In addition, the effect of the Biot number on the RCD curves is much greater than that of \(n\) (4, 5, 6) (Figures 1a and 1b, respectively). For industrial applications, a typical Freundich exponent (\(n\)) is in the range of 4-6. Therefore, \(n=4\), 5, and 6 were simulated in this study. By converting the vertical axis from CT to \((C_{T}-C_{e})/(C_{0}-C_{e})\), the effect of the fluid/solid ratio was reduced (Figure 2).