## 3 Results
### Diffusion-Based Models
#### 3.3.2 Evaluation of Pore and Surface Diffusion Coefficients
By taking into account the pore diffusion and surface diffusion intrinsically involved in the second Weber-Morris step and by using [PERSON]' theories, the total diffusion flux (J) could be expressed as [97, 98, 99, 100]:  
\[J=(1-\varepsilon_{bc})\cdot D_{s}\frac{\partial C_{s}}{\partial r}-\varepsilon_ {bc}D_{p}\frac{\partial C}{\partial r} \tag{17}\]  
where \(D_{p}\) and \(D_{s}\) are the pore diffusion coefficient and solid-phase diffusion coefficient, respectively; \(C\) and \(C_{s}\) are the concentrations of adsorbate in bulk external liquid and on the adsorbent surface per unit volume, respectively; and \(r\) represents the radial coordinate. We considered the transport of cations and mass balance equation within the pores of a spherical adsorbent by using the following expression [100]:  
\[(1/r^{2})\frac{\partial}{\partial r}(r^{2}J)+\varepsilon_{bc}\frac{\partial C }{\partial t}+(1-\varepsilon_{bc})\frac{\partial C_{s}}{\partial t}=0 \tag{18}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cations** & **D\({}_{\text{p}}\)****(a)** & **D\({}_{\text{eff}}\)****(b)** & **D\({}_{\text{eff}}\)****(b)** & **D\({}_{\text{eff}}\)****(c)** & **D\({}_{\text{s}}\)****(calc.)****(d)** & **D\({}_{\text{s}}\)****(calc.)****(d)** \\  & & **(W-M Step-2)** & **(W-M Step-3)** & **(W-M Step-3)** & **(W-M Step-2)** & **(W-M Step-3)** \\ \hline Cs\({}^{+}\) & \(1.396\times 10^{-10}\) & \(4.27\times 10^{-11}\) & \(6.84\times 10^{-12}\) & \(6.90\times 10^{-12}\) & \(4.22\times 10^{-11}\) & \(6.40\times 10^{-12}\) \\ \hline Rb\({}^{+}\) & \(1.401\times 10^{-10}\) & \(3.11\times 10^{-11}\) & \(1.18\times 10^{-11}\) & \(1.13\times 10^{-11}\) & \(3.49\times 10^{-11}\) & \(1.17\times 10^{-11}\) \\ \hline Co\({}^{2+}\) & \(8.768\times 10^{-11}\) & \(6.35\times 10^{-12}\) & \(4.79\times 10^{-12}\) & \(4.95\times 10^{-12}\) & \(6.00\times 10^{-12}\) & \(4.50\times 10^{-12}\) \\ \hline Sr\({}^{2+}\) & \(9.154\times 10^{-11}\) & \(8.88\times 10^{-12}\) & \(1.09\times 10^{-11}\) & \(1.07\times 10^{-11}\) & \(8.73\times 10^{-12}\) & \(3.89\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Surface diffusion coefficients values (_D\({}_{\text{eff}}\)_, \(D_{s}\)_, and _D\({}_{p}\)_; expressed in m\({}^{2}\cdot\)s\({}^{-1}\)) obtained by applying the homogeneous surface diffusion model (HSDM) for the adsorption of cations (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\)) on brick composite. _(a)_ _D\({}_{p}\), calculated from Equation (6); _(b)_ _D\({}_{\text{eff}}\), calculated from models established by [PERSON] et al. and discussed in ref. [93] and [PERSON]’s approximation also documented in ref. [93] at reaction time values of the different experimental points of the second stage and third stage of the Weber–Morris (W-M) kinetics model; _(c)_ _D\({}_{\text{eff}}\) calculated from models established by [PERSON] et al. or Crank (as described in ref. [93] for long adsorption times; _(d)_ _D\({}_{s}\), evaluated from Equation (20)._  
Figure 7: Plotting of \(-ln\) [\(1-X(t)^{2}\)] against time by covering kinetic data points which were used for separately determining Weber-Morris rate constants \(k_{\text{jpd}}(2)\) and \(k_{\text{jpd}}(3)\). By describing diffusion kinetics from [PERSON]'s law according to Equation (13), one could establish a mathematical expression evidencing both solid-phase diffusion contribution and pore diffusion contribution according to [100]:  
\[ \begin{array}{l}\rho_{bc}(1-\varepsilon_{bc})\frac{\partial Q}{\partial t}+ \varepsilon_{bc}\frac{\partial C}{\partial t}=\varepsilon_{bc}D_{p}[\frac{ \partial^{2}C}{\partial t^{2}r^{2}}+(2/r)\frac{\partial C}{\partial r}]\\ +\rho_{bc}(1-\varepsilon_{bc})D_{s}[\frac{\partial^{2}Q}{\partial r^{2}r}+(2/ r)\frac{\partial Q}{\partial r}]\end{array}