## 2 Experimental Design  
### Packed Fixed-Bed Column Design Considerations  
When the concentration of the effluent reaches 5%-10% of the influent, this point on a typical S-shaped breakthrough curve is commonly referred to as the 'breakthrough point' or 'breakpoint' (BP). The point of column exhaustion (EP) is when the effluent reaches maximum capacity to 90%-95% of its influent value [12; 34]. The efficiency of the column performance is related to the bed capacity at breakthrough and at exhaustion, represented by the following relationship [12; 34]:  
\[\eta=\frac{C_{\text{BP}}}{C_{\text{EP}}} \tag{1}\]  
where \(\eta\) is the column efficiency (degree of saturation), \(C_{\text{BP}}\) is the breakthrough capacity of the bed (in \(\text{meq}/\text{g}\)), and \(C_{\text{EP}}\) is the maximum capacity at exhaustion of the bed, defined by the total amount of HMI ions bound in the zeolite (or exchanged in the packed fixed-bed) (in \(\text{meq}/\text{g}\)). The breakthrough capacity and equilibrium capacity are further expressed in Equations (2) and (3), respectively [12; 34]:  
\[C_{\text{BP}}=\frac{\int_{0}^{V_{\text{BP}}}(C_{0}-C)dV}{\rho HA}=\frac{C_{0} \cdot V_{\text{BP}}}{m}=\frac{\eta_{\text{BP}}}{m} \tag{2}\]  
\[C_{\text{EP}}=\frac{\int_{0}^{V_{\text{EP}}}(C_{0}-C)dV}{\rho HA}=\frac{\int_ {0}^{V_{\text{EP}}}(C_{0}-C)dV}{m}=\frac{\eta_{\text{EP}}}{m} \tag{3}\]  
where \(V_{\text{BP}}\) is the effluent volume collected up to breakthrough point (BP) (in L), \(V_{\text{EP}}\) is the effluent volume at which the exhaustion point (EP) is reached in the zeolite bed (in L), \(C_{0}\) is the influent concentration (in \(\text{meq}/\text{L}\)), \(C\) is the effluent concentration (in \(\text{meq}/\text{L}\)), \(\rho\) is the packing density of the bed (in \(\text{g}/\text{cm}^{3}\)), H is the bed depth (in cm), A is the bed cross-sectional area (in cm\({}^{2}\)), m is the zeolite mass (in g); where \(\eta_{\text{BP}}\) and \(\eta_{\text{EP}}\) is the total amount of HMI ions removed up to BP and EP (in \(\text{meq}\)), respectively. Empty bed contact time (EBCT) is the residence time (in min) a fluid element is in contact with the bed, and is related to the systems' removal kinetics [35]. This is represented by the relationship between the bed depth (H) in the column and the feed solution velocity (v) [25; 34], as given by:  
\[\text{EBCT}=\frac{H}{v}=\frac{H}{(Q/A)}=\frac{d^{2}\pi H}{4Q} \tag{4}\]  
Research conducted by [PERSON] et al. [34] demonstrates that the breakthrough curve results reveal that the flow through the column determines the EBCT; with the same initial concentration, the increase in flow rate decreases the contact time and increases the mass transfer zone (MTZ) height. The MTZ is the restricted area where the exchange process occurs, and is defined as the zeolite layer height between the equilibrium zone and the unused bed zone [34]; where the effluent concentration varies from 5% to 95% of the influent concentration [25]. As the HMI solution is fed through the packed fixed-bed, the MTZ moves in the direction of flow and eventually reaches the exit [12; 34]. [PERSON] et al. [36] distinctly demonstrates the importance of the column bed depth on the removal of lead from aqueous solutions. The results show that as the bed depth increases, a delay in breakthrough and exhaustion occurs, with an increase of the MTZ height. The higher the bed depth, the longer the service time at various breakthrough points due to the increase in binding sites on the sorbent material (zeolite mineral) [9]. Adequate wetting of the zeolite, and ideal contact time between the zeolite and solution interface are important for mass transfer and equilibrium conditions based on the selection of the flow rate and particle size. To minimize possible wall and axial dispersion effects in the fixed-bed column, the bed depth-to-particle diameter ratio (H/d\({}_{\rm p}\)) must be greater than 20. At a higher H/d\({}_{\rm p}\) ratio, the breakthrough point appears later and the curve is steeper.