## 3 Results
### Diffusion-Based Models
#### 3.3.2 Evaluation of Pore and Surface Diffusion Coefficients  
In the second steps of the Weber-Morris plots, the (single) diffusion of the cations Cs+, Rb+, Sr2+, and Co2+ through the brick slowed down because the adsorption process became more restricted by brick pores with lower diameters. Under this condition, [PERSON]'s law might be applied to diffusion kinetics:  
\[\frac{\partial C}{\partial t}=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2 }D_{eff}\frac{\partial C}{\partial r}) \tag{13}\]  
where \(C\) is the concentration of the adsorbate; \(D_{eff}\) is the effective diffusion coefficient of the cation; and \(r\) represents the radial coordinate. Assuming that the adsorbent particles are spherical and that the pores are identical and homogeneously distributed on the solid surface, Equation (13) could be solved to Equation (14) [93] and applied to our system. \[X(t)=1-\frac{6}{\pi^{2}}\sum_{z=1}^{\alpha}\left(1/z^{2}\right)\cdot\exp(\frac {-z^{2}\pi^{2}D_{eff}t}{R_{b}^{2}}) \tag{14}\]  
where \(X(t)=Q_{t}/Q_{e}\) is the fractional attainment time \(t\); \(Q_{e}\) and \(Q_{t}\) represent the amounts of cation adsorbed on brick composite at equilibrium and time \(t\) (mg\(\cdot\)g\({}^{-1}\)), respectively; \(D_{eff}\) is the effective diffusion coefficient of the cation in the composite exchanger phase (m\({}^{2}\cdot\)s\({}^{-1}\)); \(R_{b}\) is the radius of brick grains, which are assumed to be spherical (m); and \"z\" is an integer. The following equation (valid for \(X(t)\) varying from 0 to 1) was used here as an approximation of Equation (14):  
\[X(t)=[1-\exp(\frac{\pi^{2}D_{eff}t}{R_{b}^{2}})]^{1.2} \tag{15}\]  
This equation can be transformed into:  
\[-\ln[1-X(t)^{2}]=\frac{\pi^{2}D_{eff}t}{R_{b}^{2}}+Cst \tag{16}\]  
In this equation, it was necessary to introduce a constant (noted \"Cst\") in order to take into account the reaction time t\({}_{\text{o}}\) when the adsorption kinetics process began to be controlled by surface diffusion according to the Weber-Morris model. This constant could be determined from the following formula: Cst = -ln(1-X(t\({}_{\text{o}}\))\({}^{2}\)) -\(\pi^{2}\)D\({}_{\text{eff.}}\)(t\({}_{\text{o}}\))/R\({}_{\text{b}}\)\({}^{2}\). It should be noted that the \"Cst\" value differed for each cation studied. Hence, Equation (16) ought to cover only the kinetic data points which were used for determining the Weber-Morris rate constant \(k_{ipd}\)(2). Its use permitted us to reach the effective particle diffusivity as follows. By plotting \(-ln\) [1 - X(t)\({}^{2}\)] against time, we obtained relatively good straight lines with correlation coefficients varying from 0.9741 to 0.9872 for the different adsorption reactions studied. See Figure 7.  
The slope values of these linear relations were employed to assess the effective diffusion coefficients, and the \(D_{eff}\) values are listed in Table 3.  
Likewise, from the coordinates of kinetic data points representative of the third W-M stage, we plotted \(-ln\) [1 - X(t)\({}^{2}\)] against time and estimated the effective particle diffusivity for each element by employing the same calculation procedure as that described for the second W-M stage (see Figure 7 and Table 3). In this case, Equation (16) ought to cover only the kinetic data points which were used for determining the Weber-Morris rate constant \(k_{ipd}\)(3). To interpret the results of the diffusion (D\({}_{\text{eff}}\)) measurements of the mass-transfer kinetics involved in the second Weber-Morris step, it was assumed in this work that both pore diffusion and surface diffusion might be responsible for the mass transfer resistance. Hence, in order to better model diffusion in the brick composite based on the step diffusion through mesopores and micropores, we decided to apply a heterogeneous model in which both pore diffusion and surface diffusion were considered [94, 95, 96].