## 3 Results
### Use of HSDM Formulation for Interpreting Dynamic Adsorption  
#### 3.4.1 Dynamic Adsorption Simulation  
In order to simulate dynamic fixed-bed adsorption breakthrough curves, we made the following basic assumptions: (i) the operating adsorption process was assumed to be isothermal; (ii) the liquid phase flowed according to an axial dispersion; (iii) the compressibility of the liquid phase was negligible; (iv) the liquid phase inside the pores was considered to be stationary and its flow was assumed not to be altered by movement of the mobile phase; (v) the brick bed was homogeneous and the brick particles were spherical in shape and uniform in size; and (vi) the liquid phase velocity, fluid flow rate, and column void fraction remained constant during the entire column experiment. In the modelling procedure, we further hypothesized that the adsorbate was first adsorbed on the external surface of the brick particle and then diffused inside the particle. The obtained thermodynamic equilibrium and kinetics data were used here from independent batch experiments with the aim of providing predictions of fixed-bed breakthrough behaviour (Table 4). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**HSDM Modelling**} & **Cs\({}^{*}\)** & **Rb\({}^{*}\)** & **Co\({}^{2*}\)** & **Si\({}^{2*}\)** \\ \hline \multirow{4}{*}{**Operating conditions**} & Active brick mass (m\({}_{\rm w}\), g) & 10 & 10 & 10 \\ \cline{2-5}  & Influent concentration (C\({}_{\rm w}\), mg L\({}^{-1}\)) & 118 & 85.8 & 66.1 & 174 \\ \cline{2-5}  & Flow rate (mL-min\({}^{-1}\)) & 5 & 5 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Operating conditions, adsorbent characteristics, and equilibrium/kinetics parameters used for HSDM modelling of the breakthrough curves relative to the (single) adsorption of Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\) on brick composite. Figure 9: Dependence of the surface diffusion coefficient (\(D_{\rm s}\)) relative to the adsorption of nuclides (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\)), monovalent cations (Li\({}^{+}\) and K\({}^{+}\)), and some metallic cations onto brick composites, with molar Gibbs energy of hydration of the cation M\({}^{+}\) or M\({}^{2+}\) (\(\Delta G_{\rm H}\)). As the shape of the breakthrough curve depended on the adsorption equilibrium, intra-particle mass transfer, and hydrodynamic conditions, it was preliminarily necessary to determine the ideal operational conditions (such as adsorbent particle size, flow rate and input concentration) for optimizing the system and producing steep (or ideal) breakthrough curves for optimal column performance [84]. The optimization of breakthrough curves of cations was realized under well-defined experimental conditions, which are documented in Table 4. A mathematical model of fixed-bed adsorption (HSDM) was employed herein for the simulation of experimental breakthrough curves. The homogeneous surface diffusion model (HSDM; not detailed here) has been reported extensively in the past for describing the sorption dynamics of porous materials [103; 104; 105; 106; 107]. Simulating calculations were undertaken in the present work by using the fixed-bed adsorption simulation tool (FAST 2.1) [58]. Note that, from the FAST 2.1 software, mathematical formulations applying HSDM were based upon driving forces that did not separate pore diffusion from surface diffusion by fitting experimental breakthrough curve data. Figure 10 depicts experimental breakthrough curves corresponding to the dynamic adsorption of cations onto the brick composite. In the same figure, simulated breakthrough curves obtained from the HSDM model using FAST 2.1 software are represented as well. Globally, the comparison of numerical results from the HSDM model with experimental data revealed relatively good model predictions.